Application #5

6 Data structures🔗

6.1 Gain curve structure🔗

6.1.1 Introduction🔗

A gain curve structure defines a function $GainCurve : R \to R$ .

A gain curve structure is parameterized by:

an integer $N_{cp}$ number of control points
real values $x_{i}$ , $y_{i}$ , and $m_{i}$ for $i \in Z_{N}_{cp}$ representing input, output, and slope of the control points

6.1.2 Parameter constraints🔗

The value of $N_{cp}$ shall be greater than or equal to 1 and less than or equal to 32.

For $i \in Z_{N_{cp}}$ , the value of $x_{i}$ shall be in the interval $[0, 64]$ .

For $i \in Z_{N_{cp}}$ , the value of $y_{i}$ shall be in the interval $[- 12, 12]$ .

For $i \in Z_{N_{cp} - 1}$ it shall be the case that $x_{i} \leq x_{i + 1}$ . If it is the case that $x_{i} = x_{i + 1}$ then it shall also be the case that $y_{i} = y_{i + 1}$ .

6.1.3 Function evaluation🔗

This clause describes the evaluation of the function $GainCurve$ , given an input parameter $x \in R$ .

For $i \in Z_{N_{cp} - 1}$ , let $f_{i} : [x_{i}, x_{i + 1}] \to R$ be the $i$ th component of the gain curve, as defined in Formula (1).

\begin{aligned} f_{i} (x) & = c_{3} \times t^{3} + c_{2} \times t^{2} + c_{1} \times t + c_{0} & where \\ {\hat{m}}_{i} & = (x_{i + 1} - x_{i}) \times m_{i} \\ {\hat{m}}_{i + 1} & = (x_{i + 1} - x_{i}) \times m_{i + 1} \\ c_{3} & = 2 y_{i} + {\hat{m}}_{i} - 2 y_{i + 1} + {\hat{m}}_{i + 1} \\ c_{2} & = - 3 y_{i} + 3 y_{i + 1} - 2 {\hat{m}}_{i} - {\hat{m}}_{i + 1} \\ c_{1} & = {\hat{m}}_{i} \\ c_{0} & = y_{i} \\ t & = (x - x_{i}) / (x_{i + 1} - x_{i}) \end{aligned}

(1)

The function $GainCurve$ shall be equivalent to the definition in Formula (2).

GainCurve (x) = {\begin{cases} y_{0} & : x < x_{0} \\ y_{i} & : there exists i \in Z_{N_{cp}} such that x = x_{i} \\ f_{i} (x) & : there exists i \in Z_{N_{cp} - 1} such that x_{i} < x < x_{i + 1} \\ y_{_{N_{cp} - 1}} + \log_{2} (\frac{x_{_{N_{cp} - 1}}}{x}) & : x_{_{N_{cp} - 1}} \leq x \end{cases}

(2)

6.2 Component mixing function structure🔗

6.2.1 Introduction🔗

A component mixing function structure defines a function $ComponentMixing : R^{3} \to R^{3}$ .

A component mixing function structure is parameterized by:

a real value $k_{red}$ representing the weighting of the red color component
a real value $k_{green}$ representing the weighting of the green color component
a real value $k_{blue}$ representing the weighting of the blue color component
a real value $k_{max}$ representing the weighting of the maximum color component
a real value $k_{min}$ representing the weighting of the minimum color component
a real value $k_{component}$ representing the weighting of each color component

NOTE 1 —⁠ The $ComponentMixing$ function is constructed such that for any $x \in R$ , there exists a $y \in R$ such that $ComponentMixing ([x, x, x]) = [y, y, y]$ . This means that, when applied to color values, $ComponentMixing$ will map grey values to grey values.

NOTE 2 —⁠ This function and its parameterization are the same as the adaptive gain curve input evaluator from Annex 1 of ICC Adaptive Gain Curve.

6.2.2 Parameter constraints🔗

The values of $k_{red}$ , $k_{green}$ , $k_{blue}$ , $k_{max}$ , $k_{min}$ , and $k_{component}$ shall be in the interval $[0, 1]$ .

It shall be the case that $k_{red} + k_{green} + k_{blue} + k_{max} + k_{min} + k_{component} = 1$ .

6.2.3 Function evaluation🔗

This clause describes the evaluation of the function $ComponentMixing$ , given an input parameter $C = [c_{r}, c_{g}, c_{b}] \in R^{3}$ .

Let $f_{common} : R^{3} \to R$ be the part of the component mixing function that is common to all components. It shall be equivalent to the definition in Formula (3).

\begin{aligned} f_{common} (C) = & c_{r} \times k_{red} + \\ c_{g} \times k_{green} + \\ c_{b} \times k_{blue} + \\ max (c_{r}, c_{g}, c_{b}) \times k_{max} + \\ min (c_{r}, c_{g}, c_{b}) \times k_{min} \end{aligned}

(3)

The function $ComponentMixing$ shall be equivalent to the definition in Formula (4).

ComponentMixing (C) = [\begin{matrix} c_{r} \times k_{component} + f_{common} (C) \\ c_{g} \times k_{component} + f_{common} (C) \\ c_{b} \times k_{component} + f_{common} (C) \end{matrix}]

(4)

NOTE —⁠ If the parameter $k_{component}$ is equal to zero then all three components of $ComponentMixing (C)$ will be equal.

6.3 Color gain function structure🔗

6.3.1 Introduction🔗

A color gain function structure defines a color gain function $Gain : R^{3} \to R^{3}$ .

A color gain function structure is parameterized by:

a component mixing function structure, which defines the function $ComponentMixing$
a gain curve structure, which defines the function $GainCurve$

NOTE —⁠ The resulting color gain function is equivalent to an adaptive gain curve from ICC Adaptive Gain Curve.

6.3.2 Function evaluation🔗

This clause describes the evaluation of the function $Gain$ , given an input parameter $C \in R^{3}$ .

Let $M = [m_{r}, m_{g}, m_{b}] \in R^{3}$ be the component mixing function evaluated at $C$ , as defined in Formula (5)

M = [\begin{matrix} m_{r} \\ m_{g} \\ m_{b} \end{matrix}] = ComponentMixing (C)

(5)

The function $Gain$ shall be equivalent to the definition in Formula (6).

Gain (C) = [\begin{matrix} GainCurve (m_{r}) \\ GainCurve (m_{g}) \\ GainCurve (m_{b}) \end{matrix}]

(6)

NOTE —⁠ If it is the case that all components of $M$ are equal then the function $GainCurve$ needs to be evaluated only once.

6.3.3 Zero color gain function🔗

The zero color gain function shall equal the color gain function defined in Formula (7).

Gain (C) = [0, 0, 0]

(7)

NOTE —⁠ The zero color gain function is the color gain function that is applied to the baseline image when performing headroom-adaptive tone mapping with the targeted HDR headroom parameter equal to the baseline HDR headroom.

6.3.4 Visualization🔗

Figure 2 provides a visualization of the evaluation of the $Gain$ function as described in 6.3.2.

Figure 2 — Visualization of the evaluation of the color gain function. Each arrow represents an individual color component.

