Vulkan-Docs/doc/specs/vulkan/chapters/textures.txt

// Copyright (c) 2015-2016 The Khronos Group Inc.
// Copyright notice at https://www.khronos.org/registry/speccopyright.html

[[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 level of detail
     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 level of detail
     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.
  * 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.


=== Texel Coordinate Systems

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

  * normalized texel coordinates (coordinates ranging from 0 to 1 span the
    image),
  * unnormalized texel coordinates (floating point coordinates ranging from
    0 to width/height/depth span the image), and
  * integer texel coordinates (integer coordinates ranging from 0 to
    width-1/height-1/depth-1 address the texels within the image).

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 latexmath:[$(s,t,r,q,a)$],
with the coordinates having the following meanings:

  * s: Coordinate in the first dimension of an image.
  * t: Coordinate in the second dimension of an image.
  * r: Coordinate in the third dimension of an image.
  ** (s,t,r) are interpreted as a direction vector for Cube images.
  * q: Fourth coordinate, for homogeneous (projective) coordinates.
  * 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 latexmath:[$(u,v,w,a)$],
with the coordinates having the following meanings:

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

Only the u and 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 (u [,v]), with 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 latexmath:[$(i,j,k,l,n)$], 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 (i, [,j],
[,k], [,l]), with j and 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]]
image::images/vulkantexture0.png[Title="Texel Coordinate Systems", align="left", scaledwidth="80%"]
The Texel Coordinate Systems - For the example shown of an 8x4 texel two dimensional image.

  * Normalized texel coordinates:
  ** The s coordinate goes from 0.0 to 1.0, left to right.
  ** The t coordinate goes from 0.0 to 1.0, top to bottom.
  * Unnormalized texel coordinates:
  ** The u coordinate goes from -1.0 to 9.0, left to right. The u coordinate
     within the range 0.0 to 8.0 is within the image, otherwise it is within
     the border.
  ** The v coordinate goes from -1.0 to 5.0, top to bottom. The v coordinate
     within the range 0.0 to 4.0 is within the image, otherwise it is within
     the border.
  * Integer texel coordinates:
  ** The i coordinate goes from -1 to 8, left to right. The i coordinate
     within the range 0 to 7 addresses texels within the image, otherwise it
     addresses a border texel.
  ** The j coordinate goes from -1 to 5, top to bottom. The 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 (u,v), the four texels selected are
     i0j0, i1j0, i0j1 and i1j1.
  ** The weights latexmath:[$\alpha$] and latexmath:[$\beta$].
  ** Given the offset latexmath:[$\Delta_{i}$] and latexmath:[$\Delta_{j}$],
     the four texels selected by the offset are i0j0', i1j0', i0j1' and
     i1j1'.

image::images/vulkantexture1.png[Title="Texel Coordinate Systems", align="left", scaledwidth="80%"]

The Texel Coordinate Systems - For the example shown of an 8x4 texel two
dimensional image.

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


== 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 latexmath:[$(red, green, blue)$] is transformed to a shared
exponent color latexmath:[$(red_{shared}, green_{shared}, blue_{shared},
exp_{shared})$] as follows:

First, the components latexmath:[$(red, green, blue)$] are clamped to
latexmath:[$(red_{clamped}, green_{clamped}, blue_{clamped})$] as:
[latexmath]
+++++++++++++++++++
\begin{align*}
red_{clamped}   & = \max(0,min(sharedexp_{max},red)) \\
green_{clamped} & = \max(0,min(sharedexp_{max},green)) \\
blue_{clamped}  & = \max(0,min(sharedexp_{max},blue))
\end{align*}
+++++++++++++++++++

Where:

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

[NOTE]
.Note
==================
latexmath:[$NaN$], if supported, is handled as in IEEE 754-2008 minNum() and
maxNum(). That is the result is a latexmath:[$NaN$] is mapped to zero.
==================

The largest clamped component, latexmath:[$max_{clamped}$] is determined:

[latexmath]
+++++++++++++++++++
\begin{align*}
max_{clamped} = \max(red_{clamped},green_{clamped},blue_{clamped})
\end{align*}
+++++++++++++++++++

