Vulkan-Docs/chapters/textures.txt

// Copyright (c) 2015-2018 Khronos Group. This work is licensed under a
// Creative Commons Attribution 4.0 International License; see
// http://creativecommons.org/licenses/by/4.0/

[[textures]]
= Image Operations

== Image Operations Overview

Image Operations are steps performed by SPIR-V image instructions, where
those instructions which take an code:OpTypeImage (representing a
sname:VkImageView) or code:OpTypeSampledImage (representing a
(sname:VkImageView, sname:VkSampler) pair) and texel coordinates as
operands, and return a value based on one or more neighboring texture
elements (_texels_) in the image.

[NOTE]
.Note
====
Texel is a term which is a combination of the words texture and element.
Early interactive computer graphics supported texture operations on
textures, a small subset of the image operations on images described here.
The discrete samples remain essentially equivalent, however, so we retain
the historical term texel to refer to them.
====

SPIR-V Image Instructions include the following functionality:

  * code:OpImageSample* and code:OpImageSparseSample* read one or more
    neighboring texels of the image, and <<textures-texel-filtering,filter>>
    the texel values based on the state of the sampler.
  ** Instructions with code:ImplicitLod in the name
     <<textures-level-of-detail-operation,determine>> the LOD used in the
     sampling operation based on the coordinates used in neighboring
     fragments.
  ** Instructions with code:ExplicitLod in the name
     <<textures-level-of-detail-operation,determine>> the LOD used in the
     sampling operation based on additional coordinates.
  ** Instructions with code:Proj in the name apply homogeneous
     <<textures-projection,projection>> to the coordinates.
  * code:OpImageFetch and code:OpImageSparseFetch return a single texel of
    the image.
    No sampler is used.
  * code:OpImage*code:Gather and code:OpImageSparse*code:Gather read
    neighboring texels and <<textures-gather,return a single component>> of
    each.
  * code:OpImageRead (and code:OpImageSparseRead) and code:OpImageWrite read
    and write, respectively, a texel in the image.
    No sampler is used.
ifdef::VK_NV_shader_image_footprint[]
  * code:OpImageSampleFootprintNV identifies and returns information about
    the set of texels in the image that would be accessed by an equivalent
    code:OpImageSample* instruction.
endif::VK_NV_shader_image_footprint[]
  * Instructions with code:Dref in the name apply
    <<textures-depth-compare-operation,depth comparison>> on the texel
    values.
  * Instructions with code:Sparse in the name additionally return a
    <<textures-sparse-residency,sparse residency>> code.

[[textures-texel-coordinate-systems]]
=== Texel Coordinate Systems

Images are addressed by _texel coordinates_.
There are three _texel coordinate systems_:

  * normalized texel coordinates [eq]#[0.0, 1.0]#
  * unnormalized texel coordinates [eq]#[0.0, width / height / depth)#
  * integer texel coordinates [eq]#[0, width / height / depth)#

SPIR-V code:OpImageFetch, code:OpImageSparseFetch, code:OpImageRead,
code:OpImageSparseRead, and code:OpImageWrite instructions use integer texel
coordinates.
Other image instructions can: use either normalized or unnormalized texel
coordinates (selected by the pname:unnormalizedCoordinates state of the
sampler used in the instruction), but there are
<<samplers-unnormalizedCoordinates,limitations>> on what operations, image
state, and sampler state is supported.
Normalized coordinates are logically
<<textures-normalized-to-unnormalized,converted>> to unnormalized as part of
image operations, and <<textures-normalized-operations,certain steps>> are
only performed on normalized coordinates.
The array layer coordinate is always treated as unnormalized even when other
coordinates are normalized.

Normalized texel coordinates are referred to as [eq]#(s,t,r,q,a)#, with the
coordinates having the following meanings:

  * [eq]#s#: Coordinate in the first dimension of an image.
  * [eq]#t#: Coordinate in the second dimension of an image.
  * [eq]#r#: Coordinate in the third dimension of an image.
  ** [eq]#(s,t,r)# are interpreted as a direction vector for Cube images.
  * [eq]#q#: Fourth coordinate, for homogeneous (projective) coordinates.
  * [eq]#a#: Coordinate for array layer.

The coordinates are extracted from the SPIR-V operand based on the
dimensionality of the image variable and type of instruction.
For code:Proj instructions, the components are in order (s, [t,] [r,] q)
with t and r being conditionally present based on the code:Dim of the image.
For non-code:Proj instructions, the coordinates are (s [,t] [,r] [,a]), with
t and r being conditionally present based on the code:Dim of the image and a
being conditionally present based on the code:Arrayed property of the image.
Projective image instructions are not supported on code:Arrayed images.

Unnormalized texel coordinates are referred to as [eq]#(u,v,w,a)#, with the
coordinates having the following meanings:

  * [eq]#u#: Coordinate in the first dimension of an image.
  * [eq]#v#: Coordinate in the second dimension of an image.
  * [eq]#w#: Coordinate in the third dimension of an image.
  * [eq]#a#: Coordinate for array layer.

Only the [eq]#u# and [eq]#v# coordinates are directly extracted from the
SPIR-V operand, because only 1D and 2D (non-code:Arrayed) dimensionalities
support unnormalized coordinates.
The components are in order [eq]#(u [,v])#, with [eq]#v# being conditionally
present when the dimensionality is 2D.
When normalized coordinates are converted to unnormalized coordinates, all
four coordinates are used.

Integer texel coordinates are referred to as [eq]#(i,j,k,l,n)#, and the
first four in that order have the same meanings as unnormalized texel
coordinates.
They are extracted from the SPIR-V operand in order [eq]#(i, [,j], [,k],
[,l])#, with [eq]#j# and [eq]#k# conditionally present based on the code:Dim
of the image, and l conditionally present based on the code:Arrayed property
of the image.
n is the sample index and is taken from the code:Sample image operand.

For all coordinate types, unused coordinates are assigned a value of zero.

[[textures-texel-coordinate-systems-diagrams]]
[%inline]
image::images/vulkantexture0.svg[align="center",title="Texel Coordinate Systems"]
The Texel Coordinate Systems - For the example shown of an 8{times}4 texel
two dimensional image.

  * Normalized texel coordinates:
  ** The [eq]#s# coordinate goes from 0.0 to 1.0, left to right.
  ** The [eq]#t# coordinate goes from 0.0 to 1.0, top to bottom.
  * Unnormalized texel coordinates:
  ** The [eq]#u# coordinate goes from -1.0 to 9.0, left to right.
     The [eq]#u# coordinate within the range 0.0 to 8.0 is within the image,
     otherwise it is within the border.
  ** The [eq]#v# coordinate goes from -1.0 to 5.0, top to bottom.
     The [eq]#v# coordinate within the range 0.0 to 4.0 is within the image,
     otherwise it is within the border.
  * Integer texel coordinates:
  ** The [eq]#i# coordinate goes from -1 to 8, left to right.
     The [eq]#i# coordinate within the range 0 to 7 addresses texels within
     the image, otherwise it addresses a border texel.
  ** The [eq]#j# coordinate goes from -1 to 5, top to bottom.
     The [eq]#j# coordinate within the range 0 to 3 addresses texels within
     the image, otherwise it addresses a border texel.
  * Also shown for linear filtering:
  ** Given the unnormalized coordinates [eq]#(u,v)#, the four texels
     selected are [eq]#i~0~j~0~#, [eq]#i~1~j~0~#, [eq]#i~0~j~1~#, and
     [eq]#i~1~j~1~#.
  ** The weights [eq]#{alpha}# and [eq]#{beta}#.
  ** Given the offset [eq]#{DeltaUpper}~i~# and [eq]#{DeltaUpper}~j~#, the
     four texels selected by the offset are [eq]#i~0~j'~0~#,
     [eq]#i~1~j'~0~#, [eq]#i~0~j'~1~#, and [eq]#i~1~j'~1~#.

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
[NOTE]
.Note
====
For formats with reduced-resolution channels, [eq]#{DeltaUpper}~i~# and
[eq]#{DeltaUpper}~j~# are relative to the resolution of the
highest-resolution channel, and therefore may be divided by two relative to
the unnormalized coordinate space of the lower-resolution channels.
====
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]

[%inline]
image::images/vulkantexture1.svg[align="center",title="Texel Coordinate Systems"]

The Texel Coordinate Systems - For the example shown of an 8{times}4 texel
two dimensional image.

  * Texel coordinates as above.
    Also shown for nearest filtering:
  ** Given the unnormalized coordinates [eq]#(u,v)#, the texel selected is
     [eq]#ij#.
  ** Given the offset [eq]#{DeltaUpper}~i~# and [eq]#{DeltaUpper}~j~#, the
     texel selected by the offset is [eq]#ij'#.

ifdef::VK_NV_corner_sampled_image[]
For corner-sampled images, the texel samples are located at the grid
intersections instead of the texel centers.

ifdef::editing-notes[]
[NOTE]
.editing-note
====
(dgkoch) TODO add a diagram demonstrating this.
====
endif::editing-notes[]

endif::VK_NV_corner_sampled_image[]


== Conversion Formulas

ifdef::editing-notes[]
[NOTE]
.editing-note
====
(Bill) These Conversion Formulas will likely move to Section 2.7 Fixed-Point
Data Conversions (RGB to sRGB and sRGB to RGB) and section 2.6 Numeric
Representation and Computation (RGB to Shared Exponent and Shared Exponent
to RGB)
====
endif::editing-notes[]


[[textures-RGB-sexp]]
=== RGB to Shared Exponent Conversion

An RGB color [eq]#(red, green, blue)# is transformed to a shared exponent
color [eq]#(red~shared~, green~shared~, blue~shared~, exp~shared~)# as
follows:

First, the components [eq]#(red, green, blue)# are clamped to
[eq]#(red~clamped~, green~clamped~, blue~clamped~)# as:

  :: [eq]#red~clamped~ = max(0, min(sharedexp~max~, red))#
  :: [eq]#green~clamped~ = max(0, min(sharedexp~max~, green))#
  :: [eq]#blue~clamped~ = max(0, min(sharedexp~max~, blue))#

Where:

[latexmath]
+++++++++++++++++++
\begin{aligned}
N               & = 9  & \text{number of mantissa bits per component} \\
B               & = 15 & \text{exponent bias} \\
E_{max}         & = 31 & \text{maximum possible biased exponent value} \\
sharedexp_{max} & = \frac{(2^N-1)}{2^N} \times 2^{(E_{max}-B)}
\end{aligned}
+++++++++++++++++++

[NOTE]
.Note
====
[eq]#NaN#, if supported, is handled as in <<ieee-754,IEEE 754-2008>>
`minNum()` and `maxNum()`.
That is the result is a [eq]#NaN# is mapped to zero.
====

The largest clamped component, [eq]#max~clamped~# is determined:

  :: [eq]#max~clamped~ = max(red~clamped~, green~clamped~, blue~clamped~)#

A preliminary shared exponent [eq]#exp'# is computed:
[latexmath]
+++++++++++++++++++
\begin{aligned}
exp' =
  \begin{cases}
    \left \lfloor \log_2(max_{clamped}) \right \rfloor + (B+1)
      & \text{for}\  max_{clamped} > 2^{-(B+1)} \\
    0
      & \text{for}\  max_{clamped} \leq 2^{-(B+1)}
  \end{cases}
\end{aligned}
+++++++++++++++++++

The shared exponent [eq]#exp~shared~# is computed:

[latexmath]
+++++++++++++++++++
\begin{aligned}
max_{shared} =
    \left \lfloor
        { \frac{max_{clamped}}{2^{(exp'-B-N)}} + \frac{1}{2} }
    \right \rfloor
\end{aligned}
+++++++++++++++++++

[latexmath]
+++++++++++++++++++
\begin{aligned}
exp_{shared} =
  \begin{cases}
    exp'   & \text{for}\  0 \leq max_{shared} < 2^N \\
    exp'+1 & \text{for}\  max_{shared} = 2^N
  \end{cases}
\end{aligned}
+++++++++++++++++++

Finally, three integer values in the range [eq]#0# to [eq]#2^N^# are
computed:

[latexmath]
+++++++++++++++++++
\begin{aligned}
red_{shared} & =
    \left \lfloor
        { \frac{red_{clamped}}{2^{(exp_{shared}-B-N)}}+ \frac{1}{2} }
    \right \rfloor \\
green_{shared} & =
    \left \lfloor
        { \frac{green_{clamped}}{2^{(exp_{shared}-B-N)}}+ \frac{1}{2} }
    \right \rfloor \\
blue_{shared} & =
    \left \lfloor
        { \frac{blue_{clamped}}{2^{(exp_{shared}-B-N)}}+ \frac{1}{2} }
    \right \rfloor
\end{aligned}
+++++++++++++++++++


[[textures-sexp-RGB]]
=== Shared Exponent to RGB

A shared exponent color [eq]#(red~shared~, green~shared~, blue~shared~,
exp~shared~)# is transformed to an RGB color [eq]#(red, green, blue)# as
follows:

  :: latexmath:[red = red_{shared} \times {2^{(exp_{shared}-B-N)}}]
  :: latexmath:[green = green_{shared} \times {2^{(exp_{shared}-B-N)}}]
  :: latexmath:[blue = blue_{shared} \times {2^{(exp_{shared}-B-N)}}]

Where:

  :: [eq]#N = 9# (number of mantissa bits per component)
  :: [eq]#B = 15# (exponent bias)


== Texel Input Operations

_Texel input instructions_ are SPIR-V image instructions that read from an
image.
_Texel input operations_ are a set of steps that are performed on state,
coordinates, and texel values while processing a texel input instruction,
and which are common to some or all texel input instructions.
They include the following steps, which are performed in the listed order:

  * <<textures-input-validation,Validation operations>>
  ** <<textures-operation-validation,Instruction/Sampler/Image validation>>
  ** <<textures-integer-coordinate-validation,Coordinate validation>>
  ** <<textures-sparse-validation,Sparse validation>>
  ** <<textures-layout-validation,Layout validation>>
  * <<textures-format-conversion,Format conversion>>
  * <<textures-texel-replacement,Texel replacement>>
  * <<textures-depth-compare-operation,Depth comparison>>
  * <<textures-conversion-to-rgba,Conversion to RGBA>>
  * <<textures-component-swizzle,Component swizzle>>
ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
  * <<textures-chroma-reconstruction,Chroma reconstruction>>
  * <<textures-sampler-YCbCr-conversion,Y'C~B~C~R~ conversion>>
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]

For texel input instructions involving multiple texels (for sampling or
gathering), these steps are applied for each texel that is used in the
instruction.
Depending on the type of image instruction, other steps are conditionally
performed between these steps or involving multiple coordinate or texel
values.

