## 15. Image Operations

### 15.1. Image Operations Overview

Vulkan Image Operations are operations performed by those SPIR-V 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 SPIR-V Image Instructions:

• OpImageSample* and OpImageSparseSample* 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 and OpImageSparseFetch return a single texel of the image. No sampler is used.

• OpImage*Gather and OpImageSparse*Gather read neighboring texels and return a single component of each.

• OpImageRead (and OpImageSparseRead) and OpImageWrite 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)

SPIR-V 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 SPIR-V 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 non-Proj 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 SPIR-V operand, because only 1D and 2D (non-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 (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 SPIR-V 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.

Figure 3. Texel Coordinate Systems, Linear Filtering

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 i0j0, i1j0, i0j1, and i1j1.

• The fractions α and β.

• Given the offset Δi and Δj, the four texels selected by the offset are i0j'0, i1j'0, i0j'1, and i1j'1.

Figure 4. Texel Coordinate Systems, Nearest Filtering

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 Δi and Δj, 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 (redshared, greenshared, blueshared, expshared) as follows:

First, the components (red, green, blue) are clamped to (redclamped, greenclamped, blueclamped) as:

redclamped = max(0, min(sharedexpmax, red))

greenclamped = max(0, min(sharedexpmax, green))

blueclamped = max(0, min(sharedexpmax, blue))

where:

\begin{aligned} N & = 9 & \text{number of mantissa bits per component} \\ B & = 15 & \text{exponent bias} \\ E_{max} & = 31 & \text{maximum possible biased exponent value} \\ sharedexp_{max} & = \frac{(2^N-1)}{2^N} \times 2^{(E_{max}-B)} \end{aligned}
 Note NaN, if supported, is handled as in IEEE 754-2008 minNum() and maxNum(). That is the result is a NaN is mapped to zero.

The largest clamped component, maxclamped is determined:

maxclamped = max(redclamped, greenclamped, blueclamped)

A preliminary shared exponent exp' is computed:

\begin{aligned} exp' = \begin{cases} \left \lfloor \log_2(max_{clamped}) \right \rfloor + (B+1) & \text{for}\ max_{clamped} > 2^{-(B+1)} \\ 0 & \text{for}\ max_{clamped} \leq 2^{-(B+1)} \end{cases} \end{aligned}

The shared exponent expshared is computed:

\begin{aligned} max_{shared} = \left \lfloor { \frac{max_{clamped}}{2^{(exp'-B-N)}} + \frac{1}{2} } \right \rfloor \end{aligned}
\begin{aligned} exp_{shared} = \begin{cases} exp' & \text{for}\ 0 \leq max_{shared} < 2^N \\ exp'+1 & \text{for}\ max_{shared} = 2^N \end{cases} \end{aligned}

Finally, three integer values in the range 0 to 2N are computed:

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

#### 15.2.2. Shared Exponent to RGB

A shared exponent color (redshared, greenshared, blueshared, expshared) is transformed to an RGB color (red, green, blue) as follows:

$$red = red_{shared} \times {2^{(exp_{shared}-B-N)}}$$

$$green = green_{shared} \times {2^{(exp_{shared}-B-N)}}$$

$$blue = blue_{shared} \times {2^{(exp_{shared}-B-N)}}$$

where:

N = 9 (number of mantissa bits per component)

B = 15 (exponent bias)

### 15.3. 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:

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

These cases include:

• The sampler borderColor is an integer type and the image view format 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 view format 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 or VK_BORDER_COLOR_INT_OPAQUE_BLACK) and the image view VkComponentSwizzle for any of the VkComponentMapping components is not VK_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 or OpImageSparseRead and the shaderStorageImageReadWithoutFormat feature is not enabled, or the instruction is OpImageWrite and the shaderStorageImageWriteWithoutFormat feature is not enabled, then the SPIR-V Image Format must be compatible with the image view’s format.

• The sampler unnormalizedCoordinates is VK_TRUE and any of the limitations of unnormalized coordinates are violated.