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

The shared exponent latexmath:[$exp_{shared}$] is computed:

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

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

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

[latexmath]
+++++++++++++++++++
\begin{align*}
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{align*}
+++++++++++++++++++


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

A shared exponent color latexmath:[$(red_{shared}, green_{shared},
blue_{shared}, exp_{shared})$] is transformed to an RGB color
latexmath:[$(red, green, blue)$] as follows:

[latexmath]
+++++++++++++++++++
\begin{align*}
red   & = red_{shared}\times 2^{(exp_{shared}-B-N)} \\
green & = green_{shared}\times 2^{(exp_{shared}-B-N)} \\
blue  & = blue_{shared}\times 2^{(exp_{shared}-B-N)} \\
\end{align*}
+++++++++++++++++++

Where:

[latexmath]
+++++++++++++++++++
\begin{align*}
N               & = 9  & \textrm{number of mantissa bits per component} \\
B               & = 15 & \textrm{exponent bias}
\end{align*}
+++++++++++++++++++


== 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-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>>

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.


[[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 Validation

There are a number of cases where a SPIR-V instruction can: mismatch with
the sampler, the image, or both. There are a number of cases where the
sampler can: mismatch with the image. 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
    pname:format is not one of the elink:VkFormat integer types or a stencil
    aspect of a depth/stencil format.
  * The sampler pname:borderColor is a float type and the image pname:format
    is not one of the elink:VkFormat float types or a depth aspect 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
    elink:VkComponentSwizzle for any of the slink:VkComponentMapping
    components is not ename:VK_COMPONENT_SWIZZLE_IDENTITY.
  * 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 pname:format is not one of the depth/stencil
    formats with a depth component, or the image 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's pname:samples is not equal to
     ename:VK_SAMPLE_COUNT_1_BIT, the instruction must: have code:MS = 1.


[[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 satisfy any of the conditions
[latexmath]
+++++++++++++++++++
\begin{align*}
i & < 0 & i \geq w_{s} \\
j & < 0 & j \geq h_{s} \\
k & < 0 & k \geq d_{s} \\
l & < 0 & l \geq layers \\
n & < 0 & n \geq samples
\end{align*}
+++++++++++++++++++
where:
[latexmath]
+++++++++++++++++++
\begin{align*}
& w_{s} & = \textrm{width of the image level} \\
& h_{s} & = \textrm{height of the image level} \\
& d_{s} & = \textrm{depth of the image level} \\
& layers & = \textrm{number of layers in the image} \\
& samples & = \textrm{number of samples per texel in the image}
\end{align*}
+++++++++++++++++++

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 miplevel, 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 latexmath:[$i$] or only
   latexmath:[$j$]
+
then the texel lies along an edge, so the coordinates latexmath:[$(i,j)$]
and the array layer latexmath:[$l$] 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 latexmath:[$i$] and
   latexmath:[$j$]
+
then the texel lies at the 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>>.


[[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>>.

If the image 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 ``KHR_DF_TRANSFER_SRGB'' section of the
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, or
  * 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 latexmath:[$B$]
[options="header",cols="60%,40%"]
|====
| Sampler pname:borderColor                     | Corresponding Border Color
| ename:VK_BORDER_COLOR_FLOAT_TRANSPARENT_BLACK | latexmath:[$B = (0.0, 0.0, 0.0, 0.0)$]
| ename:VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK      | latexmath:[$B = (0.0, 0.0, 0.0, 1.0)$]
| ename:VK_BORDER_COLOR_FLOAT_OPAQUE_WHITE      | latexmath:[$B = (1.0, 1.0, 1.0, 1.0)$]
| ename:VK_BORDER_COLOR_INT_TRANSPARENT_BLACK   | latexmath:[$B = (0, 0, 0, 0)$]
| ename:VK_BORDER_COLOR_INT_OPAQUE_BLACK        | latexmath:[$B = (0, 0, 0, 1)$]
| ename:VK_BORDER_COLOR_INT_OPAQUE_WHITE        | latexmath:[$B = (1, 1, 1, 1)$]
|====

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                | latexmath:[$D        = (B_{r})$]
| Stencil aspect              | latexmath:[$S        = (B_{r})$]
| One component color format  | latexmath:[$C_{r}    = (B_{r})$]
| Two component color format  | latexmath:[$C_{rg}   = (B_{r},B_{g})$]
| Three component color format| latexmath:[$C_{rgb}  = (B_{r},B_{g},B_{b})$]
| Four component color format | latexmath:[$C_{rgba} = (B_{r},B_{g},B_{b},B_{a})$]
|====

If the read operation is from a buffer resource, and the
pname:robustBufferAccess feature is enabled, an invalid texel is replaced as
described <<features-features-robustBufferAccess,here>>.