6.4 Headroom-adaptive tone mapping structure🔗

6.4.1 Introduction🔗

An headroom-adaptive tone map structure defines a function $ToneMap : R^{3} \times R_{\geq 0} \to R^{3}$ which takes as parameters an input color $C \in R^{3}$ in the gain application color space and a targeted HDR headroom $H_{target}$ , and returns a tone mapped color.

NOTE —⁠ The function $ToneMap$ performs the headroom-adaptive tone mapping, so when it is applied to the baseline image the resulting image will have an HDR headroom equal to the specified targeted HDR headroom.

This structure is parameterized by:

a real value $H_{baseline}$ representing the baseline HDR headroom
an integer $N_{alt}$ number of alternate images
real values $H_{alt, i}$ for $i \in Z_{N_{alt}}$ representing the alternate HDR headroom of each alternate image
color gain function structures that define the function ${Gain}_{alt, i}$ for each $i \in Z_{N alt}$ representing the color gain function associated with each alternate image
real valued vectors $χ_{red}$ , $χ_{green}$ , $χ_{blue}$ , and $χ_{white}$ , indicating the ISO/CIE 11664-1:2019 CIE 1931 $x y$ chromaticity coordinates from of the red, green, and blue color primaries and white point of the gain application color space

6.4.2 Parameter constraints🔗

The value of $H_{baseline}$ shall be in the interval $[0, 6]$ .

The value of $N_{alt}$ shall be greater than or equal to 0 and less than or equal to 4.

For $i \in Z_{N_{alt}}$ , the value of $H_{alt, i}$ shall be in the interval $[0, 6]$ .

For $i \in Z_{N_{alt}}$ it shall be the case that $H_{alt, i} \neq H_{baseline}$ .

For $i \in Z_{N_{alt} - 1}$ it shall be the case that $H_{alt, i} < H_{alt, i + 1}$ .

The length of the vectors $χ_{red}$ , $χ_{green}$ , $χ_{blue}$ , and $χ_{white}$ shall be 2.

The values of the components of the vectors $χ_{red}$ , $χ_{green}$ , $χ_{blue}$ , and $χ_{white}$ shall be in the interval $[0, 1]$ .

6.4.3 Gain curve constraints (Informative)🔗

This clause is informative.

This clause lists constraints that are useful for achieving coherent headroom-adaptive tone mapping behavior but are not normatively required. Implementations can adjust metadata items so that they satisfy these constraints.

For $i \in Z_{N_{alt}}$ , let ${GainCurve}_{i}$ be the function defined by the gain curve structure parameter to ${Gain}_{alt, i}$ .

When a headroom-adaptive tone mapping is performed with the targeted HDR headroom parameter set equal to an alternate HDR headroom, pixels in the baseline image with values equal to the baseline HDR headroom will preferably be mapped to values equal to that alternate HDR headroom. This is accomplished if for each $i \in Z_{N_{alt}}$ the equality in Formula (8) is true.

{GainCurve}_{i} (2^{H_{baseline}}) = \log_{2} (\frac{2^{H_{alt,i}}}{2^{H_{baseline}}}) = H_{alt,i} - H_{baseline}

(8)

Tone mapping to a targeted HDR headroom lower than the baseline HDR headroom will preferably only decrease the tone mapped result, and tone mapping to a targeted HDR headroom greater than the baseline HDR headroom will preferably only increase the tone mapped result. In symbols, this is accomplished if it is the case that for $i \in Z_{N_{alt}}$ for all $x \in R$ , if $H_{alt, i} < H_{baseline}$ then ${GainCurve}_{i} (x) \leq 0$ and if $H_{alt, i} > H_{baseline}$ then ${GainCurve}_{i} (x) \geq 0$ .

As the targeted HDR headroom of tone mapping increases, the tone mapped result will preferably only increase. In symbols, this is accomplished if it is the case that for $i \in Z_{N_{alt} - 1}$ for all $x \in R$ , ${GainCurve}_{i} (x) \leq {GainCurve}_{i + 1} (x)$ .

The tone mapping curves will preferably be monotonically increasing. In symbols, this is accomplished if it is the case that for $i \in Z_{N_{alt}}$ , the function $f (x) = x \times 2^{{GainCurve}_{i} (x)}$ is monotonically increasing.

6.4.4 Computation of alternate images🔗

When tone mapping to a targeted HDR headroom that is equal to an alternate HDR headroom the result of tone mapping will be the corresponding alternate image. In symbols, for any $i \in Z_{N_{alt}}$ , the function $ToneMap$ shall satisfy the equality in Formula (9).

ToneMap (C, H_{alt, i}) = C \times 2^{{Gain}_{alt, i} (C)}

(9)

The equality in Formula (9) is satisfied by the function defined in Formula (13).

6.4.5 Computation of the headroom-adaptive tone map 🔗

This clause describes the evaluation of the function $ToneMap$ given an input color $C \in R^{3}$ in the gain application color space and a targeted HDR headroom $H_{target} \in R_{\geq 0}$ .

Create a new list of HDR headrooms by inserting the baseline HDR headroom $H_{baseline}$ into the list of alternate HDR headrooms $H_{alt, 0}, \dots, H_{alt, N_{alt} - 1}$ in its sorted position. Let $N = N_{alt} + 1$ be the length of this new list and let $H_{0}, \dots, H_{N - 1}$ be the list. Let ${Gain}_{0}, \dots, {Gain}_{N - 1}$ be the list ${Gain}_{alt, 0}, \dots, {Gain}_{alt, N - 1}$ with the zero color gain function inserted at the position where $H_{baseline}$ was inserted.

NOTE —⁠

These lists are constructed in order to simplify the presentation by reducing the number of special cases. In this formulation the presentation does not need to distinguish between the case of interpolating between the baseline image and an alternate image and the case of interpolating between two alternate images, because they are unified into a single list.

Implementations can avoid explicitly forming these unified lists as an optimization.

A visualization of an example in which $H_{baseline}$ is inserted at the end of the list can be seen in Figure B.1. A visualization of an example in which $H_{baseline}$ is inserted at the beginning of the list can be seen in Figure B.3. A visualization of an example in which $H_{baseline}$ is the only entry in the list (because $N_{alt} = 0$ ) can be seen in Figure B.4.

Let $G \in R^{3}$ be the per-component gain in $\log_{2}$ space. This is computed as a weighted combination of color gain functions evaluated at $C$ . The weighting depends on the value of $H_{target}$ parameter.

If there exists a value $i$ such that $H_{i} = clamp (H_{target}, H_{0}, H_{N - 1})$ then $G$ shall be equivalent to the result of the definition in Formula (10).

$G = {Gain}_{i} (C)$ (10)
Otherwise, there must exist exactly one value $i \in Z_{N - 1}$ such that $H_{i} < H_{target} < H_{i + 1}$ .

Let $w_{i}$ and $w_{i + 1}$ be the weights of the $i$ -th and $i + 1$ -th color gain functions. These weights shall be computed as defined in Formula (11).