• The SPIR-V instruction is one of the OpImage*Dref* instructions and the sampler compareEnable is VK_FALSE

• The SPIR-V instruction is not one of the OpImage*Dref* instructions and the sampler compareEnable is VK_TRUE

• The SPIR-V instruction is one of the OpImage*Dref* instructions and the image view format is not one of the depth/stencil formats with a depth component, or the image view aspect is not VK_IMAGE_ASPECT_DEPTH_BIT.

• The SPIR-V instruction’s image variable’s properties are not compatible with the image view:

• Rules for viewType:

• VK_IMAGE_VIEW_TYPE_1D must have Dim = 1D, Arrayed = 0, MS = 0.

• VK_IMAGE_VIEW_TYPE_2D must have Dim = 2D, Arrayed = 0.

• VK_IMAGE_VIEW_TYPE_3D must have Dim = 3D, Arrayed = 0, MS = 0.

• VK_IMAGE_VIEW_TYPE_CUBE must have Dim = Cube, Arrayed = 0, MS = 0.

• VK_IMAGE_VIEW_TYPE_1D_ARRAY must have Dim = 1D, Arrayed = 1, MS = 0.

• VK_IMAGE_VIEW_TYPE_2D_ARRAY must have Dim = 2D, Arrayed = 1.

• VK_IMAGE_VIEW_TYPE_CUBE_ARRAY must have Dim = Cube, Arrayed = 1, MS = 0.

• If the image was created with VkImageCreateInfo::samples equal to VK_SAMPLE_COUNT_1_BIT, the instruction must have MS = 0.

• If the image was created with VkImageCreateInfo::samples not equal to VK_SAMPLE_COUNT_1_BIT, the instruction must have MS = 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 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 conversion to integer texel coordinates.

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

0 ≤ i < ws

0 ≤ j < hs

0 ≤ k < ds

0 ≤ l < layers

0 ≤ n < samples

where:

ws = width of the image level

hs = height of the image level

ds = 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:

1. 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.

2. 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.

3. 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.

4. 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 16-Bit Floating-Point Numbers, Unsigned 11-Bit Floating-Point Numbers, Unsigned 10-Bit Floating-Point Numbers, Fixed-Point Data Conversion, and Shared Exponent to RGB. Signed integer components smaller than 32 bits are sign-extended.

If the image view format is sRGB, the color components are first converted as if they are UNORM, and then sRGB to linear conversion is applied to the R, G, and B components as described in the “sRGB EOTF” section of the 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.

#### 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:

Table 15. Border Color B
Sampler borderColor Corresponding Border Color

VK_BORDER_COLOR_FLOAT_TRANSPARENT_BLACK

[Br, Bg, Bb, Ba] = [0.0, 0.0, 0.0, 0.0]

VK_BORDER_COLOR_FLOAT_OPAQUE_BLACK

[Br, Bg, Bb, Ba] = [0.0, 0.0, 0.0, 1.0]

VK_BORDER_COLOR_FLOAT_OPAQUE_WHITE

[Br, Bg, Bb, Ba] = [1.0, 1.0, 1.0, 1.0]

VK_BORDER_COLOR_INT_TRANSPARENT_BLACK

[Br, Bg, Bb, Ba] = [0, 0, 0, 0]

VK_BORDER_COLOR_INT_OPAQUE_BLACK

[Br, Bg, Bb, Ba] = [0, 0, 0, 1]

VK_BORDER_COLOR_INT_OPAQUE_WHITE

[Br, Bg, Bb, Ba] = [1, 1, 1, 1]

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

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

Table 16. Border Texel Components After Replacement
Texel Aspect or Format Component Assignment

Depth aspect

D = Br

Stencil aspect

S = Br

One component color format

Colorr = Br

Two component color format

[Colorr,Colorg] = [Br,Bg]

Three component color format

[Colorr,Colorg,Colorb] = [Br,Bg,Bb]

Four component color format

[Colorr,Colorg,Colorb,Colora] = [Br,Bg,Bb,Ba]

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 floating-point 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.