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
If <<textures-chroma-reconstruction,Chroma Reconstruction>> is implicit,
<<textures-texel-filtering, Texel Filtering>> instead takes place during
chroma reconstruction, before <<textures-sampler-YCbCr-conversion,sampler
Y'C~B~C~R~ conversion>> occurs.
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]


[[textures-input-validation]]
=== Texel Input Validation Operations

_Texel input validation operations_ inspect instruction/image/sampler state
or coordinates, and in certain circumstances cause the texel value to be
replaced or become undefined.
There are a series of validations that the texel undergoes.


[[textures-operation-validation]]
==== Instruction/Sampler/Image View Validation

There are a number of cases where a SPIR-V instruction can: mismatch with
the sampler, the image view, or both.
There are a number of cases where the sampler can: mismatch with the image
view.
In such cases the value of the texel returned is undefined.

These cases include:


  * The sampler pname:borderColor is an integer type and the image view
    pname:format is not one of the elink:VkFormat integer types or a stencil
    component of a depth/stencil format.
  * The sampler pname:borderColor is a float type and the image view
    pname:format is not one of the elink:VkFormat float types or a depth
    component of a depth/stencil format.
  * The sampler pname:borderColor is one of the opaque black colors
    (ename:VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK or
    ename:VK_BORDER_COLOR_INT_OPAQUE_BLACK) and the image view
    elink:VkComponentSwizzle for any of the slink:VkComponentMapping
    components is not ename:VK_COMPONENT_SWIZZLE_IDENTITY.
  * The elink:VkImageLayout of any subresource in the image view does not
    match that specified in slink:VkDescriptorImageInfo::pname:imageLayout
    used to write the image descriptor.
  * If the instruction is code:OpImageRead or code:OpImageSparseRead and the
    pname:shaderStorageImageReadWithoutFormat feature is not enabled, or the
    instruction is code:OpImageWrite and the
    pname:shaderStorageImageWriteWithoutFormat feature is not enabled, then
    the SPIR-V Image Format must: be <<spirvenv-image-formats,compatible>>
    with the image view's pname:format.
  * The sampler pname:unnormalizedCoordinates is ename:VK_TRUE and any of
    the <<samplers-unnormalizedCoordinates,limitations of unnormalized
    coordinates>> are violated.
  * The SPIR-V instruction is one of the code:OpImage*code:Dref*
    instructions and the sampler pname:compareEnable is ename:VK_FALSE
  * The SPIR-V instruction is not one of the code:OpImage*code:Dref*
    instructions and the sampler pname:compareEnable is ename:VK_TRUE
  * The SPIR-V instruction is one of the code:OpImage*code:Dref*
    instructions and the image view pname:format is not one of the
    depth/stencil formats with a depth component, or the image view aspect
    is not ename:VK_IMAGE_ASPECT_DEPTH_BIT.
  * The SPIR-V instruction's image variable's properties are not compatible
    with the image view:
  ** Rules for pname:viewType:
  *** ename:VK_IMAGE_VIEW_TYPE_1D must: have code:Dim = 1D, code:Arrayed =
      0, code:MS = 0.
  *** ename:VK_IMAGE_VIEW_TYPE_2D must: have code:Dim = 2D, code:Arrayed =
      0.
  *** ename:VK_IMAGE_VIEW_TYPE_3D must: have code:Dim = 3D, code:Arrayed =
      0, code:MS = 0.
  *** ename:VK_IMAGE_VIEW_TYPE_CUBE must: have code:Dim = Cube, code:Arrayed
      = 0, code:MS = 0.
  *** ename:VK_IMAGE_VIEW_TYPE_1D_ARRAY must: have code:Dim = 1D,
      code:Arrayed = 1, code:MS = 0.
  *** ename:VK_IMAGE_VIEW_TYPE_2D_ARRAY must: have code:Dim = 2D,
      code:Arrayed = 1.
  *** ename:VK_IMAGE_VIEW_TYPE_CUBE_ARRAY must: have code:Dim = Cube,
      code:Arrayed = 1, code:MS = 0.
  ** If the image was created with slink:VkImageCreateInfo::pname:samples
     equal to ename:VK_SAMPLE_COUNT_1_BIT, the instruction must: have
     code:MS = 0.
  ** If the image was created with slink:VkImageCreateInfo::pname:samples
     not equal to ename:VK_SAMPLE_COUNT_1_BIT, the instruction must: have
     code:MS = 1.
ifdef::VK_NV_corner_sampled_image[]
  * If the image was created with slink:VkImageCreateInfo::pname:flags
    containing ename:VK_IMAGE_CREATE_CORNER_SAMPLED_BIT_NV, the sampler
    addressing modes must: only use a slink:VkSamplerAddressMode of
    ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE.
endif::VK_NV_corner_sampled_image[]
ifdef::VK_NV_shader_image_footprint[]
  * The SPIR-V instruction is code:OpImageSampleFootprintNV with code:Dim =
    2D and pname:addressModeU or pname:addressModeV in the sampler is not
    ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE.
  * The SPIR-V instruction is code:OpImageSampleFootprintNV with code:Dim =
    3D and pname:addressModeU, pname:addressModeV, or pname:addressModeW in
    the sampler is not ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE.
endif::VK_NV_shader_image_footprint[]

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
Only code:OpImageSample* and code:OpImageSparseSample* can: be used with a
sampler that enables <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~
conversion>>.

code:OpImageFetch, code:OpImageSparseFetch, code:OpImage*code:Gather, and
code:OpImageSparse*code:Gather must: not be used with a sampler that enables
<<samplers-YCbCr-conversion,sampler Y\'C~B~C~R~ conversion>>.

The code:ConstOffset and code:Offset operands must: not be used with a
sampler that enables <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~
conversion>>.
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]


[[textures-integer-coordinate-validation]]
==== Integer Texel Coordinate Validation

Integer texel coordinates are validated against the size of the image level,
and the number of layers and number of samples in the image.
For SPIR-V instructions that use integer texel coordinates, this is
performed directly on the integer coordinates.
For instructions that use normalized or unnormalized texel coordinates, this
is performed on the coordinates that result after
<<textures-unnormalized-to-integer,conversion>> to integer texel
coordinates.

If the integer texel coordinates do not satisfy all of the conditions

  :: [eq]#0 {leq} i < w~s~#
  :: [eq]#0 {leq} j < h~s~#
  :: [eq]#0 {leq} k < d~s~#
  :: [eq]#0 {leq} l < layers#
  :: [eq]#0 {leq} n < samples#

where:

  :: [eq]#w~s~ =# width of the image level
  :: [eq]#h~s~ =# height of the image level
  :: [eq]#d~s~ =# depth of the image level
  :: [eq]#layers =# number of layers in the image
  :: [eq]#samples =# number of samples per texel in the image

then the texel fails integer texel coordinate validation.

There are four cases to consider:

  . Valid Texel Coordinates
+
  * If the texel coordinates pass validation (that is, the coordinates lie
     within the image),
+
then the texel value comes from the value in image memory.

  . Border Texel
+
  * If the texel coordinates fail validation, and
  * If the read is the result of an image sample instruction or image gather
     instruction, and
  * If the image is not a cube image,
+
then the texel is a border texel and <<textures-texel-replacement,texel
replacement>> is performed.

  . Invalid Texel
+
  * If the texel coordinates fail validation, and
  * If the read is the result of an image fetch instruction, image read
    instruction, or atomic instruction,
+
then the texel is an invalid texel and <<textures-texel-replacement,texel
replacement>> is performed.

  . Cube Map Edge or Corner
+
Otherwise the texel coordinates lie on the borders along the edges and
corners of a cube map image, and <<textures-cubemapedge, Cube map edge
handling>> is performed.


[[textures-cubemapedge]]
==== Cube Map Edge Handling

If the texel coordinates lie on the borders along the edges and corners of a
cube map image, the following steps are performed.
Note that this only occurs when using ename:VK_FILTER_LINEAR filtering
within a mip level, since ename:VK_FILTER_NEAREST is treated as using
ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE.

  * Cube Map Edge Texel
+
  ** If the texel lies along the border in either only [eq]#i# or only
    [eq]#j#
+
then the texel lies along an edge, so the coordinates [eq]#(i,j)# and the
array layer [eq]#l# are transformed to select the adjacent texel from the
appropriate neighboring face.

  * Cube Map Corner Texel
+
  ** If the texel lies along the border in both [eq]#i# and [eq]#j#
+
then the texel lies at a corner and there is no unique neighboring face from
which to read that texel.
The texel should: be replaced by the average of the three values of the
adjacent texels in each incident face.
However, implementations may: replace the cube map corner texel by other
methods, subject to the constraint that if the three available samples have
the same value, the replacement texel also has that value.


[[textures-sparse-validation]]
==== Sparse Validation

If the texel reads from an unbound region of a sparse image, the texel is a
_sparse unbound texel_, and processing continues with
<<textures-texel-replacement,texel replacement>>.

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]

[[textures-layout-validation]]
==== Layout Validation

If all planes of a _disjoint_ _multi-planar_ image are not in the same
<<resources-image-layouts,image layout>> when the image is sampled with
<<samplers-YCbCr-conversion,sampler Y'C~B~C~R~ conversion>>, the result of
texel reads is undefined.

endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]

[[textures-format-conversion]]
=== Format Conversion

Texels undergo a format conversion from the elink:VkFormat of the image view
to a vector of either floating point or signed or unsigned integer
components, with the number of components based on the number of components
present in the format.

  * Color formats have one, two, three, or four components, according to the
    format.
  * Depth/stencil formats are one component.
    The depth or stencil component is selected by the pname:aspectMask of
    the image view.

Each component is converted based on its type and size (as defined in the
<<features-formats-definition,Format Definition>> section for each
elink:VkFormat), using the appropriate equations in
<<fundamentals-fp16,16-Bit Floating-Point Numbers>>,
<<fundamentals-fp11,Unsigned 11-Bit Floating-Point Numbers>>,
<<fundamentals-fp10,Unsigned 10-Bit Floating-Point Numbers>>,
<<fundamentals-fixedconv,Fixed-Point Data Conversion>>, and
<<textures-sexp-RGB,Shared Exponent to RGB>>.
Signed integer components smaller than 32 bits are sign-extended.

If the image view format is sRGB, the color components are first converted
as if they are UNORM, and then sRGB to linear conversion is applied to the
R, G, and B components as described in the "`sRGB EOTF`" section of the
<<data-format,Khronos Data Format Specification>>.
The A component, if present, is unchanged.

If the image view format is block-compressed, then the texel value is first
decoded, then converted based on the type and number of components defined
by the compressed format.


[[textures-texel-replacement]]
=== Texel Replacement

A texel is replaced if it is one (and only one) of:

  * a border texel,
  * an invalid texel, or
  * a sparse unbound texel.

Border texels are replaced with a value based on the image format and the
pname:borderColor of the sampler.
The border color is:

[[textures-border-replacement-color]]
.Border Color [eq]#B#
[options="header",cols="60%,40%"]
|====
| Sampler pname:borderColor                     | Corresponding Border Color
| ename:VK_BORDER_COLOR_FLOAT_TRANSPARENT_BLACK | [eq]#B = (0.0, 0.0, 0.0, 0.0)#
| ename:VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK      | [eq]#B = (0.0, 0.0, 0.0, 1.0)#
| ename:VK_BORDER_COLOR_FLOAT_OPAQUE_WHITE      | [eq]#B = (1.0, 1.0, 1.0, 1.0)#
| ename:VK_BORDER_COLOR_INT_TRANSPARENT_BLACK   | [eq]#B = (0, 0, 0, 0)#
| ename:VK_BORDER_COLOR_INT_OPAQUE_BLACK        | [eq]#B = (0, 0, 0, 1)#
| ename:VK_BORDER_COLOR_INT_OPAQUE_WHITE        | [eq]#B = (1, 1, 1, 1)#
|====

[NOTE]
.Note
====
The names etext:VK_BORDER_COLOR_*\_TRANSPARENT_BLACK,
etext:VK_BORDER_COLOR_*\_OPAQUE_BLACK, and
etext:VK_BORDER_COLOR_*_OPAQUE_WHITE are meant to describe which components
are zeros and ones in the vocabulary of compositing, and are not meant to
imply that the numerical value of ename:VK_BORDER_COLOR_INT_OPAQUE_WHITE is
a saturating value for integers.
====

This is substituted for the texel value by replacing the number of
components in the image format

[[textures-border-replacement-table]]
.Border Texel Components After Replacement
[width="80%",options="header"]
|====
| Texel Aspect or Format      | Component Assignment
| Depth aspect                | [eq]#D       = B~r~#
| Stencil aspect              | [eq]#S       = B~r~#
| One component color format  | [eq]#C~r~    = B~r~#
| Two component color format  | [eq]#C~rg~   = (B~r~,B~g~)#
| Three component color format| [eq]#C~rgb~  = (B~r~,B~g~,B~b~)#
| Four component color format | [eq]#C~rgba~ = (B~r~,B~g~,B~b~,B~a~)#
|====

The value returned by a read of an invalid texel is undefined, unless that
read operation is from a buffer resource and the pname:robustBufferAccess
feature is enabled.
In that case, an invalid texel is replaced as described by the
<<features-features-robustBufferAccess,pname:robustBufferAccess feature>>.