If the pname:robustBufferAccess feature is not enabled, the value of an
invalid texel is undefined.

ifdef::editing-notes[]
[NOTE]
.editing-note
==================
(Bill) This is not currently catching this significant case.

For opImageFetch, which fetches from an *image* not a buffer, the
result is defined if pname:robustBufferAccess is enabled.
==================
endif::editing-notes[]

If the sname:VkPhysicalDeviceSparseProperties property
pname:residencyNonResidentStrict is 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 false, the read must: be safe, but
the value of the sparse unbound texel is undefined.


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

If the image view's format is depth and the operation is a code:Dref
instruction, a depth comparison is performed. The initial value of the
result latexmath:[$r$] is latexmath:[$0.0$], which is replaced with
latexmath:[$1.0$] if the result of the compare operation is
latexmath:[$true$]. The compare operation is selected by the pname:compareOp
member of the sampler.

[latexmath]
+++++++++++++++++++
\begin{align*}
r & = 0.0 & \textrm{initial value} \\
r & = 1.0
  \begin{cases}
    D_{ref} \leq D_{t} & \textrm{for LEQUAL}   \\
    D_{ref} \geq D_{t} & \textrm{for GEQUAL}   \\
    D_{ref} < D_{t}    & \textrm{for LESS}     \\
    D_{ref} > D_{t}    & \textrm{for GREATER}  \\
    D_{ref} = D_{t}    & \textrm{for EQUAL}    \\
    D_{ref} \neq D_{t} & \textrm{for NOTEQUAL} \\
    true               & \textrm{for ALWAYS}   \\
    false              & \textrm{for NEVER}
  \end{cases}
\end{align*}
+++++++++++++++++++

where:

[latexmath]
+++++++++++++++++++
\begin{align*}
& D_{ref} = shaderOp.D_{ref} & \textrm{(from optional SPIR-V operand)} \\
& D_{t}                      & \textrm{texel depth value}
\end{align*}
+++++++++++++++++++


[[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                | latexmath:[$C_{rgba} = (D,0,0,one)$]
| Stencil aspect              | latexmath:[$C_{rgba} = (S,0,0,one)$]
| One component color format  | latexmath:[$C_{rgba} = (C_{r},0,0,one)$]
| Two component color format  | latexmath:[$C_{rgba} = (C_{rg},0,one)$]
| Three component color format| latexmath:[$C_{rgba} = (C_{rgb},one)$]
| Four component color format | latexmath:[$C_{rgba} = C_{rgba}$]
|====

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


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

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

where:

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

For each component this is applied to, the
ename:VK_COMPONENT_SWIZZLE_IDENTITY swizzle selects the corresponding
component from latexmath:[$C_{rgba}$].

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 struct 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.


== 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 pname: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 mustnot: 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
sname:VkPhysicalDeviceSparseProperties property
pname:residencyNonResidentStrict is ename:VK_TRUE, the sparse unbound texel
write has no effect. If pname:residencyNonResidentStrict is ename:VK_FALSE,
the effect of the write is undefined but must: be safe. In addition, the
write may: have a side effect that is visible to other image instructions,
but mustnot: be written to any device memory allocation.


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

Texels 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), using the appropriate equations in
<<fundamentals-fp16,16-Bit Floating-Point Numbers>> and
<<fundamentals-fixedconf,Fixed-Point Data Conversion>>.


