Patent Publication Number: US-8971408-B2

Title: Piecewise cross color channel predictor

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present disclosure may also be related to U.S. Provisional Application Ser. No. 61/475,359, filed on Apr. 14, 2011, titled “Multiple color channel multiple regression predictor”, which was filed also as PCT Application Ser. No. PCT/US2012/033605 on 13 Apr. 2012, and is incorporated herein by reference in its entirety. This application claims priority to U.S. Provisional Patent Application Ser. No. 61/590,175, filed 24 Jan. 2012, hereby incorporated by reference in its entirety. 
    
    
     TECHNOLOGY 
     The present invention relates generally to images. More particularly, an embodiment of the present invention relates to a piecewise cross color channel predictor of high dynamic range images using standard dynamic range images. 
     BACKGROUND 
     As used herein, the term ‘dynamic range’ (DR) may relate to a capability of the human psychovisual system (HVS) to perceive a range of intensity (e.g., luminance, luma) in an image, e.g., from darkest darks to brightest brights. In this sense, DR relates to a ‘scene-referred’ intensity. DR may also relate to the ability of a display device to adequately or approximately render an intensity range of a particular breadth. In this sense, DR relates to a ‘display-referred’ intensity. Unless a particular sense is explicitly specified to have particular significance at any point in the description herein, it should be inferred that the term may be used in either sense, e.g. interchangeably. 
     As used herein, the term high dynamic range (HDR) relates to a DR breadth that spans the some 14-15 orders of magnitude of the human visual system (HVS). For example, well adapted humans with essentially normal (e.g., in one or more of a statistical, biometric or opthalmological sense) have an intensity range that spans about 15 orders of magnitude. Adapted humans may perceive dim light sources of as few as a mere handful of photons. Yet, these same humans may perceive the near painfully brilliant intensity of the noonday sun in desert, sea or snow (or even glance into the sun, however briefly to prevent damage). This span though is available to ‘adapted’ humans, e.g., those whose HVS has a time period in which to reset and adjust. 
     In contrast, the DR over which a human may simultaneously perceive an extensive breadth in intensity range may be somewhat truncated, in relation to HDR. As used herein, the terms ‘visual dynamic range’ or ‘variable dynamic range’ (VDR) may individually or interchangeably relate to the DR that is simultaneously perceivable by a HVS. As used herein, VDR may relate to a DR that spans 5-6 orders of magnitude. Thus while perhaps somewhat narrower in relation to true scene referred HDR, VDR nonetheless represents a wide DR breadth. As used herein, the term ‘simultaneous dynamic range’ may relate to VDR. 
     Until fairly recently, displays have had a significantly narrower DR than HDR or VDR. Television (TV) and computer monitor apparatus that use typical cathode ray tube (CRT), liquid crystal display (LCD) with constant fluorescent white back lighting or plasma screen technology may be constrained in their DR rendering capability to approximately three orders of magnitude. Such conventional displays thus typify a low dynamic range (LDR), also referred to as a standard dynamic range (SDR), in relation to VDR and HDR. 
     Advances in their underlying technology however allow more modern display designs to render image and video content with significant improvements in various quality characteristics over the same content, as rendered on less modern displays. For example, more modern display devices may be capable of rendering high definition (HD) content and/or content that may be scaled according to various display capabilities such as an image scaler. Moreover, some more modern displays are capable of rendering content with a DR that is higher than the SDR of conventional displays. 
     For example, some modern LCD displays have a backlight unit (BLU) that comprises a light emitting diode (LED) array. The LEDs of the BLU array may be modulated separately from modulation of the polarization states of the active LCD elements. This dual modulation approach is extensible (e.g., to N-modulation layers wherein N comprises an integer greater than two), such as with controllable intervening layers between the BLU array and the LCD screen elements. Their LED array based BLUs and dual (or N-) modulation effectively increases the display referred DR of LCD monitors that have such features. 