If the
slink:VkPhysicalDeviceSparseProperties::pname:residencyNonResidentStrict
property is ename:VK_TRUE, a sparse unbound texel is replaced with 0 or 0.0
values for integer and floating-point components of the image format,
respectively.

If pname:residencyNonResidentStrict is ename:VK_FALSE, the value of the
sparse unbound texel is undefined.


[[textures-depth-compare-operation]]
=== Depth Compare Operation

If the image view has a depth/stencil format, the depth component is
selected by the pname:aspectMask, and the operation is a code:Dref
instruction, a depth comparison is performed.
The value of the result [eq]#D# is [eq]#1.0# if the result of the compare
operation is [eq]#true#, and [eq]#0.0# otherwise.
The compare operation is selected by the pname:compareOp member of the
sampler.

[latexmath]
+++++++++++++++++++
\begin{aligned}
D & = 1.0 &
  \begin{cases}
    D_{\textit{ref}} \leq D & \text{for LEQUAL}   \\
    D_{\textit{ref}} \geq D & \text{for GEQUAL}   \\
    D_{\textit{ref}} < D    & \text{for LESS}     \\
    D_{\textit{ref}} > D    & \text{for GREATER}  \\
    D_{\textit{ref}} = D    & \text{for EQUAL}    \\
    D_{\textit{ref}} \neq D & \text{for NOTEQUAL} \\
    \textit{true}           & \text{for ALWAYS}   \\
    \textit{false}          & \text{for NEVER}
  \end{cases} \\
D & = 0.0 & \text{otherwise}
\end{aligned}
+++++++++++++++++++

where, in the depth comparison:

  :: [eq]#D~ref~ = shaderOp.D~ref~# (from optional: SPIR-V operand)
  :: [eq]#D# (texel depth value)


[[textures-conversion-to-rgba]]
=== Conversion to RGBA

The texel is expanded from one, two, or three to four components based on
the image base color:

[[textures-texel-color-rgba-conversion-table]]
.Texel Color After Conversion To RGBA
[options="header"]
|====
| Texel Aspect or Format      | RGBA Color
| Depth aspect                | [eq]#C~rgba~ = (D,0,0,one)#
| Stencil aspect              | [eq]#C~rgba~ = (S,0,0,one)#
| One component color format  | [eq]#C~rgba~ = (C~r~,0,0,one)#
| Two component color format  | [eq]#C~rgba~ = (C~rg~,0,one)#
| Three component color format| [eq]#C~rgba~ = (C~rgb~,one)#
| Four component color format | [eq]#C~rgba~ = C~rgba~#
|====

where [eq]#one = 1.0f# for floating-point formats and depth aspects, and
[eq]#one = 1# for integer formats and stencil aspects.


[[textures-component-swizzle]]
=== Component Swizzle

ifndef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
All texel input instructions apply a _swizzle_ based on the
elink:VkComponentSwizzle enums in the pname:components member of the
slink:VkImageViewCreateInfo structure for the image being read.
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
All texel input instructions apply a _swizzle_ based on:

  * the elink:VkComponentSwizzle enums in the pname:components member of the
    slink:VkImageViewCreateInfo structure for the image being read if
    <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~ conversion>> is not
    enabled, and
  * the elink:VkComponentSwizzle enums in the pname:components member of the
    slink:VkSamplerYcbcrConversionCreateInfo structure for the
    <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~ conversion>> if sampler
    Y'C~B~C~R~ conversion is enabled.

endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]

The swizzle can: rearrange the components of the texel, or substitute zero
and one for any components.
It is defined as follows for the R component, and operates similarly for the
other components.


[latexmath]
+++++++++++++++++++
\begin{aligned}
C'_{rgba}[R] & =
  \begin{cases}
    C_{rgba}[R] & \text{for RED swizzle}   \\
    C_{rgba}[G] & \text{for GREEN swizzle} \\
    C_{rgba}[B] & \text{for BLUE swizzle}  \\
    C_{rgba}[A] & \text{for ALPHA swizzle} \\
    0           & \text{for ZERO swizzle}  \\
    one         & \text{for ONE swizzle} \\
    C_{rgba}[R] & \text{for IDENTITY swizzle}
  \end{cases}
\end{aligned}
+++++++++++++++++++

where:

[latexmath]
+++++++++++++++++++
\begin{aligned}
C_{rgba}[R] & \text{is the RED component} \\
C_{rgba}[G] & \text{is the GREEN component} \\
C_{rgba}[B] & \text{is the BLUE component} \\
C_{rgba}[A] & \text{is the ALPHA component} \\
one         & = 1.0\text{f}  & \text{for floating point components} \\
one         & = 1              & \text{for integer components}
\end{aligned}
+++++++++++++++++++

For each component this is applied to, the
ename:VK_COMPONENT_SWIZZLE_IDENTITY swizzle selects the corresponding
component from [eq]#C~rgba~#.

If the border color is one of the etext:VK_BORDER_COLOR_*_OPAQUE_BLACK enums
and the elink:VkComponentSwizzle is not ename:VK_COMPONENT_SWIZZLE_IDENTITY
for all components (or the
<<resources-image-views-identity-mappings,equivalent identity mapping>>),
the value of the texel after swizzle is undefined.


[[textures-sparse-residency]]
=== Sparse Residency

code:OpImageSparse* instructions return a structure which includes a
_residency code_ indicating whether any texels accessed by the instruction
are sparse unbound texels.
This code can: be interpreted by the code:OpImageSparseTexelsResident
instruction which converts the residency code to a boolean value.


ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
[[textures-chroma-reconstruction]]
=== Chroma Reconstruction

In some color models, the color representation is defined in terms of
monochromatic light intensity (often called "`luma`") and color differences
relative to this intensity, often called "`chroma`".
It is common for color models other than RGB to represent the chroma
channels at lower spatial resolution than the luma channel.
This approach is used to take advantage of the eye's lower spatial
sensitivity to color compared with its sensitivity to brightness.
Less commonly, the same approach is used with additive color, since the
green channel dominates the eye's sensitivity to light intensity and the
spatial sensitivity to color introduced by red and blue is lower.

Lower-resolution channels are "`downsampled`" by resizing them to a lower
spatial resolution than the channel representing luminance.
The process of reconstructing a full color value for texture access involves
accessing both chroma and luma values at the same location.
To generate the color accurately, the values of the lower-resolution
channels at the location of the luma samples must be reconstructed from the
lower-resolution sample locations, an operation known here as "`chroma
reconstruction`" irrespective of the actual color model.

The location of the chroma samples relative to the luma coordinates is
determined by the pname:xChromaOffset and pname:yChromaOffset members of the
slink:VkSamplerYcbcrConversionCreateInfo structure used to create the
sampler Y'C~B~C~R~ conversion.

The following diagrams show the relationship between unnormalized (_u_,_v_)
coordinates and (_i_,_j_) integer texel positions in the luma channel (shown
in black, with circles showing integer sample positions) and the texel
coordinates of reduced-resolution chroma channels, shown as crosses in red.

[NOTE]
.Note
====
If the chroma values are reconstructed at the locations of the luma samples
by means of interpolation, chroma samples from outside the image bounds are
needed; these are determined according to <<textures-wrapping-operation>>.
These diagrams represent this by showing the bounds of the "`chroma texel`"
extending beyond the image bounds, and including additional chroma sample
positions where required for interpolation.
The limits of a sample for etext:NEAREST sampling is shown as a grid.
====

[%inline]
image::images/chromasamples_422_cosited.svg[align="center",title="422 downsampling, xChromaOffset=COSITED_EVEN"]

[%inline]
image::images/chromasamples_422_midpoint.svg[align="center",title="422 downsampling, xChromaOffset=MIDPOINT"]

[%inline]
image::images/chromasamples_420_xcosited_ycosited.svg[align="center",title="420 downsampling, xChromaOffset=COSITED_EVEN, yChromaOffset=COSITED_EVEN"]

[%inline]
image::images/chromasamples_420_xmidpoint_ycosited.svg[align="center",title="420 downsampling, xChromaOffset=MIDPOINT, yChromaOffset=COSITED_EVEN"]

[%inline]
image::images/chromasamples_420_xcosited_ymidpoint.svg[align="center",title="420 downsampling, xChromaOffset=COSITED_EVEN, yChromaOffset=MIDPOINT"]

[%inline]
image::images/chromasamples_420_xmidpoint_ymidpoint.svg[align="center",title="420 downsampling, xChromaOffset=MIDPOINT, yChromaOffset=MIDPOINT"]

Reconstruction is implemented in one of two ways:

If the format of the image that is to be sampled sets
ename:VK_FORMAT_FEATURE_SAMPLED_IMAGE_YCBCR_CONVERSION_CHROMA_RECONSTRUCTION_EXPLICIT_BIT,
or the sname:VkSamplerYcbcrConversionCreateInfo's
pname:forceExplicitReconstruction is set to ename:VK_TRUE, reconstruction is
performed as an explicit step independent of filtering, described in the
<<textures-explicit-reconstruction>> section.

If the format of the image that is to be sampled does not set
ename:VK_FORMAT_FEATURE_SAMPLED_IMAGE_YCBCR_CONVERSION_CHROMA_RECONSTRUCTION_EXPLICIT_BIT
and if the sname:VkSamplerYcbcrConversionCreateInfo's
pname:forceExplicitReconstruction is set to ename:VK_FALSE, reconstruction
is performed as an implicit part of filtering prior to color model
conversion, with no separate post-conversion texel filtering step, as
described in the <<textures-implict-reconstruction,Implicit Reconstruction>>
section.

[[textures-explicit-reconstruction]]
==== Explicit Reconstruction

  * If the pname:chromaFilter member of the
    slink:VkSamplerYcbcrConversionCreateInfo structure is
    ename:VK_FILTER_NEAREST:
    ** If the format's R and B channels are reduced in resolution in just
       width by a factor of two relative to the G channel (i.e. this is a
       "`etext:_422`" format), the latexmath:[\tau_{ijk}[level\]] values
       accessed by <<textures-texel-filtering,texel filtering>> are
       reconstructed as follows:
+
[latexmath]
++++++++++++++
\begin{aligned}
\tau_R'(i, j) & = \tau_R(\lfloor{i\times 0.5}\rfloor, j)[level] \\
\tau_B'(i, j) & = \tau_B(\lfloor{i\times 0.5}\rfloor, j)[level]
\end{aligned}
++++++++++++++

    ** If the format's R and B channels are reduced in resolution in width
       and height by a factor of two relative to the G channel (i.e. this is
       a "`etext:_420`" format), the latexmath:[\tau_{ijk}[level\]] values
       accessed by <<textures-texel-filtering,texel filtering>> are
       reconstructed as follows:
+
[latexmath]
++++++++++++++
\begin{aligned}
\tau_R'(i, j) & = \tau_R(\lfloor{i\times 0.5}\rfloor, \lfloor{j\times 0.5}\rfloor)[level] \\
\tau_B'(i, j) & = \tau_B(\lfloor{i\times 0.5}\rfloor, \lfloor{j\times 0.5}\rfloor)[level]
\end{aligned}
++++++++++++++
+
[NOTE]
.Note
====
pname:xChromaOffset and pname:yChromaOffset have no effect if
pname:chromaFilter is ename:VK_FILTER_NEAREST for explicit reconstruction.
====

  * If the pname:chromaFilter member of the
    slink:VkSamplerYcbcrConversionCreateInfo structure is
    ename:VK_FILTER_LINEAR:
  ** If the format's R and B channels are reduced in resolution in just
     width by a factor of two relative to the G channel (i.e. this is a
     "`422`" format):
  *** If pname:xChromaOffset is ename:VK_CHROMA_LOCATION_COSITED_EVEN:
+
[latexmath]
+++++
\tau_{RB}'(i,j) = \begin{cases}
\tau_{RB}(\lfloor{i\times 0.5}\rfloor,j)[level], & 0.5 \times i = \lfloor{0.5 \times i}\rfloor\\
0.5\times\tau_{RB}(\lfloor{i\times 0.5}\rfloor,j)[level] + \\
0.5\times\tau_{RB}(\lfloor{i\times 0.5}\rfloor + 1,j)[level], & 0.5 \times i \neq \lfloor{0.5 \times i}\rfloor
\end{cases}
+++++
+
  *** If pname:xChromaOffset is ename:VK_CHROMA_LOCATION_MIDPOINT:
+
[latexmath]
+++++
\tau_{RB}(i,j)' = \begin{cases}
0.25 \times \tau_{RB}(\lfloor{i\times 0.5}\rfloor - 1,j)[level] + \\
0.75 \times \tau_{RB}(\lfloor{i\times 0.5}\rfloor,j)[level], & 0.5 \times i = \lfloor{0.5 \times i}\rfloor\\
0.75 \times \tau_{RB}(\lfloor{i\times 0.5}\rfloor,j)[level] + \\
0.25 \times \tau_{RB}(\lfloor{i\times 0.5}\rfloor + 1,j)[level], & 0.5 \times i \neq \lfloor{0.5 \times i}\rfloor
\end{cases}
+++++