$\begin{aligned} w_{i} = \frac{H_{target} - H_{i + 1}}{H_{i} - H_{i + 1}} & w_{i + 1} = \frac{H_{target} - H_{i}}{H_{i + 1} - H_{i}} = 1 - w_{i} \end{aligned}$ (11)

The value of $G$ shall be equivalent to the result of the definition in Formula (12).

$G = w_{i} \times {Gain}_{i} (C) + w_{i + 1} \times {Gain}_{i + 1} (C)$ (12)

The headroom-adaptive tone mapping function shall be equivalent to the definition in Formula (13).

ToneMap (C, H_{target}) = C \times 2^{G}

(13)

6.4.6 Visualization🔗

Figure 3 provides a visualization of the computation of an alternate image as described in 6.4.4.

Figure 3 — Computation of the $i$ -th alternate image

Figure 4 provides a visualization of the evaluation of the $ToneMap$ function as described in 6.4.5.

Figure 4 — Headroom-adaptive tone map computation

6.5 Color volume transform structure🔗

6.5.1 Introduction🔗

A color volume transform structure defines the parameters for the color volume transformation informatively described in Annex A.

A color volume transform structure is parameterized by:

a real value $L_{white}$ representing the luminance of HDR reference white in cd/m²
an optional headroom-adaptive tone map structure defining a function $ToneMap : R^{3} \times R_{\geq 0} \to R^{3}$

6.5.2 Parameter constraints🔗

The value of $L_{white}$ shall be in the interval $(0, 10000]$ .

7 Application constraints🔗

7.1 Metadata set🔗

A metadata set shall comprise exactly one of each of the following:

ApplicationIdentifier
ApplicationVersion
TimeInterval
ProcessingWindow
ColorVolumeTransform

The TimeInterval group shall include exactly one of each of the following metadata items defined in Clause 8 of SMPTE ST 2094-1:2016:

TimeIntervalStart
TimeintervalDuration

The ProcessingWindow group shall include exactly one of each of the following metadata items defined in Clause 9 of SMPTE ST 2094-1:2016:

UpperLeftCorner
LowerRightCorner
WindowNumber

The ColorVolumeTransform group which shall include at most one the following metadata items:

HdrReferenceWhite, which must be present.
HeadroomAdaptiveToneMap, which may be omitted.

The HeadroomAdaptiveToneMap group shall include exactly one of each of the following metadata items:

BaselineHdrHeadroom
NumAlternateImages
AlternateHdrHeadroom[a] for each $a \in {0, . . .,$ NumAlternateImages $- 1}$
ColorGainFunction[a] for each $a \in {0, . . .,$ NumAlternateImages $- 1}$
GainApplicationChromaticities[r] for each $r \in {0, . . ., 7}$

Each ColorGainFunction[a] group shall include exactly one of each of the following metadata items:

ComponentMix[a]
GainCurve[a]

Each ComponentMix[a] group shall include exactly one of each of the following metadata items:

ComponentMixRed[a]
ComponentMixGreen[a]
ComponentMixBlue[a]
ComponentMixMax[a]
ComponentMixMin[a]
ComponentMixComponent[a]

Each GainCurve[a] group shall include exactly one of each of the following metadata items:

GainCurveNumControlPoints[a]
GainCurveControlPointX[a][c] for each $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1}$
GainCurveControlPointY[a][c] for each $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1}$
GainCurveControlPointM[a][c] for each $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1}$

7.2 Metadata set constraints🔗

This clause specifies the constraints on the metadata set from 7.1, and defines the association between the the items of the metadata set and data structures and parameters defined in Clause 6.

The ApplicationIdentifier metadata item shall equal 5.

The ApplicationVersion metadata item shall equal 0.

Thee ProcessingWindow shall be subject to the following constraints.

The WindowNumber metadata item shall equal 0.
The UpperLeftCorner metadata item shall equal [0, 0].
The LowerRightCorner metadadata item shall equal the size of the input image essence in pixels.

The ColorVolumeTransform metadata group specifies a color volume transform structure. The metadata items in this metadata group specify the parameters listed in 6.5.1, and are subject to the constraints listed in 6.5.2.

$L_{white}$ shall be assigned the value HdrReferenceWhite
$ToneMap$ shall be assigned as specified by the headroom-adaptive tone map structure parameterized by the HeadroomAdaptiveToneMap metadata group, if present

The HeadroomAdaptiveToneMap metadata group, if present, specifies a headroom-adaptive tone map structure. The metadata items in this metadata group specify the parameters listed in 6.4.1, and are subject to the constraints listed in 6.4.2.

$H_{baseline}$ shall be assigned the value BaselineHdrHeadroom
$N_{alt}$ shall be assigned the value NumAlternateImages
$H_{alt, a}$ shall be assigned the value AlternateHdrHeadroom[a]
${Gain}_{a}$ shall be assigned as specified by the gain curve structure parameterized by the ColorGainFunction[a] metadata group
$χ_{red}$ shall be assigned the value $[$ GainApplicationChromaticities[0], GainApplicationChromaticities[1] $]$
$χ_{green}$ shall be assigned the value $[$ GainApplicationChromaticities[2], GainApplicationChromaticities[3] $]$
$χ_{blue}$ shall be assigned the value $[$ GainApplicationChromaticities[4], GainApplicationChromaticities[5] $]$
$χ_{white}$ shall be assigned the value $[$ GainApplicationChromaticities[8], GainApplicationChromaticities[7] $]$

The ColorGainFunction[a] metadata group specifies a color gain function structure. The metadata items in this metadata group specify the parameters listed in 6.3.1.

$ComponentMixing$ shall be assigned as specified by the the component mixing function structure parameterized by the ComponentMix[a] metadata group
$GainCurve$ shall be assigned as specified by the the gain curve structure parameterized by the GainCurve[a] metadata group

The ComponentMix[a] metadata group specifies a component mixing function structure. The metadata items in this metadata group specify the parameters listed in 6.2.1, and are subject to the constraints listed in 6.2.2.

$k_{red}$ shall be assigned the value ComponentMixRed[a]
$k_{green}$ shall be assigned the value ComponentMixGreen[a]
$k_{blue}$ shall be assigned the value ComponentMixBlue[a]
$k_{max}$ shall be assigned the value ComponentMixMax[a]
$k_{min}$ shall be assigned the value ComponentMixMin[a]
$k_{component}$ shall be assigned the value ComponentMixComponent[a]

The GainCurve[a] metadata group specifies a gain curve structure. The metadata items in this metadata group specify the parameters listed in 6.1.1, and are subject to the constraints listed in 6.1.2.

$N_{cp}$ shall be assigned the value GainCurveNumControlPoints[a]
$x_{c}$ shall be assigned the value GainCurveControlPointX[a][c]
$y_{c}$ shall be assigned the value GainCurveControlPointY[a][c]
$m_{c}$ shall be assigned the value GainCurveControlPointM[a][c]

Annex A
Computation of color volume transformation (Informative)🔗

A.1 Introduction🔗

This annex describes the computations to apply the color volume transformation to an input image essence. Figure A.1 illustrates this specification's headroom-adaptive tone mapping algorithm in the framework of the Generalized Color Transform Model described in Annex A and visualized in Figure A.1 of SMPTE ST 2094-1:2016.

