15. Image Operations
15.1. Image Operations Overview
Vulkan Image Operations are operations performed by those SPIRV Image
Instructions which take an OpTypeImage
(representing a
VkImageView
) or OpTypeSampledImage
(representing a
(VkImageView
, VkSampler
) pair) and texel coordinates as
operands, and return a value based on one or more neighboring texture
elements (texels) in the image.
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. 
Image Operations include the functionality of the following SPIRV Image Instructions:

OpImageSample
* andOpImageSparseSample
* read one or more neighboring texels of the image, and filter the texel values based on the state of the sampler.
Instructions with
ImplicitLod
in the name determine the LOD used in the sampling operation based on the coordinates used in neighboring fragments. 
Instructions with
ExplicitLod
in the name determine the LOD used in the sampling operation based on additional coordinates. 
Instructions with
Proj
in the name apply homogeneous projection to the coordinates.


OpImageFetch
andOpImageSparseFetch
return a single texel of the image. No sampler is used. 
OpImage
*Gather
andOpImageSparse
*Gather
read neighboring texels and return a single component of each. 
OpImageRead
(andOpImageSparseRead
) andOpImageWrite
read and write, respectively, a texel in the image. No sampler is used. 
Instructions with
Dref
in the name apply depth comparison on the texel values. 
Instructions with
Sparse
in the name additionally return a sparse residency code.
15.1.1. Texel Coordinate Systems
Images are addressed by texel coordinates. There are three texel coordinate systems:

normalized texel coordinates [0.0, 1.0]

unnormalized texel coordinates [0.0, width / height / depth)

integer texel coordinates [0, width / height / depth)
SPIRV OpImageFetch
, OpImageSparseFetch
, OpImageRead
,
OpImageSparseRead
, and OpImageWrite
instructions use integer texel
coordinates.
Other image instructions can use either normalized or unnormalized texel
coordinates (selected by the unnormalizedCoordinates
state of the
sampler used in the instruction), but there are
limitations on what operations, image
state, and sampler state is supported.
Normalized coordinates are logically
converted to unnormalized as part of
image operations, and 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 (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 SPIRV operand based on the
dimensionality of the image variable and type of instruction.
For Proj
instructions, the components are in order (s [,t] [,r]
q), with t and r being conditionally present based on the
Dim
of the image.
For nonProj
instructions, the coordinates are (s [,t] [,r]
[,a]), with t and r being conditionally present based on the
Dim
of the image and a being conditionally present based on the
Arrayed
property of the image.
Projective image instructions are not supported on Arrayed
images.
Unnormalized texel coordinates are referred to as (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
SPIRV operand, because only 1D and 2D (nonArrayed
) 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 (i,j,k,l,n), with the coordinates having the following meanings:

i: Coordinate in the first dimension of an image.

j: Coordinate in the second dimension of an image.

k: Coordinate in the third dimension of an image.

l: Coordinate for array layer.

n: Coordinate for the sample index.
They are extracted from the SPIRV operand in order (i, [,j], [,k],
[,l]), with j and k conditionally present based on the Dim
of the image, and l conditionally present based on the Arrayed
property of the image.
n is conditionally present and is taken from the Sample
image
operand.
For all coordinate types, unused coordinates are assigned a value of zero.
The Texel Coordinate Systems  For the example shown of an 8×4 texel two dimensional image.

Normalized texel coordinates:

The s coordinate goes from 0.0 to 1.0.

The t coordinate goes from 0.0 to 1.0.


Unnormalized texel coordinates:

The u coordinate within the range 0.0 to 8.0 is within the image, otherwise it is outside the image.

The v coordinate within the range 0.0 to 4.0 is within the image, otherwise it is outside the image.


Integer texel coordinates:

The i coordinate within the range 0 to 7 addresses texels within the image, otherwise it is outside the image.

The j coordinate within the range 0 to 3 addresses texels within the image, otherwise it outside the image.


Also shown for linear filtering:

Given the unnormalized coordinates (u,v), the four texels selected are i_{0}j_{0}, i_{1}j_{0}, i_{0}j_{1}, and i_{1}j_{1}.

The fractions α and β.

Given the offset Δ_{i} and Δ_{j}, the four texels selected by the offset are i_{0}j'_{0}, i_{1}j'_{0}, i_{0}j'_{1}, and i_{1}j'_{1}.