     Such “HDR displays” as they are often called (although actually, their capabilities may more closely approximate the range of VDR) and the DR extension of which they are capable, in relation to conventional SDR displays represent a significant advance in the ability to display images, video content and other visual information. The color gamut that such an HDR display may render may also significantly exceed the color gamut of more conventional displays, even to the point of capably rendering a wide color gamut (WCG). Scene related HDR or VDR and WCG image content, such as may be generated by “next generation” movie and TV cameras, may now be more faithfully and effectively displayed with the “HDR” displays (hereinafter referred to as ‘HDR displays’). 
     As with the scalable video coding and HDTV technologies, extending image DR typically involves a bifurcate approach. For example, scene referred HDR content that is captured with a modern HDR capable camera may be used to generate an SDR version of the content, which may be displayed on conventional SDR displays. In one approach, generating the SDR version from the captured VDR version may involve applying a global tone mapping operator (TMO) to intensity (e.g., luminance, luma) related pixel values in the HDR content. In a second approach, as described in Patent Application PCT/US2011/048861 “Extending Image Dynamic Range”, by W. Gish et al., herein incorporated by reference for all purposes, generating an SDR image may involve applying an invertible operator (or predictor) on the VDR data. To conserve bandwidth or for other considerations, transmission of both of the actual captured VDR content and a corresponding SDR version may not be a best approach. 
     Thus, an inverse tone mapping operator (iTMO), inverted in relation to the original TMO, or an inverse operator in relation to the original predictor, may be applied to the SDR content version that was generated, which allows a version of the VDR content to be predicted. The predicted VDR content version may be compared to originally captured HDR content. For example, subtracting the predicted VDR version from the original VDR version may generate a residual image. An encoder may send the generated SDR content as a base layer (BL), and package the generated SDR content version, any residual image, and the iTMO or other predictors as an enhancement layer (EL) or as metadata. 
     Sending the EL and metadata, with its SDR content, residual and predictors, in a bitstream typically consumes less bandwidth than would be consumed in sending both the HDR and SDR contents directly into the bitstream. Compatible decoders that receive the bitstream sent by the encoder may decode and render the SDR on conventional displays. Compatible decoders however may also use the residual image, the iTMO predictors, or the metadata to compute a predicted version of the HDR content therefrom, for use on more capable displays. It is the purpose of this invention to provide novel methods for generating predictors that allow for the efficient coding, transmission, and decoding of VDR data using corresponding SDR data. 
     The approaches described in this section are approaches that could be pursued, but not necessarily approaches that have been previously conceived or pursued. Therefore, unless otherwise indicated, it should not be assumed that any of the approaches described in this section qualify as prior art merely by virtue of their inclusion in this section. Similarly, issues identified with respect to one or more approaches should not assume to have been recognized in any prior art on the basis of this section, unless otherwise indicated. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       An embodiment of the present invention is illustrated by way of example, and not in way by limitation, in the figures of the accompanying drawings and in which like reference numerals refer to similar elements and in which: 
         FIG. 1  depicts an example data flow for a VDR-SDR system, according to an embodiment of the present invention; 
         FIG. 2  depicts an example VDR encoding system according to an embodiment of the present invention; 
         FIG. 3  depicts an example piecewise cross-color channel prediction process according to an embodiment of the present invention; 
         FIG. 4  depicts an example image decoder with a predictor operating according to embodiments of this invention. 
     