  ** If the format's R and B channels are reduced in resolution in width and
     height by a factor of two relative to the G channel (i.e. this is a
     "`420`" format), a similar relationship applies.
     Due to the number of options, these formulae are expressed more
     concisely as follows:
+
[width="30%",options="header",cols="5,1"]
|====
| pname:xChromaOffset | &#948;~i~
| etext:COSITED_EVEN  | 0
| etext:MIDPOINT      | 0.5
|====
+
[width="30%",options="header",cols="5,1"]
|====
| pname:yChromaOffset | &#948;~j~
| etext:COSITED_EVEN  | 0
| etext:MIDPOINT      | 0.5
|====
+
[latexmath]
+++++
\begin{aligned}
\tau_{RB}'(i,j) = &\\
&\tau_{RB}(\lfloor 0.5\times(i-\delta_i)\rfloor, \lfloor 0.5\times(j-\delta_j)\rfloor)[level]
&& \times (1 - (0.5\times(i-\delta_i) - \lfloor 0.5\times(i-\delta_i)\rfloor))
&& \times (1 - (0.5\times(j-\delta_j) - \lfloor 0.5\times(j-\delta_j)\rfloor)) +\\
&\tau_{RB}(1+\lfloor 0.5\times(i-\delta_i)\rfloor, \lfloor 0.5\times(j-\delta_j)\rfloor)[level]
&& \times (0.5\times(i-\delta_i) - \lfloor 0.5\times(i-\delta_i)\rfloor)
&& \times (1 - (0.5\times(j-\delta_j) - \lfloor 0.5\times(j-\delta_j)\rfloor)) +\\
&\tau_{RB}(\lfloor 0.5\times(i-\delta_i)\rfloor, 1+\lfloor 0.5\times(j-\delta_j)\rfloor)[level]
&& \times (1 - (0.5\times(i-\delta_i) - \lfloor 0.5\times(i-\delta_i)\rfloor))
&& \times (0.5\times(j-\delta_j) - \lfloor 0.5\times(j-\delta_j)\rfloor) +\\
&\tau_{RB}(1+\lfloor 0.5\times(i-\delta_i)\rfloor, 1+\lfloor 0.5\times(j-\delta_j)\rfloor)[level]
&& \times (0.5\times(i-\delta_i) - \lfloor 0.5\times(i-\delta_i)\rfloor)
&& \times (0.5\times(j-\delta_j) - \lfloor 0.5\times(j-\delta_j)\rfloor)
\end{aligned}
+++++

[NOTE]
.Note
====
In the case where the texture itself is bilinearly interpolated as described
in <<textures-texel-filtering,Texel Filtering>>, thus requiring four
full-color samples for the filtering operation, and where the reconstruction
of these samples uses bilinear interpolation in the chroma channels due to
pname:chromaFilter=ename:VK_FILTER_LINEAR, up to nine chroma samples may be
required, depending on the sample location.
====


[[textures-implict-reconstruction]]
==== Implicit Reconstruction

Implicit reconstruction takes place by the samples being interpolated, as
required by the filter settings of the sampler, except that
pname:chromaFilter takes precedence for the chroma samples.

If pname:chromaFilter is ename:VK_FILTER_NEAREST, an implementation may:
behave as if pname:xChromaOffset and pname:yChromaOffset were both
ename:VK_CHROMA_LOCATION_MIDPOINT, irrespective of the values set.

[NOTE]
.Note
====
This will not have any visible effect if the locations of the luma samples
coincide with the location of the samples used for rasterization.
====

The sample coordinates are adjusted by the downsample factor of the channel
(such that, for example, the sample coordinates are divided by two if the
channel has a downsample factor of two relative to the luma channel):

[latexmath]
++++++
\begin{aligned}
u_{RB}' (422/420) &=
  \begin{cases}
     0.5\times (u + 0.5), & \textrm{xChromaOffset = COSITED}\_\textrm{EVEN} \\
     0.5\times u, & \textrm{xChromaOffset = MIDPOINT}
  \end{cases} \\
v_{RB}' (420) &=
  \begin{cases}
     0.5\times (v + 0.5), & \textrm{yChromaOffset = COSITED}\_\textrm{EVEN} \\
     0.5\times v, & \textrm{yChromaOffset = MIDPOINT}
  \end{cases}
\end{aligned}
++++++


[[textures-sampler-YCbCr-conversion]]
=== Sampler Y'C~B~C~R~ Conversion

Sampler Y'C~B~C~R~ conversion performs the following operations, which an
implementation may: combine into a single mathematical operation:

  * <<textures-sampler-YCbCr-conversion-rangeexpand,Sampler Y'C~B~C~R~ Range
    Expansion>>
  * <<textures-sampler-YCbCr-conversion-modelconversion,Sampler Y'C~B~C~R~
    Model Conversion>>

[[textures-sampler-YCbCr-conversion-rangeexpand]]
==== Sampler Y'C~B~C~R~ Range Expansion

Sampler Y'C~B~C~R~ range expansion is applied to color channel values after
all texel input operations which are not specific to sampler Y'C~B~C~R~
conversion.
For example, the input values to this stage have been converted using the
normal <<textures-format-conversion,format conversion>> rules.

Sampler Y'C~B~C~R~ range expansion is not applied if pname:ycbcrModel is
ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_RGB_IDENTITY.
That is, the shader receives the vector C'~rgba~ as output by the Component
Swizzle stage without further modification.

For other values of pname:ycbcrModel, range expansion is applied to the
texel channel values output by the <<textures-component-swizzle,Component
Swizzle>> defined by the pname:components member of
slink:VkSamplerYcbcrConversionCreateInfo.
Range expansion applies independently to each channel of the image.
For the purposes of range expansion and Y'C~B~C~R~ model conversion, the R
and B channels contain color difference (chroma) values and the G channel
contains luma.
The A channel is not modified by sampler Y'C~B~C~R~ range expansion.

The range expansion to be applied is defined by the pname:ycbcrRange member
of the sname:VkSamplerYcbcrConversionCreateInfo structure:

  * If pname:ycbcrRange is ename:VK_SAMPLER_YCBCR_RANGE_ITU_FULL, the
    following transformations are applied:
+
[latexmath]
+++++++++++++++++++
\begin{aligned}
Y' &= C'_{rgba}[G] \\
C_B &= C'_{rgba}[B] - {{2^{(n-1)}}\over{(2^n) - 1}} \\
C_R &= C'_{rgba}[R] - {{2^{(n-1)}}\over{(2^n) - 1}}
\end{aligned}
+++++++++++++++++++
+
[NOTE]
.Note
====
These formulae correspond to the "`full range`" encoding in the
<<data-format,Khronos Data Format Specification>>.

Should any future amendments be made to the ITU specifications from which
these equations are derived, the formulae used by Vulkan may: also be
updated to maintain parity.
====
  * If pname:ycbcrRange is ename:VK_SAMPLER_YCBCR_RANGE_ITU_NARROW, the
    following transformations are applied:
+
[latexmath]
+++++++++++++++++++
\begin{aligned}
Y' &= {{C'_{rgba}[G] \times (2^n-1) - 16\times 2^{n-8}}\over{219\times 2^{n-8}}} \\
C_B &= {{C'_{rgba}[B] \times \left(2^n-1\right) - 128\times 2^{n-8}}\over{224\times 2^{n-8}}} \\
C_R &= {{C'_{rgba}[R] \times \left(2^n-1\right) - 128\times 2^{n-8}}\over{224\times 2^{n-8}}}
\end{aligned}
+++++++++++++++++++
+
[NOTE]
.Note
====
These formulae correspond to the "`narrow range`" encoding in the
<<data-format,Khronos Data Format Specification>>.
====
  * _n_ is the bit-depth of the channels in the format.

The precision of the operations performed during range expansion must: be at
least that of the source format.

An implementation may: clamp the results of these range expansion operations
such that Y' falls in the range [0,1], and/or such that C~B~ and C~R~ fall
in the range [-0.5,0.5].

[[textures-sampler-YCbCr-conversion-modelconversion]]
==== Sampler Y'C~B~C~R~ Model Conversion

The range-expanded values are converted between color models, according to
the color model conversion specified in the pname:ycbcrModel member:

ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_RGB_IDENTITY::
  The color channels are not modified by the color model conversion since
  they are assumed already to represent the desired color model in which the
  shader is operating; Y'C~B~C~R~ range expansion is also ignored.
ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_YCBCR_IDENTITY::
  The color channels are not modified by the color model conversion and are
  assumed to be treated as though in Y'C~B~C~R~ form both in memory and in
  the shader; Y'C~B~C~R~ range expansion is applied to the channels as for
  other Y'C~B~C~R~ models, with the vector (C~R~,Y',C~B~,A) provided to the
  shader.
ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_YCBCR_709::
  The color channels are transformed from a Y'C~B~C~R~ representation to an
  R'G'B' representation as described in the "`BT.709 Y'C~B~C~R~ conversion`"
  section of the <<data-format,Khronos Data Format Specification>>.
ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_YCBCR_601::
  The color channels are transformed from a Y'C~B~C~R~ representation to an
  R'G'B' representation as described in the "`BT.601 Y'C~B~C~R~ conversion`"
  section of the <<data-format,Khronos Data Format Specification>>.
ename:VK_SAMPLER_YCBCR_MODEL_CONVERSION_YCBCR_2020::
  The color channels are transformed from a Y'C~B~C~R~ representation to an
  R'G'B' representation as described in the "`BT.2020 Y'C~B~C~R~
  conversion`" section of the <<data-format,Khronos Data Format
  Specification>>.

In this operation, each output channel is dependent on each input channel.

An implementation may: clamp the R'G'B' results of these conversions to the
range [0,1].

The precision of the operations performed during model conversion must: be
at least that of the source format.

The alpha channel is not modified by these model conversions.

[NOTE]
.Note
====
Sampling operations in a non-linear color space can introduce color and
intensity shifts at sharp transition boundaries.
To avoid this issue, the technically precise color correction sequence
described in the "`Introduction to Color Conversions`" chapter of the
<<data-format,Khronos Data Format Specification>> may be performed as
follows:

  * Calculate the <<textures-normalized-to-unnormalized,unnormalized texel
    coordinates>> corresponding to the desired sample position.
  * For a pname:minFilter/pname:magFilter of ename:VK_FILTER_NEAREST:
    . Calculate (_i_,_j_) for the sample location as described under the
      "`nearest filtering`" formulae in <<textures-unnormalized-to-integer>>
    . Calculate the normalized texel coordinates corresponding to these
      integer coordinates.
    . Sample using <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~
      conversion>> at this location.
  * For a pname:minFilter/pname:magFilter of ename:VK_FILTER_LINEAR:
    . Calculate (_i~[0,1]~_,_j~[0,1]~_) for the sample location as described
      under the "`linear filtering`" formulae in
      <<textures-unnormalized-to-integer>>
    . Calculate the normalized texel coordinates corresponding to these
      integer coordinates.
    . Sample using <<samplers-YCbCr-conversion,sampler Y'C~B~C~R~
      conversion>> at each of these locations.
    . Convert the non-linear AR'G'B' outputs of the Y'C~B~C~R~ conversions
      to linear ARGB values as described in the "`Transfer Functions`"
      chapter of the <<data-format,Khronos Data Format Specification>>.
    . Interpolate the linear ARGB values using the [eq]#{alpha}# and
      [eq]#{beta}# values described in the "`linear filtering`" section of
      <<textures-unnormalized-to-integer>> and the equations in
      <<textures-texel-filtering>>.

The additional calculations and, especially, additional number of sampling
operations in the ename:VK_FILTER_LINEAR case can be expected to have a
performance impact compared with using the outputs directly; since the
variation from "`correct`" results are subtle for most content, the
application author should determine whether a more costly implementation is
strictly necessary.
Note that if pname:chromaFilter and pname:minFilter/pname:magFilter are both
ename:VK_FILTER_NEAREST, these operations are redundant and sampling using
<<samplers-YCbCr-conversion,sampler Y'C~B~C~R~ conversion>> at the desired
sample coordinates will produce the "`correct`" results without further
processing.
====
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]


== Texel Output Operations

_Texel output instructions_ are SPIR-V image instructions that write to an
image.
_Texel output operations_ are a set of steps that are performed on state,
coordinates, and texel values while processing a texel output instruction,
and which are common to some or all texel output instructions.
They include the following steps, which are performed in the listed order:

  * <<textures-output-validation,Validation operations>>
  ** <<textures-format-validation,Format validation>>
  ** <<textures-output-coordinate-validation,Coordinate validation>>
  ** <<textures-output-sparse-validation,Sparse validation>>
  * <<textures-output-format-conversion,Texel output format conversion>>


[[textures-output-validation]]
=== Texel Output Validation Operations

_Texel output validation operations_ inspect instruction/image state or
coordinates, and in certain circumstances cause the write to have no effect.
There are a series of validations that the texel undergoes.


[[textures-format-validation]]
==== Texel Format Validation

If the image format of the code:OpTypeImage is not compatible with the
sname:VkImageView's pname:format, the effect of the write on the image
view's memory is undefined, but the write must: not access memory outside of
the image view.


[[textures-output-coordinate-validation]]
=== Integer Texel Coordinate Validation

The integer texel coordinates are validated according to the same rules as
for texel input <<textures-integer-coordinate-validation,coordinate
validation>>.

If the texel fails integer texel coordinate validation, then the write has
no effect.


[[textures-output-sparse-validation]]
=== Sparse Texel Operation

If the texel attempts to write to an unbound region of a sparse image, the
texel is a sparse unbound texel.
In such a case, if the
slink:VkPhysicalDeviceSparseProperties::pname:residencyNonResidentStrict
property is ename:VK_TRUE, the sparse unbound texel write has no effect.
If pname:residencyNonResidentStrict is ename:VK_FALSE, the write may: have a
side effect that becomes visible to other accesses to unbound texels in any
resource, but will not be visible to any device memory allocated by the
application.