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

where, in the depth comparison:

Dref = shaderOp.Dref (from optional SPIR-V 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:

Table 17. Texel Color After Conversion To RGBA
Texel Aspect or Format RGBA Color

Depth aspect

[Colorr,Colorg,Colorb, Colora] = [D,0,0,one]

Stencil aspect

[Colorr,Colorg,Colorb, Colora] = [S,0,0,one]

One component color format

[Colorr,Colorg,Colorb, Colora] = [Colorr,0,0,one]

Two component color format

[Colorr,Colorg,Colorb, Colora] = [Colorr,Colorg,0,one]

Three component color format

[Colorr,Colorg,Colorb, Colora] = [Colorr,Colorg,Colorb,one]

Four component color format

[Colorr,Colorg,Colorb, Colora] = [Colorr,Colorg,Colorb,Colora]

where one = 1.0f for floating-point 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:

\begin{aligned} Color'_{component} & = \begin{cases} Color_r & \text{for RED swizzle} \\ Color_g & \text{for GREEN swizzle} \\ Color_b & \text{for BLUE swizzle} \\ Color_a & \text{for ALPHA swizzle} \\ 0 & \text{for ZERO swizzle} \\ one & \text{for ONE swizzle} \\ identity & \text{for IDENTITY swizzle} \end{cases} \end{aligned}

where:

\begin{aligned} one & = \begin{cases} & 1.0\text{f} & \text{for floating point components} \\ & 1 & \text{for integer components} \\ \end{cases} \\ identity & = \begin{cases} & Color_r & \text{for}\ component = r \\ & Color_g & \text{for}\ component = g \\ & Color_b & \text{for}\ component = b \\ & Color_a & \text{for}\ component = a \\ \end{cases} \end{aligned}

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 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:

#### 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). Floating-point outputs are converted as described in Floating-Point Format Conversions and Fixed-Point Data Conversion. Integer outputs are converted such that their value is preserved. The converted value of any integer that cannot be represented in the target format is undefined.

### 15.5. Normalized Texel Coordinate Operations

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

#### 15.5.1. Projection Operation

For Proj image operations, the normalized texel coordinates (s,t,r,q,a) and (if present) the Dref coordinate are transformed as follows:

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

#### 15.5.2. 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. That is:

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

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

For explicit LOD image instructions, if the optional SPIR-V operand 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 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 λbase as described in Level-of-Detail Operation.

#### 15.5.3. Cube Map Face Selection and Transformations

For cube map image instructions, the (s,t,r) coordinates are treated as a direction vector (rx,ry,rz). The direction vector is used to select a cube map face. The direction vector is transformed to a per-face texel coordinate system (sface,tface), The direction vector is also used to transform the derivatives to per-face derivatives.

#### 15.5.4. 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 rz wins over ry and rx, and the second rule that ry wins over rx. An implementation may choose other rules, but the rules must be deterministic and depend only on (rx,ry,rz).

The layer number (corresponding to a cube map face), the coordinate selections for sc, tc, rc, and the selection of derivatives, are determined by the major axis direction as specified in the following two tables.

Table 18. Cube map face and coordinate selection
Major Axis Direction Layer Number Cube Map Face sc tc rc

+rx

0

Positive X

-rz

-ry

rx

-rx

1

Negative X

+rz

-ry

rx

+ry

2

Positive Y

+rx

+rz

ry

-ry

3

Negative Y

+rx

-rz

ry

+rz

4

Positive Z

+rx

-ry

rz

-rz

5

Negative Z

-rx

-ry

rz

Table 19. Cube map derivative selection
Major Axis Direction ∂sc / ∂x ∂sc / ∂y ∂tc / ∂x ∂tc / ∂y ∂rc / ∂x ∂rc / ∂y