    
    
     DESCRIPTION OF EXAMPLE EMBODIMENTS 
     Piecewise cross-color channel prediction is described herein. Given a pair of corresponding VDR and SDR images, that is, images that represent the same scene but at different levels of dynamic range, this section describes methods that allow an encoder to approximate the VDR image in terms of the SDR image and a piecewise cross-color channel (PCCC) predictor. In the following description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, that the present invention may be practiced without these specific details. In other instances, well-known structures and devices are not described in exhaustive detail, in order to avoid unnecessarily occluding, obscuring, or obfuscating the present invention. 
     Overview 
     Example embodiments described herein relate to coding images with high dynamic range. In one embodiment, a sequence of visual dynamic range (VDR) images may be encoded using a standard dynamic range (SDR) base layer and one or more enhancement layers. A prediction image is generated by using piecewise cross-color channel prediction (PCCC), where a color channel in the SDR input may be segmented into two or more color channel segments and each segment is assigned its own cross-color channel predictor to output a predicted VDR image. PCCC prediction models for each segment may include first order, second order, or higher order parameters. Using a minimum mean-square error criterion, a closed form solution is presented for the prediction parameters for a second-order PCCC model. Algorithms for segmenting the color channels into multiple color channel segments are also presented. Prediction-related parameters may be transmitted to a decoder using ancillary data, such as metadata. 
     In another embodiment, a decoder accesses a base SDR layer, a residual layer, and metadata related to PCCC prediction modeling. The decoder generates an output prediction image using the base layer and the PCCC prediction parameter, which may be used together with the residual layer to generate an output VDR image. 
     Example VDR-SDR System 
       FIG. 1  depicts an example data flow in a VDR-SDR system  100 , according to an embodiment of the present invention. An HDR image or video sequence is captured using HDR camera  110  or other similar means. Following capture, the captured image or video is processed by a mastering process to create a target VDR image  125 . The mastering process may incorporate a variety of processing steps, such as: editing, primary and secondary color correction, color transformation, and noise filtering. The VDR output  125  of this process typically represents the director&#39;s intent on how the captured image will be displayed on a target VDR display. 
     The mastering process may also output a corresponding SDR image  145 , representing the director&#39;s intent on how the captured image will be displayed on a legacy SDR display. The SDR output  145  may be provided directly from mastering circuit  120  or it may be generated with a separate VDR-to-SDR converter  140 . 
     In this example embodiment, the VDR  125  and SDR  145  signals are input into an encoder  130 . Purpose of encoder  130  is to create a coded bitstream that reduces the bandwidth required to transmit the VDR and SDR signals, but also allows a corresponding decoder  150  to decode and render either the SDR or VDR signals. In an example implementation, encoder  130  may be a layered encoder, such as one of those defined by the MPEG-2 and H.264 coding standards, which represents its output as a base layer, an optional enhancement layer, and metadata. As used herein, the term “metadata” relates to any auxiliary information that is transmitted as part of the coded bitstream and assists a decoder to render a decoded image. Such metadata may include, but are not limited to, such data as: color space or gamut information, dynamic range information, tone mapping information, or predictor operators, such as those described herein. 
     On the receiver, a decoder  150  uses the received coded bitstreams and metadata to render either an SDR image  157  or a VDR image  155 , according to the capabilities of the target display. For example, an SDR display may use only the base layer and the metadata to render an SDR image. In contrast, a VDR display may use information from all input layers and the metadata to render a VDR signal. 