== 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 a fragment shader.

image::images/vulkantexture2.png[Title="Implicit derivatives",align="left", scaledwidth="50%"]

Derivatives are computed as if there is a 2x2 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{align*}
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{align*}
+++++++++++++++++++

The code:Fine derivative instructions must: return the values above, for a
group of fragments in a 2x2 neighborhood. Coarse derivatives may: return
only two values. In this case, the values should: be:

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

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.


[[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
latexmath:[$(s,t,r,q,a)$] and (if present) the latexmath:[$D_{ref}$]
coordinate are transformed as follows:

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


=== Derivative Image Operations

Derivatives are used for level-of-detail 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{align*}
\partial{s}/\partial{x} & = dPdx(s), & \partial{s}/\partial{y} & = dPdy(s), & \textrm{for 1D, 2D, Cube, or 3D image} \\
\partial{t}/\partial{x} & = dPdx(t), & \partial{t}/\partial{y} & = dPdy(t), & \textrm{for 2D, Cube, or 3D image} \\
\partial{u}/\partial{x} & = dPdx(u), & \partial{u}/\partial{y} & = dPdy(u), & \textrm{for Cube or 3D image}
\end{align*}
+++++++++++++++++++

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

For explicit level-of-detail image instructions, if the optional: SPIR-V
operand latexmath:[$Grad$] 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 latexmath:[$Lod$] 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 latexmath:[$\lambda_{base}$]
as described in <<textures-level-of-detail-operation,Level-of-Detail
Operation>>.


=== Cube Map Face Selection and Transformations

For cube map image instructions, the latexmath:[$(s,t,r)$] coordinates are
treated as a direction vector latexmath:[$(r_{x},r_{y},r_{z})$]. The
direction vector is used to select a cube map face. The direction vector is
transformed to a per-face texel coordinate system
latexmath:[$(s_{face},t_{face})$]. 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 latexmath:[$r_{z}$] wins over
latexmath:[$r_{y}$] and latexmath:[$r_{x}$], and the second rule that
latexmath:[$r_{y}$] wins over latexmath:[$r_{x}$]. An implementation may:
choose other rules, but the rules must: be deterministic and depend only on
latexmath:[$(r_{x},r_{y},r_{z})$].

The layer number (corresponding to a cube map face), the coordinate
selections for latexmath:[$s_{c}$], latexmath:[$t_{c}$],
latexmath:[$r_{c}$], 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|latexmath:[$s_{c}$]|latexmath:[$t_{c}$]|latexmath:[$r_{c}$]

|latexmath:[$+r_{x}$]|latexmath:[$0$]|latexmath:[$Positive X$]|latexmath:[$-r_{z}$]|latexmath:[$-r_{y}$]|latexmath:[$r_{x}$]

|latexmath:[$-r_{x}$]|latexmath:[$1$]|latexmath:[$Negative X$]|latexmath:[$+r_{z}$]|latexmath:[$-r_{y}$]|latexmath:[$r_{x}$]

|latexmath:[$+r_{y}$]|latexmath:[$2$]|latexmath:[$Positive Y$]|latexmath:[$+r_{x}$]|latexmath:[$+r_{z}$]|latexmath:[$r_{y}$]

|latexmath:[$-r_{y}$]|latexmath:[$3$]|latexmath:[$Negative Y$]|latexmath:[$+r_{x}$]|latexmath:[$-r_{z}$]|latexmath:[$r_{y}$]

|latexmath:[$+r_{z}$]|latexmath:[$4$]|latexmath:[$Positive Z$]|latexmath:[$+r_{x}$]|latexmath:[$-r_{y}$]|latexmath:[$r_{z}$]

|latexmath:[$-r_{z}$]|latexmath:[$5$]|latexmath:[$Negative Z$]|latexmath:[$-r_{x}$]|latexmath:[$-r_{y}$]|latexmath:[$r_{z}$]

|======================


.Cube map derivative selection
[width="75%",frame="all",options="header"]
|======================
|Major Axis Direction|latexmath:[$\partial{s_{c}}/\partial{x}$]|latexmath:[$\partial{s_{c}}/\partial{y}$]|latexmath:[$\partial{t_{c}}/\partial{x}$]|latexmath:[$\partial{t_{c}}/\partial{y}$]|latexmath:[$\partial{r_{c}}/\partial{x}$]|latexmath:[$\partial{r_{c}}/\partial{y}$]