The Texel Coordinate Systems  For the example shown of an 8×4 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 Δ_{i} and Δ_{j}, the texel selected by the offset is ij'.

15.2. Conversion Formulas
15.2.1. RGB to Shared Exponent Conversion
An RGB color (red, green, blue) is transformed to a shared exponent color (red_{shared}, green_{shared}, blue_{shared}, exp_{shared}) as follows:
First, the components (red, green, blue) are clamped to (red_{clamped}, green_{clamped}, blue_{clamped}) as:

red_{clamped} = max(0, min(sharedexp_{max}, red))

green_{clamped} = max(0, min(sharedexp_{max}, green))

blue_{clamped} = max(0, min(sharedexp_{max}, blue))
where:
Note
NaN, if supported, is handled as in IEEE 7542008

The largest clamped component, max_{clamped} is determined:

max_{clamped} = max(red_{clamped}, green_{clamped}, blue_{clamped})
A preliminary shared exponent exp' is computed:
The shared exponent exp_{shared} is computed:
Finally, three integer values in the range 0 to 2^{N} are computed:
15.2.2. Shared Exponent to RGB
A shared exponent color (red_{shared}, green_{shared}, blue_{shared}, exp_{shared}) is transformed to an RGB color (red, green, blue) as follows:

\(red = red_{shared} \times {2^{(exp_{shared}BN)}}\)

\(green = green_{shared} \times {2^{(exp_{shared}BN)}}\)

\(blue = blue_{shared} \times {2^{(exp_{shared}BN)}}\)
where:

N = 9 (number of mantissa bits per component)

B = 15 (exponent bias)
15.3. Texel Input Operations
Texel input instructions are SPIRV 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:
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.
15.3.1. 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.
Instruction/Sampler/Image View Validation
There are a number of cases where a SPIRV 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
borderColor
is an integer type and the image viewformat
is not one of the VkFormat integer types or a stencil component of a depth/stencil format. 
The sampler
borderColor
is a float type and the image viewformat
is not one of the VkFormat float types or a depth component of a depth/stencil format. 
The sampler
borderColor
is one of the opaque black colors (VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK
orVK_BORDER_COLOR_INT_OPAQUE_BLACK
) and the image view VkComponentSwizzle for any of the VkComponentMapping components is notVK_COMPONENT_SWIZZLE_IDENTITY
. 
The VkImageLayout of any subresource in the image view does not match that specified in VkDescriptorImageInfo::
imageLayout
used to write the image descriptor. 
If the instruction is
OpImageRead
orOpImageSparseRead
and theshaderStorageImageReadWithoutFormat
feature is not enabled, or the instruction isOpImageWrite
and theshaderStorageImageWriteWithoutFormat
feature is not enabled, then the SPIRV Image Format must be compatible with the image view’sformat
. 
The sampler
unnormalizedCoordinates
isVK_TRUE
and any of the limitations of unnormalized coordinates are violated. 
The SPIRV instruction is one of the
OpImage
*Dref
* instructions and the samplercompareEnable
isVK_FALSE

The SPIRV instruction is not one of the
OpImage
*Dref
* instructions and the samplercompareEnable
isVK_TRUE

The SPIRV instruction is one of the
OpImage
*Dref
* instructions and the image viewformat
is not one of the depth/stencil formats with a depth component, or the image view aspect is notVK_IMAGE_ASPECT_DEPTH_BIT
. 
The SPIRV instruction’s image variable’s properties are not compatible with the image view:

Rules for
viewType
:
VK_IMAGE_VIEW_TYPE_1D
must haveDim
= 1D,Arrayed
= 0,MS
= 0. 
VK_IMAGE_VIEW_TYPE_2D
must haveDim
= 2D,Arrayed
= 0. 
VK_IMAGE_VIEW_TYPE_3D
must haveDim
= 3D,Arrayed
= 0,MS
= 0. 
VK_IMAGE_VIEW_TYPE_CUBE
must haveDim
= Cube,Arrayed
= 0,MS
= 0. 
VK_IMAGE_VIEW_TYPE_1D_ARRAY
must haveDim
= 1D,Arrayed
= 1,MS
= 0. 
VK_IMAGE_VIEW_TYPE_2D_ARRAY
must haveDim
= 2D,Arrayed
= 1. 
VK_IMAGE_VIEW_TYPE_CUBE_ARRAY
must haveDim
= Cube,Arrayed
= 1,MS
= 0.