       FIG. 2  shows in more detail an example implementation of encoder  130  incorporating the methods of this invention. In  FIG. 2 , optional SDR′  207  signal denotes an enhanced SDR signal. Typically, SDR video today is 8-bit, 4:2:0, ITU Rec. 709 data. SDR′ may have the same color space (primaries and white point) as SDR, but may use high precision, say 12-bits per pixel, with all color components at full spatial resolution (e.g., 4:4:4 RGB). From  FIG. 2 , SDR can be derived from an SDR′ signal using a set of forward transforms that may include quantization from say 12 bits per pixel to 8 bits per pixel, color transformation, say from RGB to YUV, and color subsampling, say from 4:4:4 to 4:2:0. The SDR output of converter  210  is applied to compression system  220 . Depending on the application, compression system  220  can be either lossy, such as H.264 or MPEG-2, or lossless, such as JPEG2000. The output of the compression system  220  may be transmitted as a base layer  225 . To reduce drift between the encoded and decoded signals, it is not uncommon for encoder  130  to follow compression process  220  with a corresponding decompression process  230  and inverse transforms  240 , corresponding to the forward transforms of  210 . Thus, predictor  250  may have the following inputs: VDR input  205  and either the compressed-decompressed SDR′ (or SDR) signal  245 , which corresponds to the SDR′ (or SDR) signal as it will be received by a corresponding decoder  150 , or original input SDR′  207 . Predictor  250 , using input VDR and SDR′ (or SDR) data will create signal  257  which represents an approximation or estimate of input VDR  205 . Adder  260  subtracts the predicted VDR  257  from the original VDR  205  to form output residual signal  265 . Subsequently (not shown), residual  265  may also be coded by another lossy or lossless encoder, and may be transmitted to the decoder as an enhancement layer. In some embodiments, compression unit  220  may receive directly an SDR input  215 . In such embodiments, forward transforms  210  and inverse transforms  240  units may be optional. 
     Predictor  250  may also provide the prediction parameters being used in the prediction process as metadata  255 . Since prediction parameters may change during the encoding process, for example, on a frame by frame basis, or on a scene by scene basis, these metadata may be transmitted to the decoder as part of the data that also include the base layer and the enhancement layer. 
     Since both VDR  205  and SDR′  207  (or SDR  215 ) represent the same scene, but are targeting different displays with different characteristics, such as dynamic range and color gamut, it is expected that there is a very close correlation between these two signals. In co-owned U.S. Provisional Application Ser. No. 61/475,359, filed on Apr. 14, 2011, (now PCT Application Ser. No. PCT/US2012/033605. filed on 13 Apr. 2012 titled “Multiple color channel multiple regression predictor,” from now on denoted as the &#39;359 application, incorporated herein by reference in its entirety, a novel multivariate, multi-regression (MMR) prediction model was disclosed which allowed the input VDR signal to be predicted using its corresponding SDR′ (or SDR) signal and a MMR operator. 
     The MMR predictor of the &#39;359 application may be considered a “global” cross-color predictor since it may be applied to all pixels of a frame, regardless of their individual color values. However, when translating a VDR video sequence to an SDR video sequence there are several operating factors that may degrade the efficiency of global predictors, such as color clipping and secondary color grading. 
     Under color clipping, values of some pixels in one channel or color component (e.g., the Red channel) may be clipped more severely than the values of the same pixels in other channels (say, the Green or Blue channels). Since clipping operations are non-linear operations, the predicted values of these pixels may not follow the global mapping assumptions, thus yielding large prediction errors. 
     Another factor that may affect SDR to VDR prediction is secondary color grading. In secondary color grading, the colorist may further partition each color channel into segments, such as: highlights, mid-tones, and shadows. These color boundaries may be controlled and customized during the color grading process. Estimating these color boundaries may improve overall prediction and reduce color artifacts in the decoded video. 
     Example Prediction Models 
     Example Notation and Nomenclature 
     Without loss of generality, an embodiment is considered of a piecewise cross-color channel (PCCC) predictor with two inputs: an SDR (or SDR′) input s and a VDR input v. Each of these inputs comprises multiple color channels, also commonly referred to as color components, (e.g., RGB, YCbCr, XYZ, and the like). Without loss of generality, regardless of bit depth, pixel values across each color component may be normalized to [0,1). 
     Assuming all inputs and outputs are expressed using three color components, denote the three color components of the i-th pixel in the SDR image as
 