Figure A.1 — The generalized color volume transform model

The individual stages of Figure A.1 correspond to Figure A.2, followed by Figure 4, followed by Figure A.3.

A.2 Conversion to gain application color space 🔗

This clause describes the computation of the function $ToGainApplicationSpace : R^{3} \to R^{3}$ , which converts an input signal to a color in the gain application color space.

Throughout this clause, the input signal is assumed to have three components corresponding to its red, green, and blue primaries, scaled such that the nominal peak signal is $1$ and the black signal is $0$ . It is also assumed to have transfer characteristics that determine its non-linear encoding and opto-optical transfer function. This corresponds a normalized $R^{'}, G^{'}, B^{'}$ coded signal according to Recommendation ITU-R BT.2100-3. These assumptions are to simplify the presentation, and are not a limitation on the types of input signal that can be used with this color volume transform. Very similar computations can be used for non-normalized input signals (e.g, quantized and limited range signals) and for different coded input signals (e.g, $Y^{'}, C_{B}^{'}, C_{R}^{'}$ or $I, C_{T}, C_{P}$ ).

The computation is visualized in Figure A.2, and represents the "Convert to Gain Application Color Space" block of Figure A.1.

Figure A.2 — Conversion of an input signal to gain application color space processing block. The intermediate results of the RGB components of luminance and color value in the relative linear color space with the signal's color primaries are shown at the bottom of the figure.

Let $E \in R^{3}$ be this input signal value.

NOTE 1 —⁠ Components of $E$ can be outside of the $[0, 1]$ interval. This can occur due to multiple reasons including coding and chroma format conversion. For example, a signal coded using a narrow range representation can intentionally contain values greater than the nominal peak signal or values less than the black signal as mentioned in Note 9a of Recommendation ITU-R BT.2100-3. Further information can be found in Section 2.4 of Report ITU-R BT.2408-8.

NOTE 2 —⁠ Scalar functions applied to vector arguments are applied component-wise.

Let $L_{white}$ be the value of the HDR reference white parameter specified by the HdrReferenceWhite metadata item.

Let $Eotf : R^{3} \to R^{3}$ be the electro-optical transfer function which converts an input signal value to RGB components of luminance in cd/m².

NOTE 3 —⁠ When refering to the luminance of a single color component, it means the luminance of an equivalent achromatic color with all three color components having that same value.

The function $Eotf$ for some common video signal transfer characteristics is as follows.

If the baseline image indicates Recommendation ITU-R BT.1886, Recommendation ITU-R BT.709-6, or Recommendation ITU-R BT.2020-2 transfer characteristics, then let $Eotf$ be defined as in Formula (A.1).

$Eotf (E) = L_{white} \times {(E)}^{2.4}$ (A.1)

NOTE 4 —⁠ For this and other standard dynamic range transfer characteristics, the function $Eotf$ evaluated at the nominal peak signal value $E = 1$ will equal $L_{white}$ . The resulting color value in gain application color space, after applying Formula (A.3), will be $1$ .

NOTE 5 —⁠ An input image essence that originates from video that did not indicate its transfer characteristics is often assumed to have Recommendation ITU-R BT.1886 transfer characteristics.

NOTE 6 —⁠ This formulation is equivalent to the display referred mappings described in Section 5.1.2, Section 5.1.3.1, Figure 5, and Figure 7 of Report ITU-R BT.2408-8, and in MovieLabs Best Practice: SDR to HDR Conversion. When in a context dominated by Recommendation ITU-R BT.2100-3 Hybrid Log-Gamma (HLG) content, some systems may choose to include an extra gamma adjustment step, as described in Section 5.1.3.2, Figure 6, and Figure 8 of Report ITU-R BT.2408-8. Implementations can use the formulation indicated by the content or the application context for this and other standard dynamic range transfer characteristics.
If the baseline image indicates IEC 61966-2-1 sRGB transfer characteristics then let $Eotf$ be defined as in Formula (A.2), which is $L_{white}$ multiplied by the function listed in Equations 5 and 6 of Chapter 5 of IEC 61966-2-1.

$Eotf (E) = {\begin{cases} L_{white} \times (\frac{E}{12.92}) & : E \leq 0.04045 \\ L_{white} \times {(\frac{E + 0.055}{1.055})}^{2.4} & : E > 0.04045 \end{cases}$ (A.2)
If the baseline image indicates Recommendation ITU-R BT.2100-3 Perceptual Quantization (PQ) transfer characteristics then let $Eotf$ be the PQ Reference EOTF as described in Table 4 of Recommendation ITU-R BT.2100-3.
If the baseline image indicates Recommendation ITU-R BT.2100-3 Hybrid Log-Gamma (HLG) transfer characteristics then let $Eotf$ be the HLG Reference EOTF as described in Table 5 of Recommendation ITU-R BT.2100-3, with parameters $L_{W} = 1000$ , $L_{B} = 0$ , and $γ = 1.2$ .

NOTE 7 —⁠ The HLG Reference OOTF in this formulation must be applied using the color primaries indicated in Table 2 of Recommendation ITU-R BT.2100-3.

Let $M_{input,gain}$ be the 3x3 matrix that converts color primaries from the baseline image color primaries to the color primaries of the gain application color space. The matrix is computed using chromatic adaptation (if needed) as defined in Annex E of Specification ICC.1:2022.

The function $ToGainApplicationSpace$ can be computed as as defined in Formula (A.3).

ToGainApplicationSpace (E) = M_{input,gain} \times \frac{1}{L_{white}} \times Eotf (E)

(A.3)

A.3 Headroom-adaptive tone mapping in gain application color space 🔗

This clause describes the computation of the headroom-adaptive tone mapping algorithm.

Let $H_{target} \in R_{\geq 0}$ be the targeted HDR headroom that characterizes the capabilities of target display. This can be computed as $\log_{2}$ of the ratio of the peak luminance of the display to the luminance at which HDR reference white will be displayed, or it can be computed by other means outside of the scope of this specification.

Let $ToneMap : R^{3} \times R_{\geq 0} \to R^{3}$ be defined by the headroom-adaptive tone map structure parameter specified by the HeadroomAdaptiveToneMap metadata group.

Let $C \in R^{3}$ be a color from the baseline image, in the gain application color space. Let $T \in R^{3}$ be the color resulting from applying the headroom-adaptive tone mapping algorithm to that color. The value of $T$ can be computed as defined in Formula (A.4).

T = ToneMap (C, H_{target})

(A.4)

A.4 Conversion from gain application color space to the target display🔗

This clause describes the computation of the function $ToTargetColorVolume : R^{3} \to R^{3}$ , given a tone mapped color $T \in R^{3}$ in the gain application color space. The computation is visualized in Figure A.3, and represents the "Convert to Target Color Volume" block of Figure A.1.

Figure A.3 — Conversion to target display color volume processing block

Let $L_{white,target}$ be the luminance at which HDR reference white will be displayed. Let $M_{gain,target}$ be the 3x3 matrix that converts color primaries from the gain application color space primaries to the target color volume color primaries.

The function $ToTargetColorVolume$ can be computed as defined in Formula (A.5).