+rx

-∂rz / ∂x

-∂rz / ∂y

-∂ry / ∂x

-∂ry / ∂y

+∂rx / ∂x

+∂rx / ∂y

-rx

+∂rz / ∂x

+∂rz / ∂y

-∂ry / ∂x

-∂ry / ∂y

-∂rx / ∂x

-∂rx / ∂y

+ry

+∂rx / ∂x

+∂rx / ∂y

+∂rz / ∂x

+∂rz / ∂y

+∂ry / ∂x

+∂ry / ∂y

-ry

+∂rx / ∂x

+∂rx / ∂y

-∂rz / ∂x

-∂rz / ∂y

-∂ry / ∂x

-∂ry / ∂y

+rz

+∂rx / ∂x

+∂rx / ∂y

-∂ry / ∂x

-∂ry / ∂y

+∂rz / ∂x

+∂rz / ∂y

-rz

-∂rx / ∂x

-∂rx / ∂y

-∂ry / ∂x

-∂ry / ∂y

-∂rz / ∂x

-∂rz / ∂y

#### 15.5.5. Cube Map Coordinate Transformation

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

#### 15.5.6. Cube Map Derivative Transformation

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

#### 15.5.7. Scale Factor Operation, Level-of-Detail Operation and Image Level(s) Selection

LOD selection can be either explicit (provided explicitly by the image instruction) or implicit (determined from a scale factor calculated from the derivatives). The implicit LOD selected can be queried using the SPIR-V instruction OpImageQueryLod, which gives access to the λ' and dl values, defined below. These values must be computed with mipmapPrecisionBits of accuracy and may be subject to implementation-specific maxima and minima for very large, out-of-range values.

##### Scale Factor Operation

The magnitude of the derivatives are calculated by:

mux = |∂s/∂x| × wbase

mvx = |∂t/∂x| × hbase

mwx = |∂r/∂x| × dbase

muy = |∂s/∂y| × wbase

mvy = |∂t/∂y| × hbase

mwy = |∂r/∂y| × dbase

where:

∂t/∂x = ∂t/∂y = 0 (for 1D images)

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

and:

wbase = image.w

hbase = image.h

dbase = 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 min, ρmax) should be the minor and major axes of this ellipse.

The scale factors ρx and ρy, calculated from the magnitude of the derivatives in x and y, are used to compute the minimum and maximum scale factors.

ρx and ρy may be approximated with functions fx and fy, subject to the following constraints:

\begin{aligned} & f_x \text{\ is\ continuous\ and\ monotonically\ increasing\ in\ each\ of\ } m_{ux}, m_{vx}, \text{\ and\ } m_{wx} \\ & f_y \text{\ is\ continuous\ and\ monotonically\ increasing\ in\ each\ of\ } m_{uy}, m_{vy}, \text{\ and\ } m_{wy} \end{aligned}
\begin{aligned} \max(|m_{ux}|, |m_{vx}|, |m_{wx}|) \leq f_{x} \leq \sqrt{2} (|m_{ux}| + |m_{vx}| + |m_{wx}|) \\ \max(|m_{uy}|, |m_{vy}|, |m_{wy}|) \leq f_{y} \leq \sqrt{2} (|m_{uy}| + |m_{vy}| + |m_{wy}|) \end{aligned}

The minimum and maximum scale factors minmax) are determined by:

ρmax = max(ρx, ρy)

ρmin = min(ρx, ρy)

The ratio of anisotropy is determined by:

η = min(ρmaxmin, maxAniso)

where:

sampler.maxAniso = maxAnisotropy (from sampler descriptor)

limits.maxAniso = maxSamplerAnisotropy (from physical device limits)

maxAniso = min(sampler.maxAniso, limits.maxAniso)

If ρmax = ρmin = 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 ρmax ≠ 0 and ρmin = 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 maxAniso. However, anytime the footprint is small in texel space the implementation may use a smaller value of η, even when ρmin is zero or close to zero. If either VkPhysicalDeviceFeatures::samplerAnisotropy or VkSamplerCreateInfo::anisotropyEnable are VK_FALSE, maxAniso 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.

##### Level-of-Detail Operation

The LOD parameter λ is computed as follows:

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

where:

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

and maxSamplerLodBias is the value of the VkPhysicalDeviceLimits feature maxSamplerLodBias.

##### Image Level(s) Selection

The image level(s) d, dhi, and dlo which texels are read from are determined by an image-level parameter dl, which is computed based on the LOD parameter, as follows:

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

where:

\begin{aligned} d' = level_{base} + \text{clamp}(\lambda, 0, q) \end{aligned}
\begin{aligned} nearest(d') & = \begin{cases} \left \lceil d' + 0.5\right \rceil - 1, & \text{preferred} \\ \left \lfloor d' + 0.5\right \rfloor, & \text{alternative} \end{cases} \end{aligned}

and:

levelbase = 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 = dl.