 s   i   =[s   i1   s   i2   s   i3 ],  (1)
 
     denote the three color components of the i-th pixel in the VDR input as
 
 v   i   =[v   i1   v   i2   v   i3 ],and  (2)
 
     denote the predicted three color components of the i-th pixel in predicted VDR as
 
 {circumflex over (v)}   i   =[{circumflex over (v)}   i1   {circumflex over (v)}   i2   {circumflex over (v)}   i3 ].  (3)
 
     Each color channel, say the c-th, may be sub-divided into a set of multiple, non-overlapping, color segments using a set of boundary points (e.g., u c1 , u c2 , . . . , u cU ), so that within two successive segments (e.g., u and u+1) 0≦u cu &lt;u c(u+1) &lt;1. For example, in an embodiment, each color channel may be subdivided into three segments representing shadows, midtones, and highlights, using two boundary points, u c1  and u c2 . Then, shadows will be defined in the range [0, u c1 ), midtones will be defined in the range [u c1 , u c2 ), and highlights will be defined in the range [u c2 , 1). 
     Denote the set of the pixels having values within the u-th segment in the c-th color channel as Φ c   u . Denote p c   u  as the number of pixels in Φ c   u . To facilitate the discussion and simplify the notation, the procedure is described for the u-th segment in the c-th color channel and can be repeated for all segments in all color channels. The proposed PCCC modeling may be combined with other cross-color-based models, as those described in the &#39;359 application. As an example, and without loss of generality, a second-order PCCC model is described; however, the methods can easily be extended to other prediction models as well. Example second-order PCCC Model 
     Prediction Optimization for a Segment of a Color Channel 
     For the SDR signal, denote the three color components of the i-th pixel in Φ c   u  as
 
 s   c     i     u   =[s   i1   s   i2   s   i3 ].  (4)
 
     For each SDR pixel in Φ c   u , one can find the corresponding co-located VDR pixel, denoted as
 
 v   c     i     u   =[v   ic ].  (5)
 
     As used herein, the term ‘corresponding co-located SDR and VDR pixels’ denotes two pixels, one in the SDR image and one in the VDR image, that may have different dynamic ranges, but have the same pixel coordinates within each image. For example, for an SDR pixel s(10, 20), the corresponding co-located VDR pixel is v(10, 20). 
     Denote the predicted value of the c-th color component for this VDR pixel as
 
 {circumflex over (v)}   c     i     u   =[{circumflex over (v)}   ic ].  (6)
 
     By collecting all p c   u  pixels in Φ c   u  together, one may generate the following vector expressions 
                   V   ^     c   u     =     [             v   ^       c   ⁢           ⁢   0     u                 v   ^       c   ⁢           ⁢   1     u             ⋮               v   ^         cp   c   u     -   1     u           ]       ,       S   c   u     =     [           s     c   ⁢           ⁢   0     u               s     c   ⁢           ⁢   1     u             ⋮             s       cp   c   u     -   1     u           ]       ,         
and the original VDR data
 
     
       
         
           
             
               
                 
                   
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     Given the input SDR s signal, one may define a prediction model comprising first order and second (or higher) order SDR input data, such as:
 
 sc   c     i     u   =[s   i1   ·s   i2   s   i1   ·s   i3   s   i2   ·s   i3   s   i1   ·s   i2   ·s   i3 ],  (8)
 
 s   c     i     u2   =[s   i1   2   s   i2   2   s   i3   2 ],and  (9)
 
 sc   c     i     u2   =[s   i1   2   ·s   i2   2   s   i1   2   ·s   i3   2   s   i2   2   ·s   i3   2   s   i1   2   ·s   i2   2   ·s   i3   2 ],  (10)
 
     These data vectors may be combined to form the input vector for a second-order PCCC model:
 
 s   c     i     u(2) =Ø1 s   c     i     u   sc   c     i     u   s   c     i     u2   sc   c     i     u2 ┘,  (11)
 
     Given equations (4) to (11), the VDR prediction problem may be expressed as
 
 {circumflex over (v)}   c     i     u   =s   c     i     u(2)   M   c   u ,  (12)
 
     where M c   u  denotes a prediction parameter matrix for the u-th segment within the c-th color component. Note that this is a cross-color channel prediction model. In equation (12), the c-th color component of the predicted output is expressed as a combination of all color components in the input. In other words, unlike other single-channel color predictors, where each color channel is processed on its own and independently of each other, this model may take into consideration all color components of a pixel and thus may take full advantage of any inter-color correlation and redundancy. 
     By collecting all p c   u  pixels together, one may form the corresponding data matrix 
     
       
         
           
             
               
                 
                   
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     Then, the prediction operation may be expressed in matrix form as
 
 {circumflex over (V)}   c   u   =SC   c   u(2)   ·M   c   u .  (14)
 
     In one embodiment, a solution predictor M c   u  may be obtained using least square error optimization techniques, where the elements of M c   u  are selected so that they minimize the mean square error (MSE) between the original VDR and the predicted VDR 
                     min     M   c   u       ⁢           ⁢                V   c   u     -       V   ^     c   u            2     .             (   15   )               
Under the MSE criterion, an optimum solution to equation (15) may be expressed as
 