ToTargetColorVolume (T) = L_{white,target} \times M_{gain,target} \times T

(A.5)

Displays can perform more complicated transformations from the gain application color space to the target display, for example, to adjust for non-reference ambient viewing conditions. That topic is out of the scope of this specification.

NOTE —⁠ Any colors that do not fit in the targeted display color volume (any colors with components greater than $2^{H_{target}}$ in the relative linear color space with color primaries of the targeted display color volume), can be projected (often by clamping component-wise) to the targeted display color volume.

Annex B
Examples (Informative)🔗

B.1 Simple tone mapping🔗

Figure B.1 illustrates the color gain functions and associated tone mapping functions of an headroom-adaptive tone mapping with a baseline high dynamic range headroom $H_{baseline} = \log_{2} (4) = 2$ , and a standard dynamic range alternate image with $H_{alt, 0} = \log_{2} (1) = 0$ .

On the left side of Figure B.1 are the underlying color gain functions in the $\log_{2}$ domain. The zero color gain function inserted corresponding to the baseline image is indicated with a solid line. The alternate image's color gain function is indicated with a dashed line. The control points for the gain curve parameter of the color gain function are indicated using filled circles.

On the right side of Figure B.1 are the tone mapping curves that are computed from the color gain functions. The zero color gain function results in an identity tone mapping function. The alternate image's color gain function results in a tone mapping curve that maps $2^{H_{baseline}} = 4$ to $2^{H_{altr, 0}} = 1$ , which corresponds to the standard dynamic range version. The mapping of the color gain function's gain curve's control points are indicated using empty circles.

Adaptation to a different targeted HDR headroom of $H_{target} = \log_{2} (2) = 1$ is indicated using a dotted line. Its adapted color gain function is computed as a weighted averaging of the color gain functions on the left. The weighting parameters are $w_{0} = w_{1} = \frac{1}{2}$ . The resulting adapted tone mapping curve on the right maps $2^{H_{baseline}} = 4$ to $2^{H_{target}} = 2$ .

Figure B.1 — Simple headroom-adaptive tone mapping from high dynamic range to standard dynamic range

B.2 Tone mapping with two alternate images🔗

Figure B.2 illustrates a slightly more complicated example of headroom-adaptive tone mapping with an additional alternate image with $H_{alt, 1} = \log_{2} (2) = 1$ . This new alternate image indicates that, when tone mapping to a targeted HDR headroom of $1$ , the color values below 1 in the relative linear color space (signal values below HDR reference white) are preserved and that highlights (signal values above HDR reference white) are to be compressed more aggressively.

An example adaptation to a targeted HDR headroom of $H_{target} = \log_{2} (1.5) \approx 0.58$ is included. The adapted color gain function in this instance is a weighted averaging of the two alternate images' color gain functions. The weights in this example are $w_{0} \approx 0.42$ and $w_{1} \approx 0.58$ . The resulting adapted tone mapping curve on the right maps $2^{H_{baseline}} = 4$ to $2^{H_{target}} = 1.5$ .

Figure B.2 — Headroom-adaptive tone mapping from high dynamic range to standard dynamic range with two alternate images

B.3 Inverse tone mapping🔗

Figure B.3 illustrates an example of inverse tone mapping with a standard dynamic range baseline image with $H_{baseline} = \log_{2} (1) = 0$ , and a high dynamic range alternate image with $H_{alt, 0} = \log_{2} (4) = 2$ .

This example is the inverse of the transformation performed in the example in Clause B.1. That is to say that a baseline image set to the alternate image from the example in Clause B.1 (the tone mapping to standard dynamic range), with this metadata, would appear identical to the example in Clause B.1, up to any quantization errors.

Note in the legend that in this example $H_{1} = H_{alt, 0}$ because $H_{baseline}$ is inserted at the beginning of the merged list of HDR headrooms in 6.4.5.

Adaptation to the targeted HDR headroom of $H_{target} = \log_{2} (2) = 1$ is indicated using a dotted line, as was also done in the example in Clause B.1.

Figure B.3 — Headroom-adaptive tone mapping from standard dynamic range to high dynamic range

B.4 No tone mapping (projection or clamping)🔗

The way to indicate that no tone mapping is to be performed, and that the baseline image is to be projected (clamped) to the target color volume, regardless of the targeted HDR headroom, is to specify that the number of alternate images is 0. This clause illustrates this behavior by working through the details of the computation.

In the context of the headroom-adaptive tone map computation described in in 6.4.5, this means that $N_{alt} = 0$ and therefore $N = 1$ . The only entry in the constructed list of HDR headrooms will be $H_{0} = H_{baseline}$ , and the only entry in the constructed list of color gain functions will be $G_{0}$ which will equal the zero color gain function. The value of $G$ will be set by Formula (10) and will be $G = [0, 0, 0]$ . Finally, the function $ToneMap$ as defined in Formula (13) will evaluate to be the identity, as worked out in Formula (B.1).

ToneMap (C, H_{target}) = C \times 2^{G} = C \times 2^{0} = C

(B.1)

This means that the image resulting from the headroom-adaptive tone mapping in the color volume transform step in Clause A.3 will be the unchanged baseline image. After this, the image will be converted to the target display as described in Clause A.4, and the resulting image will be projected to the targeted color volume, as described in Clause A.4, Note. The result of this operation will be to clamp the baseline image to the targeted HDR headroom.

Figure B.4 illustrates this case, where the color gain function is always equal to the zero color gain function and the headroom-adaptive tone mapping is the identity.

Figure B.4 — Headroom-adaptive tone mapping from high dynamic range to standard dynamic range with zero alternate images

Figure B.5 illustrates the result of this operation described in Clause A.4, Note, which is to project (or clamp) the baseline image to the targeted HDR headroom.

Figure B.5 — Projection of the baseline image to target color volumes with various targeted HDR headrooms

Annex C
Binary encoding (Normative)🔗

C.1 Introduction🔗

The metadata for Application #5 shall be stored in binary form according to the syntax structure in Clause C.2. The assignment of values to the items of the metadata set based on the syntax elements is in Clause C.3.

The syntax structures are presented in a tabular form. The type of each syntax element in the bitstream is indicated in the Type column. A syntax element with type u(N) is stored as an N-bit unsigned integer. The bits of each syntax element shall be read in order from most significant to least significant.

NOTE —⁠ All syntax elements that are one full byte or larger are byte aligned. For syntax elements that are 16-bit unsigned integers, this representation is equivalent to storing those syntax elements in big-endian form.

Syntax elements with the name reserved_zero shall be equal to zero.

C.2 Syntax🔗

C.2.1 Application syntax🔗

Table C.1 presents the syntax structures related to Application #5.

The smpte_st_2094_50_application_info metadata bitstream may have padding after the end of the structure. Parsers shall stop reading after the last recognized syntax element of smpte_st_2094_50_application_info and ignore all remaining data.

Table C.1 — Application #5 syntax structure
smpte_st_2094_50_application_info() {	Type
application_version	u(8)
smpte_st_2094_50_color_volume_transform()
}

C.2.2 Color volume transform syntax🔗

Table C.2 presents the syntax for specifying the ColorVolumeTransform metadata group.