[[textures-output-format-conversion]]
=== Texel Output Format Conversion

If the image format is sRGB, a linear to sRGB conversion is applied to the
R, G, and B components as described in the "`sRGB EOTF`" section of the
<<data-format,Khronos Data Format Specification>>.
The A component, if present, is unchanged.

Texels then undergo a format conversion from the floating point, signed, or
unsigned integer type of the texel data to the elink:VkFormat of the image
view.
Any unused components are ignored.

Each component is converted based on its type and size (as defined in the
<<features-formats-definition,Format Definition>> section for each
elink:VkFormat).
Floating-point outputs are converted as described in
<<fundamentals-fp-conversion,Floating-Point Format Conversions>> and
<<fundamentals-fixedconv,Fixed-Point Data Conversion>>.
Integer outputs are converted such that their value is preserved.
The converted value of any integer that cannot be represented in the target
format is undefined.


== Derivative Operations

SPIR-V derivative instructions include code:OpDPdx, code:OpDPdy,
code:OpDPdxFine, code:OpDPdyFine, code:OpDPdxCoarse, and code:OpDPdyCoarse.
Derivative instructions are only available in
ifdef::VK_NV_compute_shader_derivatives[]
compute and
endif::VK_NV_compute_shader_derivatives[]
fragment shaders.

[%inline]
image::images/vulkantexture2.svg[align="center",title="Implicit Derivatives"]

Derivatives are computed as if there is a 2{times}2 neighborhood of
fragments for each fragment shader invocation.
These neighboring fragments are used to compute derivatives with the
assumption that the values of P in the neighborhood are piecewise linear.
It is further assumed that the values of P in the neighborhood are locally
continuous, therefore derivatives in non-uniform control flow are undefined.

[latexmath]
+++++++++++++++++++
\begin{aligned}
dPdx_{i_1,j_0} & = dPdx_{i_0,j_0} & = P_{i_1,j_0} - P_{i_0,j_0}  \\
dPdx_{i_1,j_1} & = dPdx_{i_0,j_1} & = P_{i_1,j_1} - P_{i_0,j_1}  \\
\\
dPdy_{i_0,j_1} & = dPdy_{i_0,j_0} & = P_{i_0,j_1} - P_{i_0,j_0}  \\
dPdy_{i_1,j_1} & = dPdy_{i_1,j_0} & = P_{i_1,j_1} - P_{i_1,j_0}
\end{aligned}
+++++++++++++++++++

The code:Fine derivative instructions must: return the values above, for a
group of fragments in a 2{times}2 neighborhood.
Coarse derivatives may: return only two values.
In this case, the values should: be:

[latexmath]
+++++++++++++++++++
\begin{aligned}
dPdx & =
  \begin{cases}
    dPdx_{i_0,j_0} & \text{preferred}\\
    dPdx_{i_0,j_1}
  \end{cases} \\
dPdy & =
  \begin{cases}
    dPdy_{i_0,j_0} & \text{preferred}\\
    dPdy_{i_1,j_0}
  \end{cases}
\end{aligned}
+++++++++++++++++++

code:OpDPdx and code:OpDPdy must: return the same result as either
code:OpDPdxFine or code:OpDPdxCoarse and either code:OpDPdyFine or
code:OpDPdyCoarse, respectively.
Implementations must: make the same choice of either coarse or fine for both
code:OpDPdx and code:OpDPdy, and implementations should: make the choice
that is more efficient to compute.

ifdef::VK_NV_compute_shader_derivatives[]
[[texture-derivatives-compute]]
=== Compute Shader Derivatives

For compute shaders, derivatives are also evaluated using a 2{times}2
logical neighborhood of compute shader invocations.
Compute shader invocations are arranged into neighborhoods according to one
of two SPIR-V execution modes.
For the code:DerivativeGroupQuadsNV execution mode, each neighborhood is
assembled from a 2{times}2{times}1 region of invocations based on the
code:LocalInvocationId built-in.
For the code:DerivativeGroupLinearNV execution mode, each neighborhood is
assembled from a group of four invocations based on the
code:LocalInvocationIndex built-in.
The <<texture-derivatives-compute-group>> table specifies the
code:LocalInvocationId or code:LocalInvocationIndex values for the four
values of P in each neighborhood, where __x__ and __y__ are per-neighborhood
integer values.

[[texture-derivatives-compute-group]]
.Compute shader derivative group assignments
[width="75%",frame="all",options="header",cols="^2,^6,^6"]
|===
|Value
|DerivativeGroupQuadsNV
|DerivativeGroupLinearNV

|P~i0,j0~ | (2__x__ + 0, 2__y__ + 0, z) | 4__x__ + 0
|P~i1,j0~ | (2__x__ + 1, 2__y__ + 0, z) | 4__x__ + 1
|P~i0,j1~ | (2__x__ + 0, 2__y__ + 1, z) | 4__x__ + 2
|P~i1,j1~ | (2__x__ + 1, 2__y__ + 1, z) | 4__x__ + 3
|===

endif::VK_NV_compute_shader_derivatives[]

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
For multi-planar formats, the derivatives are computed based on the plane
with the largest dimensions.
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]


[[textures-normalized-operations]]
== Normalized Texel Coordinate Operations

If the image sampler instruction provides normalized texel coordinates, some
of the following operations are performed.


[[textures-projection]]
=== Projection Operation

For code:Proj image operations, the normalized texel coordinates
[eq]#(s,t,r,q,a)# and (if present) the [eq]#D~ref~# coordinate are
transformed as follows:

[latexmath]
+++++++++++++++++++
\begin{aligned}
s       & = \frac{s}{q},       & \text{for 1D, 2D, or 3D image} \\
\\
t       & = \frac{t}{q},       & \text{for 2D or 3D image} \\
\\
r       & = \frac{r}{q},       & \text{for 3D image} \\
\\
D_{\textit{ref}} & = \frac{D_{\textit{ref}}}{q}, & \text{if provided}
\end{aligned}
+++++++++++++++++++


[[textures-derivative-image-operations]]
=== Derivative Image Operations

Derivatives are used for LOD selection.
These derivatives are either implicit (in an code:ImplicitLod image
instruction in a fragment shader) or explicit (provided explicitly by shader
to the image instruction in any shader).

For implicit derivatives image instructions, the derivatives of texel
coordinates are calculated in the same manner as derivative operations
above.
That is:

[latexmath]
+++++++++++++++++++
\begin{aligned}
\partial{s}/\partial{x} & = dPdx(s), & \partial{s}/\partial{y} & = dPdy(s), & \text{for 1D, 2D, Cube, or 3D image} \\
\partial{t}/\partial{x} & = dPdx(t), & \partial{t}/\partial{y} & = dPdy(t), & \text{for 2D, Cube, or 3D image} \\
\partial{u}/\partial{x} & = dPdx(u), & \partial{u}/\partial{y} & = dPdy(u), & \text{for Cube or 3D image}
\end{aligned}
+++++++++++++++++++

Partial derivatives not defined above for certain image dimensionalities are
set to zero.

For explicit LOD image instructions, if the optional: SPIR-V operand
[eq]#Grad# is provided, then the operand values are used for the
derivatives.
The number of components present in each derivative for a given image
dimensionality matches the number of partial derivatives computed above.

If the optional: SPIR-V operand [eq]#Lod# is provided, then derivatives are
set to zero, the cube map derivative transformation is skipped, and the
scale factor operation is skipped.
Instead, the floating point scalar coordinate is directly assigned to
[eq]#{lambda}~base~# as described in <<textures-level-of-detail-operation,
Level-of-Detail Operation>>.

For implicit derivative image instructions, the partial derivative values
may: be computed by linear approximation using a 2{times}2 neighborhood of
shader invocations (known as a _quad_), as described above.
If the instruction is in control flow that is not uniform across the quad,
then the derivative values and hence the implicit LOD values are undefined.

ifdef::VK_EXT_descriptor_indexing[]
If the image or sampler object used by an implicit derivative image
instruction is not uniform across the quad and
<<features-limits-quadDivergentImplicitLod,pname:quadDivergentImplicitLod>>
is not supported, then the derivative and LOD values are undefined.
Implicit derivatives are well-defined when the image and sampler and control
flow are uniform across the quad, even if they diverge between different
quads.

If
<<features-limits-quadDivergentImplicitLod,pname:quadDivergentImplicitLod>>
is supported, then derivatives and implicit LOD values are well-defined even
if the image or sampler object are not uniform within a quad.
The derivatives are computed as specified above, and the implicit LOD
calculation proceeds for each shader invocation using its respective image
and sampler object.

For the purposes of implicit derivatives, code:Flat fragment input variables
are uniform within a quad.
endif::VK_EXT_descriptor_indexing[]


=== Cube Map Face Selection and Transformations

For cube map image instructions, the [eq]#(s,t,r)# coordinates are treated
as a direction vector [eq]#(r~x~,r~y~,r~z~)#.
The direction vector is used to select a cube map face.
The direction vector is transformed to a per-face texel coordinate system
[eq]#(s~face~,t~face~)#, The direction vector is also used to transform the
derivatives to per-face derivatives.


=== Cube Map Face Selection

The direction vector selects one of the cube map's faces based on the
largest magnitude coordinate direction (the major axis direction).
Since two or more coordinates can: have identical magnitude, the
implementation must: have rules to disambiguate this situation.

The rules should: have as the first rule that [eq]#r~z~# wins over
[eq]#r~y~# and [eq]#r~x~#, and the second rule that [eq]#r~y~# wins over
[eq]#r~x~#.
An implementation may: choose other rules, but the rules must: be
deterministic and depend only on [eq]#(r~x~,r~y~,r~z~)#.

The layer number (corresponding to a cube map face), the coordinate
selections for [eq]#s~c~#, [eq]#t~c~#, [eq]#r~c~#, and the selection of
derivatives, are determined by the major axis direction as specified in the
following two tables.

.Cube map face and coordinate selection
[width="75%",frame="all",options="header"]
|====
| Major Axis Direction | Layer Number | Cube Map Face | [eq]#s~c~#  | [eq]#t~c~#  | [eq]#r~c~#
| [eq]#+r~x~#          | [eq]#0#      | Positive X    | [eq]#-r~z~# | [eq]#-r~y~# | [eq]#r~x~#
| [eq]#-r~x~#          | [eq]#1#      | Negative X    | [eq]#+r~z~# | [eq]#-r~y~# | [eq]#r~x~#
| [eq]#+r~y~#          | [eq]#2#      | Positive Y    | [eq]#+r~x~# | [eq]#+r~z~# | [eq]#r~y~#
| [eq]#-r~y~#          | [eq]#3#      | Negative Y    | [eq]#+r~x~# | [eq]#-r~z~# | [eq]#r~y~#
| [eq]#+r~z~#          | [eq]#4#      | Positive Z    | [eq]#+r~x~# | [eq]#-r~y~# | [eq]#r~z~#
| [eq]#-r~z~#          | [eq]#5#      | Negative Z    | [eq]#-r~x~# | [eq]#-r~y~# | [eq]#r~z~#
|====


.Cube map derivative selection
[width="75%",frame="all",options="header"]
|====
| Major Axis Direction | [eq]#{partial}s~c~ / {partial}x# | [eq]#{partial}s~c~ / {partial}y# | [eq]#{partial}t~c~ / {partial}x# | [eq]#{partial}t~c~ / {partial}y# | [eq]#{partial}r~c~ / {partial}x# | [eq]#{partial}r~c~ / {partial}y#

| [eq]#+r~x~#
| [eq]#-{partial}r~z~ / {partial}x# | [eq]#-{partial}r~z~ / {partial}y#
| [eq]#-{partial}r~y~ / {partial}x# | [eq]#-{partial}r~y~ / {partial}y#
| [eq]#+{partial}r~x~ / {partial}x# | [eq]#+{partial}r~x~ / {partial}y#

| [eq]#-r~x~#
| [eq]#+{partial}r~z~ / {partial}x# | [eq]#+{partial}r~z~ / {partial}y#
| [eq]#-{partial}r~y~ / {partial}x# | [eq]#-{partial}r~y~ / {partial}y#
| [eq]#-{partial}r~x~ / {partial}x# | [eq]#-{partial}r~x~ / {partial}y#

| [eq]#+r~y~#
| [eq]#+{partial}r~x~ / {partial}x# | [eq]#+{partial}r~x~ / {partial}y#
| [eq]#+{partial}r~z~ / {partial}x# | [eq]#+{partial}r~z~ / {partial}y#
| [eq]#+{partial}r~y~ / {partial}x# | [eq]#+{partial}r~y~ / {partial}y#

| [eq]#-r~y~#
| [eq]#+{partial}r~x~ / {partial}x# | [eq]#+{partial}r~x~ / {partial}y#
| [eq]#-{partial}r~z~ / {partial}x# | [eq]#-{partial}r~z~ / {partial}y#
| [eq]#-{partial}r~y~ / {partial}x# | [eq]#-{partial}r~y~ / {partial}y#

| [eq]#+r~z~#
| [eq]#+{partial}r~x~ / {partial}x# | [eq]#+{partial}r~x~ / {partial}y#
| [eq]#-{partial}r~y~ / {partial}x# | [eq]#-{partial}r~y~ / {partial}y#
| [eq]#+{partial}r~z~ / {partial}x# | [eq]#+{partial}r~z~ / {partial}y#

| [eq]#-r~z~#
| [eq]#-{partial}r~x~ / {partial}x# | [eq]#-{partial}r~x~ / {partial}y#
| [eq]#-{partial}r~y~ / {partial}x# | [eq]#-{partial}r~y~ / {partial}y#
| [eq]#-{partial}r~z~ / {partial}x# | [eq]#-{partial}r~z~ / {partial}y#
|====