|latexmath:[$+r_{x}$]
|latexmath:[$-\partial{r_{z}}/\partial{x}$]|latexmath:[$-\partial{r_{z}}/\partial{y}$]
|latexmath:[$-\partial{r_{y}}/\partial{x}$]|latexmath:[$-\partial{r_{y}}/\partial{y}$]
|latexmath:[$+\partial{r_{x}}/\partial{x}$]|latexmath:[$+\partial{r_{x}}/\partial{y}$]

|latexmath:[$-r_{x}$]
|latexmath:[$+\partial{r_{z}}/\partial{x}$]|latexmath:[$+\partial{r_{z}}/\partial{y}$]
|latexmath:[$-\partial{r_{y}}/\partial{x}$]|latexmath:[$-\partial{r_{y}}/\partial{y}$]
|latexmath:[$-\partial{r_{x}}/\partial{x}$]|latexmath:[$-\partial{r_{x}}/\partial{y}$]

|latexmath:[$+r_{y}$]
|latexmath:[$+\partial{r_{x}}/\partial{x}$]|latexmath:[$+\partial{r_{x}}/\partial{y}$]
|latexmath:[$+\partial{r_{z}}/\partial{x}$]|latexmath:[$+\partial{r_{z}}/\partial{y}$]
|latexmath:[$+\partial{r_{y}}/\partial{x}$]|latexmath:[$+\partial{r_{y}}/\partial{y}$]

|latexmath:[$-r_{y}$]
|latexmath:[$+\partial{r_{x}}/\partial{x}$]|latexmath:[$+\partial{r_{x}}/\partial{y}$]
|latexmath:[$-\partial{r_{z}}/\partial{x}$]|latexmath:[$-\partial{r_{z}}/\partial{y}$]
|latexmath:[$-\partial{r_{y}}/\partial{x}$]|latexmath:[$-\partial{r_{y}}/\partial{y}$]

|latexmath:[$+r_{z}$]
|latexmath:[$+\partial{r_{x}}/\partial{x}$]|latexmath:[$+\partial{r_{x}}/\partial{y}$]
|latexmath:[$-\partial{r_{y}}/\partial{x}$]|latexmath:[$-\partial{r_{y}}/\partial{y}$]
|latexmath:[$+\partial{r_{z}}/\partial{x}$]|latexmath:[$+\partial{r_{z}}/\partial{y}$]

|latexmath:[$-r_{z}$]
|latexmath:[$-\partial{r_{x}}/\partial{x}$]|latexmath:[$-\partial{r_{x}}/\partial{y}$]
|latexmath:[$-\partial{r_{y}}/\partial{x}$]|latexmath:[$-\partial{r_{y}}/\partial{y}$]
|latexmath:[$-\partial{r_{z}}/\partial{x}$]|latexmath:[$-\partial{r_{z}}/\partial{y}$]
|======================

=== Cube Map Coordinate Transformation

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


=== Cube Map Derivative Transformation

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\frac{\partial{s_{face}}}{\partial{x}} &=
    \frac{\partial}{\partial{x}} \left ( \frac{1}{2} \times \frac{s_{c}}{|r_{c}|}
    + \frac{1}{2}\right ) \\
\frac{\partial{s_{face}}}{\partial{x}} &=
    \frac{1}{2} \times \frac{\partial}{\partial{x}}
    \left ( \frac{s_{c}}{|r_{c}|}  \right ) \\
\frac{\partial{s_{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{align*}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\frac{\partial{s_{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_{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_{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{align*}
++++++++++++++++++++++++

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 latexmath:[$s_{face}$] and latexmath:[$t_{face}$] 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

Level-of-detail selection can: be either explicit (provided explicitly by
the image instruction) or implicit (determined from a scale factor
calculated from the derivatives).