Table C.2 — Color volume transform syntax structure
smpte_st_2094_50_color_volume_transform() {	Type
has_custom_hdr_reference_white_flag	u(1)
has_adaptive_tone_map_flag	u(1)
reserved_zero	u(6)
if (has_custom_hdr_reference_white_flag == 1) {
hdr_reference_white	u(16)
}
if (has_adaptive_tone_map_flag == 1) {
smpte_st_2094_50_adaptive_tone_map()
}
}

C.2.3 Adaptive tone mapping syntax🔗

Table C.3 presents the syntax for specifying the HeadroomAdaptiveToneMap metadata group.

Table C.3 — Headroom-adaptive tone map structure syntax
smpte_st_2094_50_adaptive_tone_map() {	Type
baseline_hdr_headroom	u(16)
use_reference_white_tone_mapping_flag	u(1)
if (use_reference_white_tone_mapping_flag == 0) {
num_alternate_images	u(3)
gain_application_space_chromaticities_flag	u(2)
has_common_component_mix_params_flag	u(1)
has_common_curve_params_flag	u(1)
if (gain_application_space_chromaticities_flag == 3) {
for (r = 0; r < 8; r++) {
gain_application_space_chromaticities[r]	u(16)
}
}
for (a = 0; a < min(num_alternate_images, 4); a++) {
alternate_hdr_headrooms[a]	u(16)
smpte_st_2094_50_component_mixing(a)
smpte_st_2094_50_gain_curve(a)
}
} else {
reserved_zero	u(7)
}
}

C.2.4 Component mixing syntax🔗

Table C.4 presents the syntax for specifying a ComponentMix metadata group.

Table C.4 — Component mixing function structure syntax
smpte_st_2094_50_component_mixing(a) {	Type
if (a == 0 \|\| has_common_component_mix_params_flag == 0) {
component_mixing_type[a]	u(2)
if (component_mixing_type[a] != 3) {
reserved_zero	u(6)
} else {
for (k = 0; k < 6; k++) {
has_component_mixing_coefficient_flag[a][k]	u(1)
}
for (k = 0; k < 6; k++) {
if (has_component_mixing_coefficient_flag[a][k] == 1) {
component_mixing_coefficient[a][k]	u(16)
} else {
component_mixing_coefficient[a][k] = 0
}
}
}
} else {
component_mixing_type[a] = component_mixing_type[0]
if (component_mixing_type[a] == 3) {
for (k = 0; k < 6; k++) {
component_mixing_coefficient[a][k] = component_mixing_coefficient[0][k]
}
}
}
}

C.2.5 Gain curve syntax🔗

Table C.5 presents the syntax for specifying a GainCurve metadata group.

Table C.5 — Gain curve structure syntax
smpte_st_2094_50_gain_curve(a) {	Type
if (a == 0 \|\| has_common_curve_params_flag == 0) {
gain_curve_num_control_points_minus_1[a]	u(5)
gain_curve_use_pchip_slope_flag[a]	u(1)
reserved_zero	u(2)
for (c = 0; c < gain_curve_num_control_points_minus_1[a] + 1; c++) {
gain_curve_control_points_x[a][c]	u(16)
}
} else {
gain_curve_use_pchip_slope_flag[a] = gain_curve_use_pchip_slope_flag[0]
gain_curve_num_control_points_minus_1[a] = gain_curve_num_control_points_minus_1[0]
for (c = 0; c < gain_curve_num_control_points_minus_1[a] + 1; c++) {
gain_curve_control_points_x[a][c] = gain_curve_control_points_x[0][c]
}
}
for (c = 0; c < gain_curve_num_control_points_minus_1[a] + 1; c++) {
gain_curve_control_points_y[a][c]	u(16)
}
if (gain_curve_use_pchip_slope_flag[a] == 0) {
for (c = 0; c < gain_curve_num_control_points_minus_1[a] + 1; c++) {
gain_curve_control_points_theta[a][c]	u(16)
}
}
}

C.3 Semantics🔗

C.3.1 Introduction🔗

This clause describes how to assign values to the metadata items described in 7.1 given the syntax elements from Clause C.2.

C.3.2 Application semantics🔗

The ApplicationIdentifier metadata item shall be assigned the value 5.

The ApplicationVersion metadata item shall be assigned the value application_version.

The ProcessingWindow metadata group shall be assigned values as follows.

The WindowNumber metadadata item shall be assigned the value 0.
The UpperLeftCorner metadadata item shall be assigned the value $[0, 0]$ .
The LowerRightCorner metadadata item shall be assigned the value of the size of the input image essence in pixels.

The ColorVolumeTransform metadata group shall be assigned values as specified in C.3.3.

C.3.3 Color volume transform semantics🔗

If has_custom_hdr_reference_white_flag==0, then:

The HdrReferenceWhite metadata item shall be assigned the value 203.

If has_custom_hdr_reference_white_flag==1, then:

The HdrReferenceWhite metadata item shall be assigned the value $clamp ($ hdr_reference_white $, 1, 50000) \times \frac{1}{5}$ .

If has_adaptive_tone_map_flag==0, then:

The HeadroomAdaptiveToneMap metadata group shall be absent.

If has_adaptive_tone_map_flag==1, then:

The HeadroomAdaptiveToneMap metadata group shall be present and shall have values assigned as specified in C.3.4.

C.3.4 Adaptive tone mapping semantics🔗

The BaselineHdrHeadroom metadata item shall be assigned the value $min ($ baseline_hdr_headroom $, 60000) \times \frac{1}{10000}$ .

If use_reference_white_tone_mapping_flag==0, then:

The GainApplicationChromaticities metadata item shall be assigned values as specified in C.3.5.
The NumAlternateImages metadata item shall be assigned the value $min ($ num_alternate_images $, 4)$ .
For each value $a \in {0, . . .,$ NumAlternateImages $- 1}$ :
- The AlternateHdrHeadroom[a] metadata item shall be assigned the value $min ($ alternate_hdr_headrooms[a] $, 60000) \times \frac{1}{10000}$ .
- The ColorGainFunction[a] metadata group shall be assigned values as follows.
  - The ComponentMix[a] metadata group shall be assigned values as specified in C.3.6.
  - The GainCurve[a] metadata group shall be assigned values as specified in C.3.7.

If use_reference_white_tone_mapping_flag==1, then:

The remaining items in the HeadroomAdaptiveToneMap metadata group shall be assigned values as described in C.3.8.

C.3.5 Gain application color space chromaticity semantics🔗

The GainApplicationChromaticities, metadata item shall be assigned a value as follows.

If gain_application_space_chromaticities_flag==0, then:

The GainApplicationChromaticities metadata item shall be assigned the value $[0.64, 0.33, 0.3, 0.6, 0.15, 0.06, 0.3127, 0.329]$
This corresponds to the color primaries and white point specified in Section 1 of Recommendation ITU-R BT.709-6.

If gain_application_space_chromaticities_flag==1, then:

The GainApplicationChromaticities metadata item shall be assigned the value $[0.68, 0.32, 0.265, 0.69, 0.15, 0.06, 0.3127, 0.329]$
This corresponds to the color primaries specified in Table 1 of SMPTE ST 2113:2018 and the D65 white point specified in Clause 6.2.