If the sampler’s mipmapMode is VK_SAMPLER_MIPMAP_MODE_LINEAR, two neighboring levels are selected:

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

δ is the fractional value, quantized to the number of mipmap precision bits, used for linear filtering between levels.

#### 15.5.8. (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 dhi and dlo).

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

where:

widthscale = widthlevel

heightscale = heightlevel

depthscale = depthlevel

and where i, Δj, Δk) 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.

### 15.6. Unnormalized Texel Coordinate Operations

#### 15.6.1. (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:

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

The sample index n is assigned the value zero.

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

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

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 i0 or i1, j0 or j1, k0 or k1, as well as weights α, β, and γ.

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

where:

shift = 0.5

and where:

$\mathbin{frac}(x) = x - \lfloor x \rfloor$

where the number of fraction bits retained is specified by VkPhysicalDeviceLimits::subTexelPrecisionBits.

### 15.7. Integer Texel Coordinate Operations

The OpImageFetch and OpImageFetchSparse SPIR-V instructions may supply a LOD from which texels are to be fetched using the optional SPIR-V operand Lod. Other integer-coordinate operations must not. If the Lod is provided then it must be an integer.

The image level selected is:

\begin{aligned} d & = level_{base} + \begin{cases} Lod & \text{(from optional SPIR-V operand)} \\ 0 & \text{otherwise} \end{cases} \\ \end{aligned}

If d does not lie in the range [baseMipLevel, baseMipLevel + levelCount) then any values fetched are undefined.

### 15.8. Image Sample Operations

#### 15.8.1. 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.

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

where:

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

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.

#### 15.8.2. Texel Gathering

SPIR-V 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 four-component 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:

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

If the operation does use the ConstOffsets image operand then the offsets allow a custom filter to be defined:

\begin{aligned} \tau[R] &= \tau_{i0j0 + \Delta_0}[level_{base}][comp] \\ \tau[G] &= \tau_{i0j0 + \Delta_1}[level_{base}][comp] \\ \tau[B] &= \tau_{i0j0 + \Delta_2}[level_{base}][comp] \\ \tau[A] &= \tau_{i0j0 + \Delta_3}[level_{base}][comp] \end{aligned}

where:

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

#### 15.8.3. Texel Filtering

Texel filtering is first performed for each level (either d or dhi and dlo).

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.

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

\begin{aligned} w_{i_0} &= (1-\alpha) \\ w_{i_1} &= (\alpha) \\ w_{j_0} &= (1-\beta) \\ w_{j_1} &= (\beta) \\ w_{k_0} &= (1-\gamma) \\ w_{k_1} &= (\gamma) \end{aligned}

The values of multiple texels, together with their weights, are combined using a weighted average to produce a filtered value:

\begin{aligned} \tau_{3D} &= \sum_{k=k_0}^{k_1}\sum_{j=j_0}^{j_1}\sum_{i=i_0}^{i_1}(w_{i})(w_{j})(w_{k})\tau_{ijk} \\ \tau_{2D} &= \sum_{j=j_0}^{j_1}\sum_{i=i_0}^{i_1}(w_{i})(w_{j})\tau_{ij} \\ \tau_{1D} &= \sum_{i=i_0}^{i_1}(w_{i})\tau_{i} \end{aligned}
##### 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:

\begin{aligned} w_{hi} &= (1-\delta) \\ w_{lo} &= (\delta) \\ \end{aligned}

The values of multiple mipmap levels together with their linear weights, are combined using a weighted average to produce a final filtered value:

\begin{aligned} \tau &= (w_{hi})\tau[hi]+(w_{lo})\tau[lo] \end{aligned}
##### 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 τ2Daniso is defined using the single isotropic τ2D(u,v) at level d.

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

### 15.9. 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, Level-of-Detail Operation and Image Level(s) Selection, and Texel Anisotropic Filtering: Performed by all OpImageSample* and 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 OpImageSample, OpImageSparseSample, and OpImage*Gather instructions.

• Texel Gathering: Performed by OpImage*Gather instructions.

• Texel Filtering: Performed by all OpImageSample* and OpImageSparseSample* instructions.

• Sparse Residency: Performed by all OpImageSparse* instructions.