If the image was created with VkImageCreateInfo::
samples
equal toVK_SAMPLE_COUNT_1_BIT
, the instruction must haveMS
= 0. 
If the image was created with VkImageCreateInfo::
samples
not equal toVK_SAMPLE_COUNT_1_BIT
, the instruction must haveMS
= 1.

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 SPIRV 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 conversion to integer texel coordinates.
If the integer texel coordinates do not satisfy all of the conditions

0 ≤ i < w_{s}

0 ≤ j < h_{s}

0 ≤ k < d_{s}

0 ≤ l < layers

0 ≤ n < samples
where:

w_{s} = width of the image level

h_{s} = height of the image level

d_{s} = depth of the image level

layers = number of layers in the image

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 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 texel replacement is performed.


Cube Map Edge or Corner
Otherwise the texel coordinates lie beyond the edges or corners of the selected cube map face, and Cube map edge handling is performed.
Cube Map Edge Handling
If the texel coordinates lie beyond the edges or corners of the selected
cube map face, the following steps are performed.
Note that this does not occur when using VK_FILTER_NEAREST
filtering
within a mip level, since VK_FILTER_NEAREST
is treated as using
VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE
.

Cube Map Edge Texel

If the texel lies beyond the selected cube map face in either only i or only j, then the coordinates (i,j) and the array layer l are transformed to select the adjacent texel from the appropriate neighboring face.


Cube Map Corner Texel

If the texel lies beyond the selected cube map face in both i and j, then 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. The methods are subject to the constraint that if the three available texels have the same value, the resulting filtered texel must have that value.

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 texel replacement.
15.3.2. Format Conversion
Texels undergo a format conversion from the 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
aspectMask
of the image view.
Each component is converted based on its type and size (as defined in the Format Definition section for each VkFormat), using the appropriate equations in 16Bit FloatingPoint Numbers, Unsigned 11Bit FloatingPoint Numbers, Unsigned 10Bit FloatingPoint Numbers, FixedPoint Data Conversion, and Shared Exponent to RGB. Signed integer components smaller than 32 bits are signextended.
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 Khronos Data Format Specification. The A component, if present, is unchanged.
If the image view format is blockcompressed, then the texel value is first decoded, then converted based on the type and number of components defined by the compressed format.
15.3.3. 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
borderColor
of the sampler.
The border color is:
Sampler borderColor 
Corresponding Border Color 


[B_{r}, B_{g}, B_{b}, B_{a}] = [0.0, 0.0, 0.0, 0.0] 

[B_{r}, B_{g}, B_{b}, B_{a}] = [0.0, 0.0, 0.0, 1.0] 

[B_{r}, B_{g}, B_{b}, B_{a}] = [1.0, 1.0, 1.0, 1.0] 

[B_{r}, B_{g}, B_{b}, B_{a}] = [0, 0, 0, 0] 

[B_{r}, B_{g}, B_{b}, B_{a}] = [0, 0, 0, 1] 

[B_{r}, B_{g}, B_{b}, B_{a}] = [1, 1, 1, 1] 
Note
The names 
This is substituted for the texel value by replacing the number of components in the image format
Texel Aspect or Format  Component Assignment 

Depth aspect 
D = B_{r} 
Stencil aspect 
S = B_{r} 
One component color format 
Color_{r} = B_{r} 
Two component color format 
[Color_{r},Color_{g}] = [B_{r},B_{g}] 
Three component color format 
[Color_{r},Color_{g},Color_{b}] = [B_{r},B_{g},B_{b}] 
Four component color format 
[Color_{r},Color_{g},Color_{b},Color_{a}] = [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 robustBufferAccess
feature is enabled.
In that case, an invalid texel is replaced as described by the
robustBufferAccess
feature.
If the
VkPhysicalDeviceSparseProperties::residencyNonResidentStrict
property is VK_TRUE
, a sparse unbound texel is replaced with 0 or 0.0
values for integer and floatingpoint components of the image format,
respectively.
If residencyNonResidentStrict
is VK_FALSE
, the value of the
sparse unbound texel is undefined.
15.3.4. Depth Compare Operation
If the image view has a depth/stencil format, the depth component is
selected by the aspectMask
, and the operation is a Dref
instruction, a depth comparison is performed.
The value of the result D is 1.0 if the result of the compare
operation is true, and 0.0 otherwise.
The compare operation is selected by the compareOp
member of the
sampler.
where, in the depth comparison:

D_{ref} = shaderOp.D_{ref} (from optional SPIRV operand)

D (texel depth value)
15.3.5. Conversion to RGBA
The texel is expanded from one, two, or three components to four components based on the image base color:
Texel Aspect or Format  RGBA Color 

Depth aspect 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [D,0,0,one] 
Stencil aspect 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [S,0,0,one] 
One component color format 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [Color_{r},0,0,one] 
Two component color format 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [Color_{r},Color_{g},0,one] 
Three component color format 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [Color_{r},Color_{g},Color_{b},one] 
Four component color format 
[Color_{r},Color_{g},Color_{b}, Color_{a}] = [Color_{r},Color_{g},Color_{b},Color_{a}] 
where one = 1.0f for floatingpoint formats and depth aspects, and one = 1 for integer formats and stencil aspects.
15.3.6. Component Swizzle
All texel input instructions apply a swizzle based on the
VkComponentSwizzle enums in the components
member of the
VkImageViewCreateInfo structure for the image being read.
The swizzle can rearrange the components of the texel, or substitute zero or one for any components. It is defined as follows for each color component:
where:
If the border color is one of the VK_BORDER_COLOR_*_OPAQUE_BLACK
enums
and the VkComponentSwizzle is not VK_COMPONENT_SWIZZLE_IDENTITY
for all components (or the
equivalent identity mapping),
the value of the texel after swizzle is undefined.
15.3.7. Sparse Residency
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 OpImageSparseTexelsResident
instruction which converts the residency code to a boolean value.
15.4. Texel Output Operations
Texel output instructions are SPIRV 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:
15.4.1. 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.
Texel Format Validation
If the image format of the OpTypeImage
is not compatible with the
VkImageView
’s format
, the write causes the contents of the
image’s memory to become undefined.
15.4.2. Integer Texel Coordinate Validation
The integer texel coordinates are validated according to the same rules as for texel input coordinate validation.
If the texel fails integer texel coordinate validation, then the write has no effect.
15.4.3. 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
VkPhysicalDeviceSparseProperties::residencyNonResidentStrict
property is VK_TRUE
, the sparse unbound texel write has no effect.
If residencyNonResidentStrict
is 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.
15.4.4. 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 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 VkFormat of the image view. Any unused components are ignored.
Each component is converted based on its type and size (as defined in the Format Definition section for each VkFormat). Floatingpoint outputs are converted as described in FloatingPoint Format Conversions and FixedPoint 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.
15.5. Derivative Operations
SPIRV derivative instructions include OpDPdx
, OpDPdy
,
OpDPdxFine
, OpDPdyFine
, OpDPdxCoarse
, and OpDPdyCoarse
.
Derivative instructions are only available in
fragment shaders.
Derivatives are computed as if there is a 2×2 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. Applications must not use derivative instructions in nonuniform control flow.
For a 2×2 neighborhood, for the four fragments labled 0, 1, 2 and 3,
the Fine
derivative instructions must return:
Coarse derivatives may return only two values. In this case, the values should be:
OpDPdx
and OpDPdy
must return the same result as either
OpDPdxFine
or OpDPdxCoarse
and either OpDPdyFine
or
OpDPdyCoarse
, respectively.
Implementations must make the same choice of either coarse or fine for both
OpDPdx
and OpDPdy
, and implementations should make the choice
that is more efficient to compute.
15.6. Normalized Texel Coordinate Operations
If the image sampler instruction provides normalized texel coordinates, some of the following operations are performed.
15.6.1. Projection Operation
For Proj
image operations, the normalized texel coordinates
(s,t,r,q,a) and (if present) the D_{ref} coordinate are
transformed as follows:
15.6.2. Derivative Image Operations
Derivatives are used for LOD selection.
These derivatives are either implicit (in an 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:
Partial derivatives not defined above for certain image dimensionalities are set to zero.
For explicit LOD image instructions, if the optional SPIRV operand 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 SPIRV operand 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 λ_{base} as described in LevelofDetail Operation.
For implicit derivative image instructions, the partial derivative values may be computed by linear approximation using a 2×2 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.
15.6.3. Cube Map Face Selection and Transformations
For cube map image instructions, the (s,t,r) coordinates are treated as a direction vector (r_{x},r_{y},r_{z}). The direction vector is used to select a cube map face. The direction vector is transformed to a perface texel coordinate system (s_{face},t_{face}), The direction vector is also used to transform the derivatives to perface derivatives.
15.6.4. 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 r_{z} wins over r_{y} and r_{x}, and the second rule that r_{y} wins over r_{x}. An implementation may choose other rules, but the rules must be deterministic and depend only on (r_{x},r_{y},r_{z}).
The layer number (corresponding to a cube map face), the coordinate selections for s_{c}, t_{c}, r_{c}, and the selection of derivatives, are determined by the major axis direction as specified in the following two tables.
Major Axis Direction  Layer Number  Cube Map Face  s_{c}  t_{c}  r_{c} 