=== Cube Map Coordinate Transformation

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
s_{\textit{face}} & =
    \frac{1}{2} \times \frac{s_c}{|r_c|} + \frac{1}{2} \\
t_{\textit{face}} & =
    \frac{1}{2} \times \frac{t_c}{|r_c|} + \frac{1}{2} \\
\end{aligned}
++++++++++++++++++++++++


=== Cube Map Derivative Transformation

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\frac{\partial{s_{\textit{face}}}}{\partial{x}} &=
    \frac{\partial}{\partial{x}} \left ( \frac{1}{2} \times \frac{s_{c}}{|r_{c}|}
    + \frac{1}{2}\right ) \\
\frac{\partial{s_{\textit{face}}}}{\partial{x}} &=
    \frac{1}{2} \times \frac{\partial}{\partial{x}}
    \left ( \frac{s_{c}}{|r_{c}|}  \right ) \\
\frac{\partial{s_{\textit{face}}}}{\partial{x}} &=
    \frac{1}{2} \times
    \left (
    \frac{
      |r_{c}| \times \partial{s_c}/\partial{x}
      -s_c \times {\partial{r_{c}}}/{\partial{x}}}
    {\left ( r_{c} \right )^2}
    \right )
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\frac{\partial{s_{\textit{face}}}}{\partial{y}} &=
    \frac{1}{2} \times
    \left (
    \frac{
      |r_{c}| \times \partial{s_c}/\partial{y}
      -s_c \times {\partial{r_{c}}}/{\partial{y}}}
    {\left ( r_{c} \right )^2}
    \right )\\
\frac{\partial{t_{\textit{face}}}}{\partial{x}} &=
    \frac{1}{2} \times
    \left (
    \frac{
      |r_{c}| \times \partial{t_c}/\partial{x}
      -t_c \times {\partial{r_{c}}}/{\partial{x}}}
    {\left ( r_{c} \right )^2}
    \right ) \\
\frac{\partial{t_{\textit{face}}}}{\partial{y}} &=
    \frac{1}{2} \times
    \left (
    \frac{
       |r_{c}| \times \partial{t_c}/\partial{y}
      -t_c \times {\partial{r_{c}}}/{\partial{y}}}
    {\left ( r_{c} \right )^2}
    \right )
\end{aligned}
++++++++++++++++++++++++

ifdef::editing-notes[]
[NOTE]
.editing-note
====
(Bill) Note that we never revisited ARB_texture_cubemap after we introduced
dependent texture fetches (ARB_fragment_program and ARB_fragment_shader).

The derivatives of [eq]#s~face~# and [eq]#t~face~# are only valid for
non-dependent texture fetches (pre OpenGL 2.0).
====
endif::editing-notes[]


=== Scale Factor Operation, Level-of-Detail Operation and Image Level(s) Selection

LOD selection can: be either explicit (provided explicitly by the image
instruction) or implicit (determined from a scale factor calculated from the
derivatives).
The implicit LOD selected can: be queried using the SPIR-V instruction
code:OpImageQueryLod, which gives access to the [eq]#{lambda}#' and
[eq]#d~l~# values, defined below.


[[textures-scale-factor]]
==== Scale Factor Operation

The magnitude of the derivatives are calculated by:

  :: [eq]#m~ux~ = {vert}{partial}s/{partial}x{vert} {times} w~base~#
  :: [eq]#m~vx~ = {vert}{partial}t/{partial}x{vert} {times} h~base~#
  :: [eq]#m~wx~ = {vert}{partial}r/{partial}x{vert} {times} d~base~#

  :: [eq]#m~uy~ = {vert}{partial}s/{partial}y{vert} {times} w~base~#
  :: [eq]#m~vy~ = {vert}{partial}t/{partial}y{vert} {times} h~base~#
  :: [eq]#m~wy~ = {vert}{partial}r/{partial}y{vert} {times} d~base~#


where:

  :: [eq]#{partial}t/{partial}x = {partial}t/{partial}y = 0# (for 1D images)
  :: [eq]#{partial}r/{partial}x = {partial}r/{partial}y = 0# (for 1D, 2D or
     Cube images)

and

  :: [eq]#w~base~ = image.w#
  :: [eq]#h~base~ = image.h#
  :: [eq]#d~base~ = image.d#

(for the pname:baseMipLevel, from the image descriptor).

ifdef::VK_NV_corner_sampled_image[]

For corner-sampled images, the [eq]#w~base~#, [eq]#h~base~#, and
[eq]#d~base~# are instead:

  :: [eq]#w~base~ = image.w - 1#
  :: [eq]#h~base~ = image.h - 1#
  :: [eq]#d~base~ = image.d - 1#

endif::VK_NV_corner_sampled_image[]

A point sampled in screen space has an elliptical footprint in texture
space.
The minimum and maximum scale factors [eq]#({rho}~min~, {rho}~max~)# should:
be the minor and major axes of this ellipse.

The _scale factors_ [eq]#{rho}~x~# and [eq]#{rho}~y~#, calculated from the
magnitude of the derivatives in x and y, are used to compute the minimum and
maximum scale factors.

[eq]#{rho}~x~# and [eq]#{rho}~y~# may: be approximated with functions
[eq]#f~x~# and [eq]#f~y~#, subject to the following constraints:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
& f_x \text{\ is\ continuous\ and\ monotonically\ increasing\ in\ each\ of\ }
    m_{ux},
    m_{vx}, \text{\ and\ }
    m_{wx} \\
& f_y \text{\ is\ continuous\ and\ monotonically\ increasing\ in\ each\ of\ }
    m_{uy},
    m_{vy}, \text{\ and\ }
    m_{wy}
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\max(|m_{ux}|, |m_{vx}|, |m_{wx}|) \leq f_{x}
\leq \sqrt{2} (|m_{ux}| + |m_{vx}| + |m_{wx}|) \\
\max(|m_{uy}|, |m_{vy}|, |m_{wy}|) \leq f_{y}
\leq \sqrt{2} (|m_{uy}| + |m_{vy}| + |m_{wy}|)
\end{aligned}
++++++++++++++++++++++++


ifdef::editing-notes[]
[NOTE]
.editing-note
====
(Bill) For reviewers only - anticipating questions.

We only support implicit derivatives for normalized texel coordinates.

So we are documenting the derivatives in s,t,r (normalized texel
coordinates) rather than u,v,w (unnormalized texel coordinates) as in OpenGL
and OpenGL ES specifications.
(I know, u,v,w is the way it has been documented since OpenGL V1.0.)

Also there is no reason to have conditional application of [eq]#w~base~,
h~base~, d~base~# for rectangle textures either, since they do not support
implicit derivatives.
====
endif::editing-notes[]


The minimum and maximum scale factors [eq]#({rho}~min~,{rho}~max~)# are
determined by:

  :: [eq]#{rho}~max~ = max({rho}~x~, {rho}~y~)#
  :: [eq]#{rho}~min~ = min({rho}~x~, {rho}~y~)#

The ratio of anisotropy is determined by:

  :: [eq]#{eta} = min({rho}~max~/{rho}~min~, max~Aniso~)#

where:

  :: [eq]#sampler.max~Aniso~ = pname:maxAnisotropy# (from sampler
     descriptor)
  :: [eq]#limits.max~Aniso~ = pname:maxSamplerAnisotropy# (from physical
     device limits)
  :: [eq]#max~Aniso~ = min(sampler.max~Aniso~, limits.max~Aniso~)#

If [eq]#{rho}~max~ = {rho}~min~ = 0#, then all the partial derivatives are
zero, the fragment's footprint in texel space is a point, and [eq]#N#
should: be treated as 1.
If [eq]#{rho}~max~ {neq} 0# and [eq]#{rho}~min~ = 0# then all partial
derivatives along one axis are zero, the fragment's footprint in texel space
is a line segment, and [eq]#{eta}# should: be treated as [eq]#max~Aniso~#.
However, anytime the footprint is small in texel space the implementation
may: use a smaller value of [eq]#{eta}#, even when [eq]#{rho}~min~# is zero
or close to zero.
If either slink:VkPhysicalDeviceFeatures::pname:samplerAnisotropy or
slink:VkSamplerCreateInfo::pname:anisotropyEnable are ename:VK_FALSE,
[eq]#max~Aniso~# is set to 1.

If [eq]#{eta} = 1#, sampling is isotropic.
If [eq]#{eta} > 1#, sampling is anisotropic.

The sampling rate ([eq]#N#) is derived as:

  :: [eq]#N = {lceil}{eta}{rceil}#

An implementation may: round [eq]#N# up to the nearest supported sampling
rate.
An implementation may: use the value of [eq]#N# as an approximation of
[eq]#{eta}#.


[[textures-level-of-detail-operation]]
==== Level-of-Detail Operation

The LOD parameter [eq]#{lambda}# is computed as follows:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\lambda_{base}(x,y) & =
  \begin{cases}
    shaderOp.Lod                                    & \text{(from optional SPIR-V operand)} \\
    \log_2 \left ( \frac{\rho_{max}}{\eta} \right ) & \text{otherwise}
  \end{cases} \\
\lambda'(x,y)       & = \lambda_{base} + \mathbin{clamp}(sampler.bias + shaderOp.bias,-maxSamplerLodBias,maxSamplerLodBias) \\
\lambda             & =
  \begin{cases}
    lod_{max}, & \lambda' > lod_{max} \\
    \lambda',  & lod_{min} \leq \lambda' \leq lod_{max} \\
    lod_{min}, & \lambda' < lod_{min} \\
    \textit{undefined}, & lod_{min} > lod_{max}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
sampler.bias       & = mipLodBias & \text{(from sampler descriptor)} \\
shaderOp.bias      & =
  \begin{cases}
    Bias & \text{(from optional SPIR-V operand)} \\
    0    & \text{otherwise}
  \end{cases} \\
sampler.lod_{min}  & = minLod & \text{(from sampler descriptor)} \\
shaderOp.lod_{min} & =
  \begin{cases}
    MinLod & \text{(from optional SPIR-V operand)} \\
    0      & \text{otherwise}
  \end{cases} \\
\\
lod_{min}          & = \max(sampler.lod_{min}, shaderOp.lod_{min}) \\
lod_{max}          & = maxLod & \text{(from sampler descriptor)}
\end{aligned}
++++++++++++++++++++++++

and [eq]#maxSamplerLodBias# is the value of the slink:VkPhysicalDeviceLimits
feature <<features-limits-maxSamplerLodBias,pname:maxSamplerLodBias>>.


[[textures-image-level-selection]]
==== Image Level(s) Selection

The image level(s) [eq]#d#, [eq]#d~hi~#, and [eq]#d~lo~# which texels are
read from are determined by an image-level parameter [eq]#d~l~#, which is
computed based on the LOD parameter, as follows:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
d_{l} =
  \begin{cases}
    nearest(d'),  & \text{mipmapMode is VK\_SAMPLER\_MIPMAP\_MODE\_NEAREST} \\
    d',           & \text{otherwise}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
d' = level_{base} + \text{clamp}(\lambda, 0, q)
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
nearest(d') & =
  \begin{cases}
    \left \lceil d' + 0.5\right \rceil - 1, &
        \text{preferred} \\
    \left \lfloor d' + 0.5\right \rfloor,   &
        \text{alternative}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

and

  :: [eq]#level~base~ = pname:baseMipLevel#
  :: [eq]#q = pname:levelCount - 1#

pname:baseMipLevel and pname:levelCount are taken from the
pname:subresourceRange of the image view.

If the sampler's pname:mipmapMode is ename:VK_SAMPLER_MIPMAP_MODE_NEAREST,
then the level selected is [eq]#d = d~l~#.

If the sampler's pname:mipmapMode is ename:VK_SAMPLER_MIPMAP_MODE_LINEAR,
two neighboring levels are selected:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
d_{hi} & = \lfloor d_{l} \rfloor \\
d_{lo} & = min( d_{hi} + 1, q ) \\
\delta & = d_{l} - d_{hi}
\end{aligned}
++++++++++++++++++++++++

[eq]#{delta}# is the fractional value used for <<textures-texel-filtering,
linear filtering>> between levels.


[[textures-normalized-to-unnormalized]]
=== (s,t,r,q,a) to (u,v,w,a) Transformation

The normalized texel coordinates are scaled by the image level dimensions
and the array layer is selected.
This transformation is performed once for each level ([eq]#d# or [eq]#d~hi~#
and [eq]#d~lo~#) used in <<textures-texel-filtering,filtering>>.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
u(x,y) & = s(x,y) \times width_{scale} \\
v(x,y) & =
  \begin{cases}
    0                         & \text{for 1D images} \\
    t(x,y) \times height_{scale} & \text{otherwise}
  \end{cases} \\
w(x,y) & =
  \begin{cases}
    0                         & \text{for 2D or Cube images} \\
    r(x,y) \times depth_{scale}  & \text{otherwise}
  \end{cases} \\
\\
a(x,y) & =
  \begin{cases}
    a(x,y)                    & \text{for array images} \\
    0                         & \text{otherwise}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

where

  :: [eq]#width~scale~ = width~level~#
  :: [eq]#height~scale~ = height~level~#
  :: [eq]#depth~scale~ = depth~level~#

for conventional images
ifdef::VK_NV_corner_sampled_image[]
, and

  :: [eq]#width~scale~ = width~level~ - 1#
  :: [eq]#height~scale~ = height~level~ - 1#
  :: [eq]#depth~scale~ = depth~level~ - 1#

for corner-sampled images
endif::VK_NV_corner_sampled_image[]
.

Operations then proceed to Unnormalized Texel Coordinate Operations.


== Unnormalized Texel Coordinate Operations


[[textures-unnormalized-to-integer]]
=== (u,v,w,a) to (i,j,k,l,n) Transformation And Array Layer Selection

The unnormalized texel coordinates are transformed to integer texel
coordinates relative to the selected mipmap level.