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

The magnitude of the derivatives are calculated by:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
m_{ux} & = \left | \partial s / \partial x \right | \times w_{base} \\
m_{vx} & = \left | \partial t / \partial x \right | \times h_{base} \\
m_{wx} & = \left | \partial r / \partial x \right | \times d_{base} \\
\\
m_{uy} & = \left | \partial s / \partial y \right | \times w_{base} \\
m_{vy} & = \left | \partial t / \partial y \right | \times h_{base} \\
m_{wy} & = \left | \partial r / \partial y \right | \times d_{base}
\end{align*}
++++++++++++++++++++++++


where:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\partial t / \partial x & = \partial t / \partial y = 0 & \textrm{(for 1D image)} \\
\partial r / \partial x & = \partial r / \partial y = 0 & \textrm{(for 1D, 2D or Cube image)} \\
\\
w_{base}                & = image.w \\
h_{base}                & = image.h \\
d_{base}                & = image.d \\
                        & \textrm{of the } baseMipLevel & \textrm{(from image descriptor)}
\end{align*}
++++++++++++++++++++++++

The _scale factors_ latexmath:[$(\rho_{x}, \rho{y})$] should: be calculated
by:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\rho_{x} & = \sqrt{ m_{ux} ^{2} + m_{vx} ^{2} + m_{wx} ^{2} } \\
\rho_{y} & = \sqrt{ m_{uy} ^{2} + m_{vy} ^{2} + m_{wy} ^{2} }
\end{align*}
++++++++++++++++++++++++

The ideal functions latexmath:[$\rho_{x}$] and latexmath:[$\rho_{y}$] may:
be approximated with functions latexmath:[$f_x$] and latexmath:[$f_y$],
subject to the following constraints:

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

\begin{align*}
\max \left (
    \left | m_{ux} \right | ,
    \left | m_{vx} \right | ,
    \left | m_{wx} \right |
    \right ) \leq
& f_x \leq
    \left | m_{ux} \right | +
    \left | m_{vx} \right | +
    \left | m_{wx} \right | \\
\max \left (
    \left | m_{uy} \right | ,
    \left | m_{vy} \right | ,
    \left | m_{wy} \right |
    \right ) \leq
& f_y \leq
    \left | m_{uy} \right | +
    \left | m_{vy} \right | +
    \left | m_{wy} \right |
\end{align*}
++++++++++++++++++++++++

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's no reason to have conditional application of
latexmath:[$w_{base} , h_{base} , d_{base}$] for rectangle textures either,
since they don't support implicit derivatives.
==================
endif::editing-notes[]


The minimum and maximum scale factors latexmath:[$(\rho_{min},\rho_{max})$]
are determined by:
[latexmath]
++++++++++++++++++++++++
\begin{align*}
\rho_{max} & = \max( \rho_{x} , \rho_{y} ) \\
\rho_{min} & = \min( \rho_{x} , \rho_{y} )
\end{align*}
++++++++++++++++++++++++

The sampling rate is determined by:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
N & = \min \left (\left \lceil \frac{\rho_{max}}{\rho_{min}}  \right \rceil ,max_{Aniso} \right )
\end{align*}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
sampler.max_{Aniso} & = maxAnisotropy & \textrm{(from sampler descriptor)} \\
limits.max_{Aniso}  & = maxSamplerAnisotropy & \textrm{(from physical device limits)} \\
max_{Aniso}         & = \min \left ( sampler.max_{Aniso}, limits.max_{Aniso} \right )
\end{align*}
++++++++++++++++++++++++

If latexmath:[$\rho_{max} = \rho_{min} = 0$], then all the partial
derivatives are zero, the fragment's footprint in texel space is a point,
and latexmath:[$N$] should: be treated as 1. If
latexmath:[$\rho_{max} \neq 0 \textrm{ and } \rho_{min} = 0$] then all
partial derivatives along one axis are zero, the fragment's footprint in
texel space is a line segment, and latexmath:[$N$] should: be treated as
latexmath:[$max_{Aniso}$]. However, anytime the footprint is small in texel
space the implementation may: use a smaller value of latexmath:[$N$], even
when latexmath:[$\rho_{min}$] is zero or close to zero.

An implementation may: round latexmath:[$N$] up to the nearest supported
sampling rate.

If latexmath:[$N=1$], sampling is isotropic. If latexmath:[$N>1$], sampling
is anistropic.