If gain_application_space_chromaticities_flag==2, then

The GainApplicationChromaticities metadata item shall be assigned the value $[0.708, 0.292, 0.17, 0.797, 0.131, 0.046, 0.3127, 0.329]$
This corresponds to the color primaries and white point specified in Table 3 of Recommendation ITU-R BT.2020-2.

If gain_application_space_chromaticities_flag==3, then:

For each $r \in Z_{8}$ , the GainApplicationChromaticities[r] metadata item shall be assigned the value $χ_{r} = min ($ gain_application_space_chromaticities[r] $, 50000) \times \frac{1}{50000}$ .

C.3.6 Component mixing semantics🔗

The ComponentMix[a] metadata group shall be assigned values as follows.

If component_mixing_type[a]==0 then:

The ComponentMixMax[a] metadata item shall be assigned the value 1.
The ComponentMixRed[a], ComponentMixGreen[a], ComponentMixBlue[a], ComponentMixMin[a], and ComponentMixComponent[a] metadata items shall be assigned the value 0.

If component_mixing_type[a]==1 then:

The ComponentMixComponent[a] metadata item shall be assigned the value 1.
The ComponentMixRed[a], ComponentMixGreen[a], ComponentMixBlue[a], ComponentMixMax[a], and ComponentMixMin[a] metadata items shall be assigned the value 0.

If component_mixing_type[a]==2 then:

The ComponentMixRed[a], ComponentMixGreen[a], and ComponentMixBlue[a] metadata items shall be assigned the value $\frac{1}{6}$ .
The ComponentMixMax[a] metadata item shall be assigned the value $\frac{1}{2}$ .
The ComponentMixMin[a] and ComponentMixComponent[a] metadata items shall be assigned the value 0.

If component_mixing_type[a]==3 then:

For $k \in {0, . . ., 5}$ , let $p_{k} = min ($ component_mixing_coefficient[a][k] $, 50000) \times \frac{1}{50000}$ .
The ComponentMixRed[a], ComponentMixGreen[a], ComponentMixBlue[a], ComponentMixMax[a], ComponentMixMin[a], and ComponentMixComponent[a] metadata items shall be assigned the values $p_{0}$ , $p_{1}$ , $p_{2}$ , $p_{3}$ , $p_{4}$ , and $p_{5}$ respectively.

C.3.7 Gain curve semantics🔗

The GainCurve[a] metadata group shall be assigned values as follows.

The GainCurveNumControlPoints[a] metadata item shall be assigned the value gain_curve_num_control_points_minus_1[a] $+ 1$ .

For each value $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1)$ :

The GainCurveControlPointX[a][c] metadata item shall be assigned the value $min ($ gain_curve_control_points_x[a][c] $, 64000) \times \frac{1}{1000}$ .
The GainCurveControlPointY[a][c] metadata item shall be assigned the value $min ($ gain_curve_control_points_y[a][c] $, 48000) \times \frac{1}{4000} - 6$ .

If gain_curve_use_pchip_slope_flag[a]==0, then:

For each value $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1)$ :
- Let $θ_{a, c} =$ gain_curve_control_points_theta[a][c].
- The GainCurveControlPointM[a][c] metadata item shall be assigned the value $\tan (clamp (θ_{a, c}, 1, 35999) \times \frac{π}{36000} - \frac{π}{2})$ .

If gain_curve_use_pchip_slope_flag[a]==1, then:

For each value $c \in {0, . . .,$ GainCurveNumControlPoints[a] $- 1)$ , the metadata item GainCurveControlPointM[a][c] shall be assigned a value following the computation specified in C.3.9.

C.3.8 Reference white adaptive tone mapping computation🔗

This clause specifies the metadata item values to assign in the HeadroomAdaptiveToneMap metadata group when the use_reference_white_tone_mapping_flag syntax element indicates that the parameters for headroom-adaptive tone mapping should be computed (as opposed to being specified).

NOTE —⁠ The results of this computation are equivalent to the Reference White Tone Mapping Operator described in ISO/TC 22028-5:2023.

Let $H_{baseline} =$ BaselineHdrHeadroom.

The GainApplicationChromaticities, metadata item shall be assigned the value $[0.708, 0.292, 0.17, 0.797, 0.131, 0.046, 0.3127, 0.329]$ . This corresponds to the color primaries and white point specified in Table 3 of Recommendation ITU-R BT.2020-2.

If $H_{baseline} = 0$ then:

The NumAlternateImages metadata item shall be assigned the value 0.

If $H_{baseline} > 0$ then:

The NumAlternateImages metadata item shall be assigned the value 2.
The AlternateHdrHeadroom[0] metadata item shall be assigned the value $H_{alt, 0} = 0$ .
The AlternateHdrHeadroom[1] metadata item shall be assigned the value $H_{alt, 1}$ as defined in Formula (C.1).

$H_{alt, 1} = \log_{2} (\frac{8}{3}) \times min (\frac{H_{baseline}}{\log_{2} (1000 / 203)}, 1)$ (C.1)

For $a \in {0, . . .,$ NumAlternateImages $- 1}$ , the ColorGainFunction[a] metadata group shall be assigned values as follows.

The ComponentMix[a] metadata group shall be assigned values as follows.
- The ComponentMixMax[a] metadata item shall be assigned the value 1.
- The ComponentMixRed[a], ComponentMixGreen[a], ComponentMixBlue[a], ComponentMixMin[a], and ComponentMixComponent[a] metadata items shall be assigned the value 0.
The GainCurve[a] metadata group shall be assigned values as follows.
- Let $N_{cp} = 8$ .
  
  The GainCurveNumControlPoints[a] metadata item shall be assigned the value $N_{cp}$ .
- Let $y_{white, 0}$ be as defined in Formula (C.2).
  
  $y_{white, 0} = 1 - \frac{1}{2} \times min (\frac{H_{baseline}}{\log_{2} (1000 / 203)}, 1)$ (C.2)
  
  Let $y_{white, 1} = 1$ .
  
  The control point values will be defined by the functions $G_{x}, G_{y}, G_{m} : [0, 1] \times R^{4} \times [0, 1] \to R$ , defined in Formula (C.6). These functions create a gain curve based on a knee in relative linear color space, which will be at the point $[1, y_{white, a}]$ .
  
  For each value $c \in Z_{N_{_{cp}}}$ :
  - The GainCurveControlPointX[a][c] metadata item shall be assigned the value $G_{x} (\frac{c}{N_{cp} - 1}, 1, y_{white, a}, 2^{H_{baseline}}, 2^{H_{alt, a}}, 0.65)$ .
  - The GainCurveControlPointY[a][c] metadata item shall be assigned the value $G_{y} (\frac{c}{N_{cp} - 1}, 1, y_{white, a}, 2^{H_{baseline}}, 2^{H_{alt, a}}, 0.65)$ .
  - The GainCurveControlPointM[a][c] metadata item shall be assigned the value $G_{m} (\frac{c}{N_{cp} - 1}, 1, y_{white, a}, 2^{H_{baseline}}, 2^{H_{alt, a}}, 0.65)$ .