The layer index [eq]#l# is computed as:

  :: [eq]#l = clamp(RNE(a), 0, pname:layerCount - 1) {plus}
     pname:baseArrayLayer#

where pname:layerCount is the number of layers in the image subresource
range of the image view, pname:baseArrayLayer is the first layer from the
subresource range, and where:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\mathbin{RNE}(a) & =
  \begin{cases}
    \mathbin{roundTiesToEven}(a)                  & \text{preferred, from IEEE Std 754-2008 Floating-Point Arithmetic} \\
    \left \lfloor a + 0.5 \right \rfloor & \text{alternative}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

The sample index n is assigned the value zero.

Nearest filtering (ename:VK_FILTER_NEAREST) computes the integer texel
coordinates that the unnormalized coordinates lie within:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
i &= \lfloor u + shift \rfloor \\
j &= \lfloor v + shift \rfloor \\
k &= \lfloor w + shift \rfloor
\end{aligned}
++++++++++++++++++++++++
where

  :: [eq]#shift = 0.0#

for conventional images
ifdef::VK_NV_corner_sampled_image[]
, and

  :: [eq]#shift = 0.5#

for corner-sampled images
endif::VK_NV_corner_sampled_image[]
.

Linear filtering (ename:VK_FILTER_LINEAR) computes a set of neighboring
coordinates which bound the unnormalized coordinates.
The integer texel coordinates are combinations of [eq]#i~0~# or [eq]#i~1~#,
[eq]#j~0~# or [eq]#j~1~#, [eq]#k~0~# or [eq]#k~1~#, as well as weights
[eq]#{alpha}, {beta}#, and [eq]#{gamma}#.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
i_0 &= \lfloor u - shift \rfloor \\
i_1 &= i_0 + 1 \\
j_0 &= \lfloor v - shift \rfloor \\
j_1 &= j_0 + 1 \\
k_0 &= \lfloor w - shift \rfloor \\
k_1 &= k_0 + 1 \\
\alpha &= \left(u - shift\right) - i_0 \\
\beta &= \left(v - shift\right) - j_0 \\
\gamma &= \left(w - shift\right) - k_0
\end{aligned}
++++++++++++++++++++++++

where

  :: [eq]#shift = 0.5#

for conventional images
ifdef::VK_NV_corner_sampled_image[]
, and

  :: [eq]#shift = 0.0#

for corner-sampled images
endif::VK_NV_corner_sampled_image[]
.

ifdef::VK_IMG_filter_cubic[]
Cubic filtering (ename:VK_FILTER_CUBIC_IMG) computes a set of neighboring
coordinates which bound the unnormalized coordinates.
The integer texel coordinates are combinations of [eq]#i~0~#, [eq]#i~1~#,
[eq]#i~2~# or [eq]#i~3~#, [eq]#j~0~#, [eq]#j~1~#, [eq]#j~2~# or [eq]#j~3~#,
as well as weights [eq]#{alpha}# and [eq]#{beta}#.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
i_{0}  & = \left \lfloor u - \frac{3}{2} \right \rfloor & i_{1} & = i_{0} + 1 & i_{2} & = i_{1} + 1 & i_{3} & = i_{2} + 1 \\[1em]
j_{0}  & = \left \lfloor v - \frac{3}{2} \right \rfloor & j_{1} & = j_{0} + 1 & j_{2} & = j_{1} + 1 & j_{3} & = j_{2} + 1 \\
\\
\alpha & = \mathbin{frac} \left ( u - \frac{1}{2} \right ) \\[1em]
\beta  & = \mathbin{frac} \left ( v - \frac{1}{2} \right )
\end{aligned}
++++++++++++++++++++++++
endif::VK_IMG_filter_cubic[]

If the image instruction includes a [eq]#ConstOffset# operand, the constant
offsets [eq]#({DeltaUpper}~i~, {DeltaUpper}~j~, {DeltaUpper}~k~)# are added
to [eq]#(i,j,k)# components of the integer texel coordinates.


[[textures-integer-coordinate-operations]]
== Integer Texel Coordinate Operations

ifdef::VK_AMD_shader_image_load_store_lod[]
Integer texel coordinate operations may: supply a LOD which texels are to be
read from or written to using the optional SPIR-V operand code:Lod.
endif::VK_AMD_shader_image_load_store_lod[]
ifndef::VK_AMD_shader_image_load_store_lod[]
The code:OpImageFetch and code:OpImageFetchSparse SPIR-V instructions may:
supply a LOD from which texels are to be fetched using the optional SPIR-V
operand code:Lod.
Other integer-coordinate operations must: not.
endif::VK_AMD_shader_image_load_store_lod[]
If the code:Lod is provided then it must: be an integer.

The image level selected is:
[latexmath]
++++++++++++++++++++++++
\begin{aligned}
d & = level_{base} +
  \begin{cases}
    Lod & \text{(from optional SPIR-V operand)} \\
    0   & \text{otherwise}
  \end{cases} \\
\end{aligned}
++++++++++++++++++++++++

If [eq]#d# does not lie in the range [eq]#[pname:baseMipLevel,
pname:baseMipLevel {plus} pname:levelCount)# then any values fetched are
ifndef::VK_AMD_shader_image_load_store_lod[undefined.]
ifdef::VK_AMD_shader_image_load_store_lod[]
undefined, and any writes are discarded.
endif::VK_AMD_shader_image_load_store_lod[]


[[textures-sample-operations]]
== Image Sample Operations


[[textures-wrapping-operation]]
=== Wrapping Operation

code:Cube images ignore the wrap modes specified in the sampler.
Instead, if ename:VK_FILTER_NEAREST is used within a mip level then
ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE is used, and if
ename:VK_FILTER_LINEAR is used within a mip level then sampling at the edges
is performed as described earlier in the <<textures-cubemapedge,Cube map
edge handling>> section.

The first integer texel coordinate i is transformed based on the
pname:addressModeU parameter of the sampler.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
i &=
  \begin{cases}
    i \bmod size                                & \text{for repeat} \\
    (size - 1) - \mathbin{mirror}
        ((i \bmod (2 \times size)) - size)      & \text{for mirrored repeat} \\
    \mathbin{clamp}(i,0,size-1)                  & \text{for clamp to edge} \\
    \mathbin{clamp}(i,-1,size)                   & \text{for clamp to border} \\
    \mathbin{clamp}(\mathbin{mirror}(i),0,size-1) & \text{for mirror clamp to edge}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
& \mathbin{mirror}(n) =
  \begin{cases}
    n      & \text{for}\  n \geq 0 \\
    -(1+n) & \text{otherwise}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++

[eq]#j# (for 2D and Cube image) and [eq]#k# (for 3D image) are similarly
transformed based on the pname:addressModeV and pname:addressModeW
parameters of the sampler, respectively.


[[textures-gather]]
=== Texel Gathering

SPIR-V instructions with code:Gather in the name return a vector derived
from a 2{times}2 rectangular region of texels in the base level of the image
view.
The rules for the ename:VK_FILTER_LINEAR minification filter are applied to
identify the four selected texels.
Each texel is then converted to an RGBA value according to
<<textures-conversion-to-rgba,conversion to RGBA>> and then
<<textures-component-swizzle,swizzled>>.
A four-component vector is then assembled by taking the component indicated
by the code:Component value in the instruction from the swizzled color value
of the four texels:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau[R] &= \tau_{i0j1}[level_{base}][comp] \\
\tau[G] &= \tau_{i1j1}[level_{base}][comp] \\
\tau[B] &= \tau_{i1j0}[level_{base}][comp] \\
\tau[A] &= \tau_{i0j0}[level_{base}][comp]
\end{aligned}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau[level_{base}][comp] &=
  \begin{cases}
    \tau[level_{base}][R], & \text{for}\  comp = 0 \\
    \tau[level_{base}][G], & \text{for}\  comp = 1 \\
    \tau[level_{base}][B], & \text{for}\  comp = 2 \\
    \tau[level_{base}][A], & \text{for}\  comp = 3
  \end{cases}\\
comp & \,\text{from SPIR-V operand Component}
\end{aligned}
++++++++++++++++++++++++

ifdef::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]
code:OpImage*Gather must: not be used on a sampled image with
<<samplers-YCbCr-conversion,sampler Y'C~B~C~R~ conversion>> enabled.
endif::VK_VERSION_1_1,VK_KHR_sampler_ycbcr_conversion[]


[[textures-texel-filtering]]
=== Texel Filtering

If [eq]#{lambda}# is less than or equal to zero, the texture is said to be
_magnified_, and the filter mode within a mip level is selected by the
pname:magFilter in the sampler.
If [eq]#{lambda}# is greater than zero, the texture is said to be
_minified_, and the filter mode within a mip level is selected by the
pname:minFilter in the sampler.

Within a mip level, ename:VK_FILTER_NEAREST filtering selects a single value
using the [eq]#(i, j, k)# texel coordinates, with all texels taken from
layer l.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau[level] &=
  \begin{cases}
     \tau_{ijk}[level], & \text{for 3D image} \\
     \tau_{ij}[level],  & \text{for 2D or Cube image} \\
     \tau_{i}[level],   & \text{for 1D image}
   \end{cases}
\end{aligned}
++++++++++++++++++++++++

Within a mip level, ename:VK_FILTER_LINEAR filtering combines 8 (for 3D), 4
(for 2D or Cube), or 2 (for 1D) texel values, using the weights computed
earlier:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau_{3D}[level] & = reduce((1-\alpha)(1-\beta)(1-\gamma),\tau_{i0j0k0}[level], \\
          & \,              (\alpha)(1-\beta)(1-\gamma),\tau_{i1j0k0}[level], \\
          & \,              (1-\alpha)(\beta)(1-\gamma),\tau_{i0j1k0}[level], \\
          & \,              (\alpha)(\beta)(1-\gamma),\tau_{i1j1k0}[level],   \\
          & \,              (1-\alpha)(1-\beta)(\gamma),\tau_{i0j0k1}[level], \\
          & \,              (\alpha)(1-\beta)(\gamma),\tau_{i1j0k1}[level],   \\
          & \,              (1-\alpha)(\beta)(\gamma),\tau_{i0j1k1}[level],   \\
          & \,              (\alpha)(\beta)(\gamma),\tau_{i1j1k1}[level])
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau_{2D}[level] & = reduce((1-\alpha)(1-\beta),\tau_{i0j0}[level], \\
          & \,              (\alpha)(1-\beta),\tau_{i1j0}[level], \\
          & \,              (1-\alpha)(\beta),\tau_{i0j1}[level], \\
          & \,              (\alpha)(\beta),\tau_{i1j1}[level])
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau_{1D}[level] & = reduce((1-\alpha),\tau_{i0}[level], \\
          & \,              (\alpha),\tau_{i1}[level])
\end{aligned}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau[level] &=
  \begin{cases}
     \tau_{3D}[level], & \text{for 3D image} \\
     \tau_{2D}[level], & \text{for 2D or Cube image} \\
     \tau_{1D}[level], & \text{for 1D image}
   \end{cases}
\end{aligned}
++++++++++++++++++++++++

The function [eq]#reduce()# is defined to operate on pairs of weights and
texel values as follows.
When using linear or anisotropic filtering, the values of multiple texels
are combined using a weighted average to produce a filtered texture value.
ifdef::VK_EXT_sampler_filter_minmax[]
However, a filtered texture value can: also be produced by computing
per-component minimum and maximum values over the set of texels that would
normally be averaged.
The slink:VkSamplerReductionModeCreateInfoEXT::pname:reductionMode controls
the process by which multiple texels are combined to produce a filtered
texture value.
When set to ename:VK_SAMPLER_REDUCTION_MODE_WEIGHTED_AVERAGE_EXT, a weighted
average is computed.
If the reduction mode is ename:VK_SAMPLER_REDUCTION_MODE_MIN_EXT or
ename:VK_SAMPLER_REDUCTION_MODE_MAX_EXT, [eq]#reduce()# computes a
component-wise minimum or maximum, respectively, of the components of the
set of provided texels with non-zero weights.
endif::VK_EXT_sampler_filter_minmax[]

ifdef::VK_IMG_filter_cubic[]
Within a mip level, ename:VK_FILTER_CUBIC_IMG filtering computes a weighted
average of 16 (for 2D), or 4 (for 1D) texel values, using the weights
computed during texel selection.