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

The _level-of-detail_ parameter latexmath:[$\lambda$] is computed as
follows:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\lambda_{base}(x,y) & =
  \begin{cases}
    shaderOp.Lod                               & \textrm{(from optional SPIR-V operand)} \\
    \log_2 \left ( \frac{\rho_{max}}{N} \right ) & \textrm{otherwise}
  \end{cases} \\
\lambda'(x,y)       & = \lambda_{base} + \operatorname{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} \\
    undefined, & lod_{min} > lod_{max} \\
  \end{cases}
\end{align*}
++++++++++++++++++++++++

where:

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

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


==== Image Level(s) Selection

The image level(s) latexmath:[$d, d_{hi},\textrm{ and }d_{lo}$] which texels
are read from are selected based on the level-of-detail parameter, as
follows. If the sampler's pname:mipmapMode is
ename:VK_SAMPLER_MIPMAP_MODE_NEAREST, then level d is used:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
d =
  \begin{cases}
    level_{base},     & \lambda \leq \frac{1}{2} \\
    nearest(\lambda), & \lambda > \frac{1}{2},
                        level_{base} + \lambda \leq
                        q + \frac{1}{2} \\
    q,                & \lambda > \frac{1}{2}, level_{base} + \lambda > q + \frac{1}{2}
  \end{cases}
\end{align*}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
nearest(\lambda) & =
  \begin{cases}
    \left \lceil level_{base}+\lambda + \frac{1}{2}\right \rceil - 1, &
        \textrm{preferred} \\
    \left \lfloor level_{base}+\lambda + \frac{1}{2}\right \rfloor,   &
        \textrm{alternative}
  \end{cases}
\end{align*}
++++++++++++++++++++++++

and where q is the pname:levelCount from the pname:subresourceRange of the
image view.

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

[latexmath]
++++++++++++++++++++++++
\begin{align*}
d_{hi} & =
  \begin{cases}
    q,                                                 & level_{base} + \lambda \geq q \\
    \left \lfloor level_{base}+\lambda \right \rfloor, & \textrm{otherwise}
  \end{cases} \\
d_{lo} & =
  \begin{cases}
    q,        & level_{base} + \lambda \geq q \\
    d_{hi}+1, & \textrm{otherwise}
  \end{cases}
\end{align*}
++++++++++++++++++++++++

latexmath:[$\delta$] is the fractional value used for linear filtering
between levels.

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\delta & = \operatorname{frac}(\lambda)
\end{align*}
++++++++++++++++++++++++

[[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 (latexmath:[$d\textrm{ or }d_{hi}\textrm{ and }d_{lo}$]) used in
<<textures-texel-filtering,filtering>>.

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

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 l is computed as:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
l & = \operatorname{clamp}( \operatorname{RNE}(a), 0, layerCount - 1 ) + baseArrayLayer
\end{align*}
++++++++++++++++++++++++

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

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\operatorname{RNE}(a) & =
  \begin{cases}
    \operatorname{roundTiesToEven}(a)         & \textrm{preferred, from IEEE Std 754-2008 Floating-Point Arithmetic} \\
    \left \lfloor a + \frac{1}{2} \right \rfloor & \textrm{alternative}
  \end{cases}
\end{align*}
++++++++++++++++++++++++

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{align*}
i & = \left \lfloor u \right \rfloor \\
j & = \left \lfloor v \right \rfloor \\
k & = \left \lfloor w \right \rfloor
\end{align*}
++++++++++++++++++++++++

Linear filtering (ename:VK_FILTER_LINEAR) computes a set of neighboring
coordinates which bound the unnormalized coordinates. The integer texel
coordinates are combinations of
latexmath:[$i_0\textrm{ or }i_1,j_0\textrm{ or }j_1,k_0\textrm{ or }k_1$],
as well as weights latexmath:[$\alpha, \beta, and \gamma$].