The functions $G_{x}, G_{y}, G_{m}$ define a parametric gain curve and its slope, given a knee point $[x_{knee}, y_{knee}] \in R^{2}$ , a maximum point $[x_{max}, y_{max}] \in R^{2}$ , and a highlight compression factor $κ \in [0, 1]$ .

The gain curve is $\log_{2}$ of the gain of a tone curve defined by a 2D Bézier in relative linear color space, visualized on the left in Figure C.1. The control points of the Bézier curve are $[x_{knee}, y_{knee}]$ , $[x_{mid}, y_{mid}]$ , and $[x_{max}, y_{max}]$ . The middle Bézier control point $[x_{mid}, y_{mid}]$ is computed using the highlight compression factor $κ$ in Formula (C.3).

\begin{aligned} x_{mid} & = (1 - κ) \times x_{knee} + κ \times (\frac{x_{knee} \times y_{max}}{y_{knee}}) \\ y_{mid} & = (1 - κ) \times y_{knee} + κ \times y_{max} \end{aligned}

(C.3)

The parametric Bézier curve and its slope are given by the scalar functions $x, y, m : [0, 1] \to R$ defined in Formula (C.4).

\begin{aligned} x (t) & = a_{x} \times t^{2} + b_{x} \times t + c_{x} \\ y (t) & = a_{y} \times t^{2} + b_{y} \times t + c_{y} \\ m (t) & = \frac{2 \times a_{y} \times t + b_{y}}{2 \times a_{x} \times t + b_{x}} \end{aligned}

(C.4)

The quadratic parameters $a_{x}, b_{x}, c_{x} \in R$ and $a_{y}, b_{y}, c_{y} \in R$ in Formula (C.4) are defined in Formula (C.5).

\begin{aligned} a_{x} & = x_{knee} - 2 \times x_{mid} + x_{max} & b_{x} & = 2 \times x_{mid} - 2 \times x_{knee} & c_{x} & = x_{knee} \\ a_{y} & = y_{knee} - 2 \times y_{mid} + y_{max} & b_{y} & = 2 \times y_{mid} - 2 \times y_{knee} & c_{y} & = y_{knee} \end{aligned}

(C.5)

Finally, the functions $G_{x}, G_{y}, G_{m}$ are defined in Formula (C.6) by computing $\log_{2}$ of the gain of the tone curve defined by the functions $x, y, m$ .

\begin{aligned} G_{x} (t, x_{knee}, y_{knee}, x_{max}, y_{max}, κ) & = x (t) \\ G_{y} (t, x_{knee}, y_{knee}, x_{max}, y_{max}, κ) & = \log_{2} (\frac{y (t)}{x (t)}) \\ G_{m} (t, x_{knee}, y_{knee}, x_{max}, y_{max}, κ) & = \frac{x (t) \times m (t) - y (t)}{\log (2) \times x (t) \times y (t)} \end{aligned}

(C.6)

Figure C.1 illustrates the interpretation of the equations in this clause. The Bézier control points in Formula (C.3) and the resulting Bézier curve from Formula (C.4) are displayed on the left. The resulting gain curve from Formula (C.6) is displayed on the right. The extrapolation behavior beyond the control points, specified by Formula (2), is indicated in both plots.

Figure C.1 — Illustration of Formula (C.6) and its components Formula (C.3) and Formula (C.4) with example parameters $x_{knee} = 1$ , $y_{knee} = 0.8$ , $x_{max} = 4$ , $y_{max} = 2$ , and $κ = 0.65$ .

C.3.9 Piecewise cubic hermite interpolation package slope computation🔗

This clause specifies, for a given value of $a$ , the metadata item values to assign in the GainCurve[a] metadata group when the gain_curve_use_pchip_slope_flag syntax element indicates that the slopes of the $a$ th gain curve's control points should be computed (as opposed to being specified).

NOTE —⁠ The results of this computation are equivalent to the Piecewise Cubic Hermite Interpolation Package (PCHIP) algorithm described in DOI:10.1137/0905021.

Let $N_{cp} =$ GainCurveNumControlPoints[a]. For each $i \in Z_{N_{_{cp}}}$ , let $x_{i} =$ GainCurveControlPointX[a][c] and let $y_{i} =$ GainCurveControlPointY[a][c].

For $i \in Z_{N_{cp} - 1}$ , let $h_{i} = x_{i + 1} - x_{i}$ be the length of the $i$ -th interval and let $s_{i} = \frac{y_{i + 1} - y_{i}}{x_{i + 1} - x_{i}}$ be the slope of the line connecting its control points.

Let $m_{0}$ be defined by Formula (C.7) if $N_{cp} \geq 3$ . Otherwise if $N_{cp} = 2$ then let $m_{0} = s_{0}$ and if $N_{cp} = 1$ then $m_{0}$ will not be used.

m_{0} = \frac{(2 \times h_{0} + h_{1}) \times s_{0} - h_{0} \times s_{1}}{h_{0} + h_{1}}

(C.7)

Let $m_{N_{cp} - 1}$ be defined by Formula (C.8) if $N_{cp} \geq 3$ . Otherwise if $N_{cp} = 2$ then let $m_{_{N_{cp} - 1}} = s_{0}$ .

m_{N_{cp} - 1} = \frac{(2 \times h_{N_{cp} - 2} + h_{N_{cp} - 3}) \times s_{N_{cp} - 2} - h_{N_{cp} - 2} \times s_{N_{cp} - 3}}{h_{N_{cp} - 2} + h_{N_{cp} - 3}}

(C.8)

For $i \in {1, . . ., N_{cp} - 2}$ , let $m_{i}$ be defined by Formula (C.9).

m_{i} = {\begin{cases} 0 & : sign (s_{i - 1}) \neq sign (s_{i}) \\ \frac{3 \times (h_{i - 1} + h_{i}) \times s_{i - 1} \times s_{i}}{(2 \times h_{i - 1} + h_{i}) \times s_{i - 1} + (h_{i - 1} + 2 \times h_{i}) \times s_{i}} & : otherwise \end{cases}

(C.9)

For each $c \in Z_{N_{_{cp}}}$ , GainCurveControlPointM[a][c] metadata item shall be assigned the value $m_{c}$ .

metadata_itu_t_t35() {	Type
itu_t_t35_country_code	u(8)
if (itu_t_t35_country_code == 0xB5) {
itu_t_t35_us_terminal_provider_code	u(16)
if (itu_t_t35_us_terminal_provider_code == 0x0090) {
itu_t_t35_smpte_terminal_provider_oriented_code	u(16)
if (itu_t_t35_smpte_terminal_provider_oriented_code == 0x0001) {
smpte_st_2094_50_application_info()
} else {
// Handling of other terminal provider oriented codes
}
} else {
// Handling of other terminal provider codes
}
} else {
// Handling of other country codes
}
}

Dynamic metadata for color volume transform — Application #5

Table of contents🔗

Foreword🔗

Introduction🔗

1 Scope🔗

2 Conformance🔗

3 Normative references🔗

4 Terms and definitions🔗

5 Mathematical notation🔗