Catmull-Rom Spine interpolation of four points is defined by the equation:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
cinterp(\tau_0, \tau_1, \tau_2, \tau_3, \omega) =
\frac{1}{2}
\begin{bmatrix}1 & \omega & \omega^2 & \omega^3 \end{bmatrix}
\times
\begin{bmatrix}
 0 &  2 &  0 &  0 \\
-1 &  0 &  1 &  0 \\
 2 & -5 &  4 &  1 \\
-1 &  3 & -3 &  1
\end{bmatrix}
\times
\begin{bmatrix}
\tau_0 \\
\tau_1 \\
\tau_2 \\
\tau_3
\end{bmatrix}
\end{aligned}
++++++++++++++++++++++++

Using the values calculated in texel selection, this equation is applied to
the four points in 1D images.
For 2D images, the this equation is evaluated first for each row, and the
result is then fed back into the equation and interpolated again:

  :: [eq]#{tau}~1D~[level] = cinterp({tau}~i0~[level], {tau}~i1~[level],
     {tau}~i2~[level], {tau}~i3~[level], {alpha})#

  :: [eq]#{tau}~j0~[level] = cinterp({tau}~i0j0~[level], {tau}~i1j0~[level],
     {tau}~i2j0~[level], {tau}~i3j0~[level], {alpha})#
  :: [eq]#{tau}~j1~[level] = cinterp({tau}~i0j1~[level], {tau}~i1j1~[level],
     {tau}~i2j1~[level], {tau}~i3j1~[level], {alpha})#
  :: [eq]#{tau}~j2~[level] = cinterp({tau}~i0j2~[level], {tau}~i1j2~[level],
     {tau}~i2j2~[level], {tau}~i3j2~[level], {alpha})#
  :: [eq]#{tau}~j3~[level] = cinterp({tau}~i0j3~[level], {tau}~i1j3~[level],
     {tau}~i2j3~[level], {tau}~i3j3~[level], {alpha})#
  :: [eq]#{tau}~2D~[level] = cinterp({tau}~j0~[level], {tau}~j1~[level],
     {tau}~j2~[level], {tau}~j3~[level], {beta})#

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau[level] &=
  \begin{cases}
     \tau_{2D}[level], & \text{for 2D image} \\
     \tau_{1D}[level], & \text{for 1D image}
   \end{cases}
\end{aligned}
++++++++++++++++++++++++
endif::VK_IMG_filter_cubic[]

Finally, mipmap filtering either selects a value from one mip level or
computes a weighted average between neighboring mip levels:

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau &=
  \begin{cases}
    \tau[d], & \text{for mip mode BASE or NEAREST} \\
    reduce((1-\delta),\tau[d_{hi}],\delta,\tau[d_{lo}]), & \text{for mip mode LINEAR}
  \end{cases}
\end{aligned}
++++++++++++++++++++++++


[[textures-texel-anisotropic-filtering]]
=== Texel Anisotropic Filtering

Anisotropic filtering is enabled by the pname:anisotropyEnable in the
sampler.
When enabled, the image filtering scheme accounts for a degree of
anisotropy.

The particular scheme for anisotropic texture filtering is implementation
dependent.
Implementations should: consider the pname:magFilter, pname:minFilter and
pname:mipmapMode of the sampler to control the specifics of the anisotropic
filtering scheme used.
In addition, implementations should: consider pname:minLod and pname:maxLod
of the sampler.

The following describes one particular approach to implementing anisotropic
filtering for the 2D Image case, implementations may: choose other methods:

Given a pname:magFilter, pname:minFilter of ename:VK_FILTER_LINEAR and a
pname:mipmapMode of ename:VK_SAMPLER_MIPMAP_MODE_NEAREST:

Instead of a single isotropic sample, N isotropic samples are be sampled
within the image footprint of the image level [eq]#d# to approximate an
anisotropic filter.
The sum [eq]#{tau}~2Daniso~# is defined using the single isotropic
[eq]#{tau}~2D~(u,v)# at level [eq]#d#.

[latexmath]
++++++++++++++++++++++++
\begin{aligned}
\tau_{2Daniso} & =
     \frac{1}{N}\sum_{i=1}^{N}
     {\tau_{2D}\left (
       u \left ( x - \frac{1}{2} + \frac{i}{N+1} , y \right ),
         \left ( v \left (x-\frac{1}{2}+\frac{i}{N+1} \right ), y
\right )
     \right )},
     & \text{when}\  \rho_{x} > \rho_{y} \\
\tau_{2Daniso} &=
     \frac{1}{N}\sum_{i=1}^{N}
     {\tau_{2D}\left (
        u \left  ( x, y - \frac{1}{2} + \frac{i}{N+1} \right ),
          \left ( v \left (x,y-\frac{1}{2}+\frac{i}{N+1} \right )
\right )
     \right )},
     & \text{when}\  \rho_{y} \geq \rho_{x}
\end{aligned}
++++++++++++++++++++++++

ifdef::VK_EXT_sampler_filter_minmax[]

When slink:VkSamplerReductionModeCreateInfoEXT::pname:reductionMode is set
to ename:VK_SAMPLER_REDUCTION_MODE_WEIGHTED_AVERAGE_EXT, the above summation
is used.
If the reduction mode is ename:VK_SAMPLER_REDUCTION_MODE_MIN_EXT or
ename:VK_SAMPLER_REDUCTION_MODE_MAX_EXT, then the value is instead computed
as [eq]#\tau_{2Daniso} = reduce(\tau_1, ..., \tau_N)#, combining all texel
values with non-zero weights.

endif::VK_EXT_sampler_filter_minmax[]


ifdef::VK_NV_shader_image_footprint[]
[[textures-footprint]]
== Texel Footprint Evaluation

The SPIR-V instruction code:OpImageSampleFootprintNV evaluates the set of
texels from a single mip level that would be accessed during a
<<textures-texel-filtering, texel filtering>> operation.
In addition to the inputs that would be accepted by an equivalent
code:OpImageSample* instruction, code:OpImageSampleFootprintNV accepts two
additional inputs.
The code:Granularity input is an integer identifying the size of texel
groups used to evaluate the footprint.
Each bit in the returned footprint mask corresponds to an aligned block of
texels whose size is given by the following table:

.Texel footprint granularity values
[width="50%",options="header"]
|=====
| code:Granularity  | code:Dim = 2D |  code:Dim = 3D
|         0         |  unsupported  |  unsupported
|         1         |      2x2      |     2x2x2
|         2         |      4x2      |  unsupported
|         3         |      4x4      |     4x4x2
|         4         |      8x4      |  unsupported
|         5         |      8x8      |  unsupported
|         6         |     16x8      |  unsupported
|         7         |     16x16     |  unsupported
|         8         |  unsupported  |  unsupported
|         9         |  unsupported  |  unsupported
|         10        |  unsupported  |    16x16x16
|         11        |     64x64     |    32x16x16
|         12        |    128x64     |    32x32x16
|         13        |    128x128    |    32x32x32
|         14        |    256x128    |    64x32x32
|         15        |    256x256    |  unsupported
|=====

The code:Coarse input is used to select between the two mip levels that may:
be accessed during texel filtering when using a pname:mipmapMode of
ename:VK_SAMPLER_MIPMAP_MODE_LINEAR.
When filtering between two mip levels, a code:Coarse value of ename:true
requests the footprint in the lower-resolution mip level (higher level
number), while ename:false requests the footprint in the higher-resolution
mip level.
If texel filtering would access only a single mip level, the footprint in
that level would be returned when code:Coarse is set to ename:false; an
empty footprint would be returned when code:Coarse is set to ename:true.

The footprint for code:OpImageSampleFootprintNV is returned in a structure
with six members:

* The first member is a boolean value that is true if the texel filtering
  operation would access only a single mip level.
* The second member is a two- or three-component integer vector holding the
  footprint anchor location.
  For two-dimensional images, the returned components are in units of eight
  texel groups.
  For three-dimensional images, the returned components are in units of four
  texel groups.
* The third member is a two- or three-component integer vector holding a
  footprint offset relative to the anchor.
  All returned components are in units of texel groups.
* The fourth member is a two-component integer vector mask, which holds a
  bitfield identifying the set of texel groups in an 8x8 or 4x4x4
  neighborhood relative to the anchor and offset.
* The fifth member is an integer identifying the mip level containing the
  footprint identified by the anchor, offset, and mask.
* The sixth member is an integer identifying the granularity of the returned
  footprint.

For footprints in two-dimensional images (code:Dim2D), the mask returned by
code:OpImageSampleFootprintNV indicates whether each texel group in a 8x8
local neighborhood of texel groups would have one or more texels accessed
during texel filtering.
In the mask, the texel group with local group coordinates
latexmath:[(lgx,lgy)] is considered covered if and only if
[latexmath]
+++++++++++++++++++
\begin{aligned}
0 \neq ((mask.x + (mask.y << 32)) \text{ \& } (1 << (lgy \times 8 + lgx)))
\end{aligned}
+++++++++++++++++++
where

  * latexmath:[0<=lgx<8] and latexmath:[0<=lgy<8]; and
  * latexmath:[mask] is the returned two-component mask.

The local group with coordinates latexmath:[(lgx,lgy)] in the mask is
considered covered if and only if the texel filtering operation would access
one or more texels latexmath:[\tau_{ij}] in the returned miplevel where
[latexmath]
+++++++++++++++++++
\begin{aligned}
i0 & =
  \begin{cases}
    gran.x \times (8 \times anchor.x + lgx), & \text{if } lgx + offset.x < 8 \\
    gran.x \times (8 \times (anchor.x - 1) + lgx), & \text{otherwise}
  \end{cases} \\
i1 & = i0 + gran.x - 1 \\
j0 & =
  \begin{cases}
    gran.y \times (8 \times anchor.y + lgy), & \text{if } lgy + offset.y < 8 \\
    gran.y \times (8 \times (anchor.y - 1) + lgy), & otherwise
  \end{cases} \\
j1 & = j0 + gran.y - 1
\end{aligned}
+++++++++++++++++++
and

  * latexmath:[i0<=i<=i1] and latexmath:[j0<=j<=j1];
  * latexmath:[gran] is a two-component vector holding the width and height
    of the texel group identified by the granularity;
  * latexmath:[anchor] is the returned two-component anchor vector; and
  * latexmath:[offset] is the returned two-component offset vector.

For footprints in three-dimensional images (code:Dim3D), the mask returned
by code:OpImageSampleFootprintNV indicates whether each texel group in a
4x4x4 local neighborhood of texel groups would have one or more texels
accessed during texel filtering.
In the mask, the texel group with local group coordinates
latexmath:[(lgx,lgy,lgz)], is considered covered if and only if:
[latexmath]
+++++++++++++++++++
\begin{aligned}
0 \neq ((mask.x + (mask.y << 32)) \text{ \& } (1 << (lgz \times 16 + lgy \times 4 + lgx)))
\end{aligned}
+++++++++++++++++++
where

  * latexmath:[0<=lgx<4], latexmath:[0<=lgy<4], and latexmath:[0<=lgz<4];
    and
  * latexmath:[mask] is the returned two-component mask.

The local group with coordinates latexmath:[(lgx,lgy,lgz)] in the mask is
considered covered if and only if the texel filtering operation would access
one or more texels latexmath:[\tau_{ijk}] in the returned miplevel where
[latexmath]
+++++++++++++++++++
\begin{aligned}
i0 & =
  \begin{cases}
    gran.x \times (4 \times anchor.x + lgx), & \text{if } lgx + offset.x < 4 \\
    gran.x \times (4 \times (anchor.x - 1) + lgx), & \text{otherwise}
  \end{cases} \\
i1 & = i0 + gran.x - 1 \\
j0 & =
  \begin{cases}
    gran.y \times (4 \times anchor.y + lgy), & \text{if } lgy + offset.y < 4 \\
    gran.y \times (4 \times (anchor.y - 1) + lgy), & otherwise
  \end{cases} \\
j1 & = j0 + gran.y - 1 \\
k0 & =
  \begin{cases}
    gran.z \times (4 \times anchor.z + lgz), & \text{if } lgz + offset.z < 4 \\
    gran.z \times (4 \times (anchor.z - 1) + lgz), & otherwise
  \end{cases} \\
k1 & = k0 + gran.z - 1
\end{aligned}
+++++++++++++++++++
and

  * latexmath:[i0<=i<=i1], latexmath:[j0<=j<=j1], latexmath:[k0<=k<=k1];
  * latexmath:[gran] is a three-component vector holding the width, height,
    and depth of the texel group identified by the granularity;
  * latexmath:[anchor] is the returned three-component anchor vector; and
  * latexmath:[offset] is the returned three-component offset vector.

If the sampler used by code:OpImageSampleFootprintNV enables anisotropic
texel filtering via pname:anisotropyEnable, it is possible that the set of
texel groups accessed in a mip level may be too large to be expressed using
an 8x8 or 4x4x4 mask using the granularity requested in the instruction.
In this case, the implementation uses a texel group larger than the
requested granularity.
When a larger texel group size is used, code:OpImageSampleFootprintNV
returns an integer granularity value that can: be interpreted in the same
manner as the granularity value provided to the instruction to determine the
texel group size used.
If anisotropic texel filtering is disabled in the sampler, or if an
anisotropic footprint can be represented as an 8x8 or 4x4x4 mask with the
requested granularity, code:OpImageSampleFootprintNV will use the requested
granularity as-is and return a granularity value of zero.

code:OpImageSampleFootprintNV supports only two- and three-dimensional image
accesses (code:Dim2D and code:Dim3D) and the footprint returned is undefined
if a sampler uses an addressing mode other than
ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE.

endif::VK_NV_shader_image_footprint[]


[[textures-instructions]]
== Image Operation Steps

Each step described in this chapter is performed by a subset of the image
instructions:

  * Texel Input Validation Operations, Format Conversion, Texel Replacement,
    Conversion to RGBA, and Component Swizzle: Performed by all instructions
    except code:OpImageWrite.
  * Depth Comparison: Performed by code:OpImage*code:Dref instructions.
  * All Texel output operations: Performed by code:OpImageWrite.
  * Projection: Performed by all code:OpImage*code:Proj instructions.
  * Derivative Image Operations, Cube Map Operations, Scale Factor
    Operation, Level-of-Detail Operation and Image Level(s) Selection, and
    Texel Anisotropic Filtering: Performed by all code:OpImageSample* and
    code:OpImageSparseSample* instructions.
  * (s,t,r,q,a) to (u,v,w,a) Transformation, Wrapping, and (u,v,w,a) to
    (i,j,k,l,n) Transformation And Array Layer Selection: Performed by all
    code:OpImageSample, code:OpImageSparseSample, and
    code:OpImage*code:Gather instructions.
  * Texel Gathering: Performed by code:OpImage*code:Gather instructions.
ifdef::VK_NV_shader_image_footprint[]
  * Texel Footprint Evaluation: Performed by code:OpImageSampleFootprint
    instructions.
endif::VK_NV_shader_image_footprint[]
  * Texel Filtering: Performed by all code:OpImageSample* and
    code:OpImageSparseSample* instructions.
  * Sparse Residency: Performed by all code:OpImageSparse* instructions.