+r_{x} 
0 
Positive X 
r_{z} 
r_{y} 
r_{x} 
r_{x} 
1 
Negative X 
+r_{z} 
r_{y} 
r_{x} 
+r_{y} 
2 
Positive Y 
+r_{x} 
+r_{z} 
r_{y} 
r_{y} 
3 
Negative Y 
+r_{x} 
r_{z} 
r_{y} 
+r_{z} 
4 
Positive Z 
+r_{x} 
r_{y} 
r_{z} 
r_{z} 
5 
Negative Z 
r_{x} 
r_{y} 
r_{z} 
Major Axis Direction  ∂s_{c} / ∂x  ∂s_{c} / ∂y  ∂t_{c} / ∂x  ∂t_{c} / ∂y  ∂r_{c} / ∂x  ∂r_{c} / ∂y 

+r_{x} 
∂r_{z} / ∂x 
∂r_{z} / ∂y 
∂r_{y} / ∂x 
∂r_{y} / ∂y 
+∂r_{x} / ∂x 
+∂r_{x} / ∂y 
r_{x} 
+∂r_{z} / ∂x 
+∂r_{z} / ∂y 
∂r_{y} / ∂x 
∂r_{y} / ∂y 
∂r_{x} / ∂x 
∂r_{x} / ∂y 
+r_{y} 
+∂r_{x} / ∂x 
+∂r_{x} / ∂y 
+∂r_{z} / ∂x 
+∂r_{z} / ∂y 
+∂r_{y} / ∂x 
+∂r_{y} / ∂y 
r_{y} 
+∂r_{x} / ∂x 
+∂r_{x} / ∂y 
∂r_{z} / ∂x 
∂r_{z} / ∂y 
∂r_{y} / ∂x 
∂r_{y} / ∂y 
+r_{z} 
+∂r_{x} / ∂x 
+∂r_{x} / ∂y 
∂r_{y} / ∂x 
∂r_{y} / ∂y 
+∂r_{z} / ∂x 
+∂r_{z} / ∂y 
r_{z} 
∂r_{x} / ∂x 
∂r_{x} / ∂y 
∂r_{y} / ∂x 
∂r_{y} / ∂y 
∂r_{z} / ∂x 
∂r_{z} / ∂y 
15.6.5. Cube Map Coordinate Transformation
15.6.6. Cube Map Derivative Transformation
15.6.7. Scale Factor Operation, LevelofDetail 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 SPIRV instruction
OpImageQueryLod
, which gives access to the λ' and
d_{l} values, defined below.
These values must be computed with mipmapPrecisionBits
of accuracy
and may be subject to implementationspecific maxima and minima for very
large, outofrange values.
Scale Factor Operation
The magnitude of the derivatives are calculated by:

m_{ux} = ∂s/∂x × w_{base}

m_{vx} = ∂t/∂x × h_{base}

m_{wx} = ∂r/∂x × d_{base}

m_{uy} = ∂s/∂y × w_{base}

m_{vy} = ∂t/∂y × h_{base}

m_{wy} = ∂r/∂y × d_{base}
where:

∂t/∂x = ∂t/∂y = 0 (for 1D images)