[latexmath]
++++++++++++++++++++++++
\begin{align*}
i_{0}  & = \left \lfloor u - \frac{1}{2} \right \rfloor & i_{1} & = i_{0} + 1 \\
j_{0}  & = \left \lfloor v - \frac{1}{2} \right \rfloor & j_{1} & = j_{0} + 1 \\
k_{0}  & = \left \lfloor w - \frac{1}{2} \right \rfloor & k_{1} & = k_{0} + 1 \\
\\
\alpha & = \operatorname{frac} \left ( u - \frac{1}{2} \right ) \\
\beta  & = \operatorname{frac} \left ( v - \frac{1}{2} \right ) \\
\gamma & = \operatorname{frac} \left ( w - \frac{1}{2} \right )
\end{align*}
++++++++++++++++++++++++

If the image instruction includes a latexmath:[$ConstOffset$] operand, the
constant offsets latexmath:[$(\Delta_{i},\Delta_{j},\Delta_{k})$] are added
to latexmath:[$(i,j,k)$] components of the integer texel coordinates.


== 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 miplevel then
ename:VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE is used, and if
ename:VK_FILTER_LINEAR is used within a miplevel 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{align*}
i &=
  \begin{cases}
    i \operatorname{mod} size                                & \textrm{for repeat} \\
    (size-1) - \operatorname{mirror}((i \operatorname{mod} (2 \times size)) - size) & \textrm{for mirrored repeat} \\
    \operatorname{clamp}(i,0,size-1)                         & \textrm{for clamp to edge} \\
    \operatorname{clamp}(i,-1,size)                          & \textrm{for clamp to border} \\
    \operatorname{clamp}(\operatorname{mirror}(i),0,size-1)  & \textrm{for mirror clamp to edge}
  \end{cases}
\end{align*}
++++++++++++++++++++++++

where:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
&\operatorname{mirror}(n) =
  \begin{cases}
    n & \textrm{for }n \geq 0 \\
    -(1+n) &\textrm{otherwise} \\
  \end{cases}
\end{align*}
++++++++++++++++++++++++

latexmath:[$j$] (for 2D and Cube image) and latexmath:[$k$] (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 2x2 rectangular region of texels in the base level of the image view.
The rules for the etext: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{align*}
\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{align*}
++++++++++++++++++++++++

where:

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


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

If latexmath:[$\lambda$] 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 latexmath:[$\lambda$] 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 miplevel, etext:NEAREST filtering selects a single value using the
latexmath:[$(i,j,k)$] texel coordinates, with all texels taken from layer l.

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

Within a miplevel, etext:LINEAR filtering computes a weighted average of 8
(for 3D), 4 (for 2D or Cube), or 2 (for 1D) texel values, using the weights
computed earlier:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\tau_{3D}[level] & = (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{align*}
++++++++++++++++++++++++

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\tau_{2D}[level] & = (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{align*}
++++++++++++++++++++++++

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

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

Finally, mipmap filtering either selects a value from one miplevel or
computes a weighted average between neighboring miplevels:

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\tau &=
  \begin{cases}
    \tau[d], &\textrm{for mipmode BASE or NEAREST} \\
    (1-\delta)\tau[d_{hi}]+\delta\tau[d_{lo}], &\textrm{for mipmode LINEAR}
  \end{cases}
\end{align*}
++++++++++++++++++++++++


[[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 etext:LINEAR and a
pname:mipmapMode of etext:NEAREST,

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

[latexmath]
++++++++++++++++++++++++
\begin{align*}
\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 )},
     &\textrm{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 )},
     &\textrm{when } \rho_{y} \geq \rho_{x}
\end{align*}
++++++++++++++++++++++++

ifdef::editing-notes[]
[NOTE]
.editing-note
==================
(Bill) EXT_texture_filter_anisotropic has not been updated since 2000,
except for ES extension number (2007) and a minor speeling (sic) correction
(2014), neither of which are functional changes. It is showing its age.

In particular, there's an open issue about 3D textures.

There are no interactions with ARB_texture_cube_map (approved 1999, promoted
to core OpenGL 1.3 in 2001), let alone interactions with
ARB_seamless_cube_map (approved and promoted to core OpenGL 3.2 in 2009).

There are no interactions with texture offsets or texture gather.
==================
endif::editing-notes[]


[[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.
  * Texel Filtering: Performed by all code:OpImageSample* and
    code:OpImageSparseSample* instructions.
  * Sparse Residency: Performed by all code:OpImageSparse* instructions.