∂r/∂x = ∂r/∂y = 0 (for 1D, 2D or Cube images)
and:

w_{base} = image.w

h_{base} = image.h

d_{base} = image.d
(for the baseMipLevel
, from the image descriptor).
A point sampled in screen space has an elliptical footprint in texture space. The minimum and maximum scale factors (ρ_{min}, ρ_{max}) should be the minor and major axes of this ellipse.
The scale factors ρ_{x} and ρ_{y}, calculated from the magnitude of the derivatives in x and y, are used to compute the minimum and maximum scale factors.
ρ_{x} and ρ_{y} may be approximated with functions f_{x} and f_{y}, subject to the following constraints:
The minimum and maximum scale factors (ρ_{min},ρ_{max}) are determined by:

ρ_{max} = max(ρ_{x}, ρ_{y})

ρ_{min} = min(ρ_{x}, ρ_{y})
The ratio of anisotropy is determined by:

η = min(ρ_{max}/ρ_{min}, max_{Aniso})
where:

sampler.max_{Aniso} =
maxAnisotropy
(from sampler descriptor) 
limits.max_{Aniso} =
maxSamplerAnisotropy
(from physical device limits) 
max_{Aniso} = min(sampler.max_{Aniso}, limits.max_{Aniso})
If ρ_{max} = ρ_{min} = 0, then all the partial derivatives are
zero, the fragment’s footprint in texel space is a point, and N
should be treated as 1.
If ρ_{max} ≠ 0 and ρ_{min} = 0 then all partial
derivatives along one axis are zero, the fragment’s footprint in texel space
is a line segment, and η should be treated as max_{Aniso}.
However, anytime the footprint is small in texel space the implementation
may use a smaller value of η, even when ρ_{min} is zero
or close to zero.
If either VkPhysicalDeviceFeatures::samplerAnisotropy
or
VkSamplerCreateInfo::anisotropyEnable
are VK_FALSE
,
max_{Aniso} is set to 1.
If η = 1, sampling is isotropic. If η > 1, sampling is anisotropic.
The sampling rate (N) is derived as:

N = ⌈η⌉
An implementation may round N up to the nearest supported sampling rate. An implementation may use the value of N as an approximation of η.
LevelofDetail Operation
The LOD parameter λ is computed as follows:
where:
and maxSamplerLodBias is the value of the VkPhysicalDeviceLimits
feature maxSamplerLodBias
.
Image Level(s) Selection
The image level(s) d, d_{hi}, and d_{lo} which texels are read from are determined by an imagelevel parameter d_{l}, which is computed based on the LOD parameter, as follows:
where:
and:

level_{base} =
baseMipLevel

q =
levelCount
 1
baseMipLevel
and levelCount
are taken from the
subresourceRange
of the image view.
If the sampler’s mipmapMode
is VK_SAMPLER_MIPMAP_MODE_NEAREST
,
then the level selected is d = d_{l}.
If the sampler’s mipmapMode
is VK_SAMPLER_MIPMAP_MODE_LINEAR
,
two neighboring levels are selected:
δ is the fractional value, quantized to the number of mipmap precision bits, used for linear filtering between levels.
15.6.8. (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 used in filtering (either d, or d_{hi} and d_{lo}).
where:

width_{scale} = width_{level}

height_{scale} = height_{level}

depth_{scale} = depth_{level}
and where (Δ_{i}, Δ_{j}, Δ_{k}) are
taken from the image instruction if it includes a ConstOffset
or
Offset
operand, otherwise they are taken to be zero.
Operations then proceed to Unnormalized Texel Coordinate Operations.
15.7. Unnormalized Texel Coordinate Operations
15.7.1. (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:

l = clamp(RNE(a), 0,
layerCount
 1) +baseArrayLayer
where layerCount
is the number of layers in the image subresource
range of the image view, baseArrayLayer
is the first layer from the
subresource range, and where:
The sample index n is assigned the value zero.
Nearest filtering (VK_FILTER_NEAREST
) computes the integer texel
coordinates that the unnormalized coordinates lie within:
where:

shift = 0.0
Linear filtering (VK_FILTER_LINEAR
) computes a set of neighboring
coordinates which bound the unnormalized coordinates.
The integer texel coordinates are combinations of i_{0} or i_{1},
j_{0} or j_{1}, k_{0} or k_{1}, as well as weights
α, β, and γ.
where:

shift = 0.5
and where:
where the number of fraction bits retained is specified by
VkPhysicalDeviceLimits
::subTexelPrecisionBits
.
15.8. Integer Texel Coordinate Operations
The OpImageFetch
and OpImageFetchSparse
SPIRV instructions may
supply a LOD from which texels are to be fetched using the optional SPIRV
operand Lod
.
Other integercoordinate operations must not.
If the Lod
is provided then it must be an integer.
The image level selected is:
If d does not lie in the range [baseMipLevel
,
baseMipLevel
+ levelCount
) then any values fetched are
undefined.
15.9. Image Sample Operations
15.9.1. Wrapping Operation
Cube
images ignore the wrap modes specified in the sampler.
Instead, if VK_FILTER_NEAREST
is used within a mip level then
VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE
is used, and if
VK_FILTER_LINEAR
is used within a mip level then sampling at the edges
is performed as described earlier in the Cube map
edge handling section.
The first integer texel coordinate i is transformed based on the
addressModeU
parameter of the sampler.
where:
j (for 2D and Cube image) and k (for 3D image) are similarly
transformed based on the addressModeV
and addressModeW
parameters of the sampler, respectively.
15.9.2. Texel Gathering
SPIRV instructions with Gather
in the name return a vector derived
from 4 texels in the base level of the image view.
The rules for the VK_FILTER_LINEAR
minification filter are applied to
identify the four selected texels.
Each texel is then converted to an RGBA value according to
conversion to RGBA and then
swizzled.
A fourcomponent vector is then assembled by taking the component indicated
by the Component
value in the instruction from the swizzled color value
of the four texels.
If the operation does not use the ConstOffsets
image operand then the
four texels form the 2 × 2 rectangle used for texture filtering:
If the operation does use the ConstOffsets
image operand then the
offsets allow a custom filter to be defined:
where:
15.9.3. Texel Filtering
Texel filtering is first performed for each level (either d or d_{hi} and d_{lo}).
If λ 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
magFilter
in the sampler.
If λ is greater than zero, the texture is said to be
minified, and the filter mode within a mip level is selected by the
minFilter
in the sampler.
Texel Nearest Filtering
Within a mip level, VK_FILTER_NEAREST
filtering selects a single value
using the (i, j, k) texel coordinates, with all texels taken from
layer l.
Texel Linear Filtering
Within a mip level, VK_FILTER_LINEAR
filtering combines 8 (for 3D), 4
(for 2D or Cube), or 2 (for 1D) texel values, together with their linear
weights.
The linear weights are derived from the fractions computed earlier:
The values of multiple texels, together with their weights, are combined using a weighted average to produce a filtered value:
Texel Mipmap Filtering
VK_SAMPLER_MIPMAP_MODE_NEAREST
filtering returns the value of a single
mipmap level,
τ = τ[d].
VK_SAMPLER_MIPMAP_MODE_LINEAR
filtering combines the values of
multiple mipmap levels (τ[hi] and τ[lo]), together with their linear
weights.
The linear weights are derived from the fraction computed earlier:
The values of multiple mipmap levels together with their linear weights, are combined using a weighted average to produce a final filtered value:
Texel Anisotropic Filtering
Anisotropic filtering is enabled by the 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 magFilter
, minFilter
and
mipmapMode
of the sampler to control the specifics of the anisotropic
filtering scheme used.
In addition, implementations should consider minLod
and 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 magFilter
, minFilter
of VK_FILTER_LINEAR
and a
mipmapMode
of 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 d to approximate an anisotropic filter. The sum τ_{2Daniso} is defined using the single isotropic τ_{2D}(u,v) at level d.
15.10. 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
OpImageWrite
. 
Depth Comparison: Performed by
OpImage
*Dref
instructions. 
All Texel output operations: Performed by
OpImageWrite
. 
Projection: Performed by all
OpImage
*Proj
instructions. 
Derivative Image Operations, Cube Map Operations, Scale Factor Operation, LevelofDetail Operation and Image Level(s) Selection, and Texel Anisotropic Filtering: Performed by all
OpImageSample
* andOpImageSparseSample
* 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
OpImageSample
,OpImageSparseSample
, andOpImage
*Gather
instructions. 
Texel Gathering: Performed by
OpImage
*Gather
instructions. 
Texel Filtering: Performed by all
OpImageSample
* andOpImageSparseSample
* instructions. 
Sparse Residency: Performed by all
OpImageSparse
* instructions.