 M   c   u =( SC   c   u(2)     T     SC   c   u(2) ) −1   SC   c   u(2)     T     V   c   u .  (16)
 
     The above formulation derives a predictor for a specific segment within one of the color channels, assuming the boundaries of these segments within a color channel are known. However, in practice, the specific boundary points of each channel segment may not be available and may need to be derived during the encoding process. 
       FIG. 3  depicts an example prediction process according to an embodiment of this invention. In step  310 , a predictor accesses input VDR and SDR signals. In step  320 , each color channel in the input SDR signal may be segmented into two or more non-overlapping segments. The boundaries of these segments may be received as part of the input data, say from the VDR to SDR color grading process, or they may be determined from the input data using techniques as those to be described in the next Section. In step  330 , for each color segment in each of the color channels, using a cross-color prediction model, for example the second-order PCCC model of equations (4) to (14), and an optimization criterion, such as minimizing the prediction MSE, a prediction parameter matrix (e.g., M c   u ) is determined. In step  340 , a predicted VDR output is computed. In addition to computing a predicted VDR image, the prediction parameter matrix may be communicated to a decoder using ancillary data, such as metadata. 
     Prediction Optimization for the Whole Color Channel 
     Consider the problem of optimizing the prediction for all pixels across all segments within the c-th color channel. For all p pixels within this channel, denote the predicted VDR as 
                         V   ^     c     =     [             v   ^       c   ⁢           ⁢   0                   v   ^       c   ⁢           ⁢   1               ⋮               v   ^       cp   -   1             ]       ,           (   17   )               
and denote the original VDR data
 
     
       
         
           
             
               
                 
                   
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     The optimization problem for the c-th color channel may be formulated as a MSE minimization problem to find
 
min∥ V   c   −{circumflex over (V)}   c ∥ 2 .  (19)
 
     Given a set of boundary points u c     i   , the whole-channel parameter optimization problem can be decomposed into several sub-problems, one for each segment of the c-th color channel, and a solution for each sub-problem can be derived using equation (16). More specifically, given a set of U color segments, equation (19) can be expressed as 
     
       
         
           
             
               
                 
                   
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     Given a set of boundary points u c     i   , the total distortion for a set of prediction parameters may be given by: 
     
       
         
           
             
               
                 
                   
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                         } 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         u 
                         = 
                         1 
                       
                       U 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                            
                           
                             
                               V 
                               c 
                               u 
                             
                             - 
                             
                               
                                 
                                   V 
                                   ^ 
                                 
                                 c 
                                 u 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     M 
                                     ^ 
                                   
                                   c 
                                   u 
                                 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     When the value of any of the boundary point changes, the above overall distortion changes, too. Therefore, the goal is to identify those boundary points for which the overall distortion in the c-th channel is minimized. 
                     min     {     u   ci     }       ⁢           ⁢       J   ⁡     (     {     u   ci     }     )       .             (   23   )               
Example Solutions
 
Color Channel with only Two Segments
 
     As most SDR content is normally limited to 8 bits, if one excludes the values 0 and 255, the total number of boundary points is limited to 2 8 −2=254. If a color channel comprises only two color segments, then one needs to identify a single boundary (u c1 ) within the range [1, 255). In one embodiment, a full search may compute J({u c1 }) for all possible 254 boundaries points and then select as boundary point u c1  the boundary point for which J({u c1 }) is minimum. 
     In another embodiment, one may derive the best boundary point using a heuristic, iterative, search technique that may expedite the search time but may not necessarily yield optimal boundary values. For example, in one embodiment, the original SDR range may be subdivided into K segments (e.g., K=8). Then, assuming the boundary u c1  is approximately in the middle of each of these segments, one may compute equation (22) K times. Let k c  denote the segment with the minimum prediction error among all K segments. Then within the k c  segment, one can perform either full search or similar hierarchical searches to identify a locally optimum boundary point. The steps of this two-step search algorithm are summarized in pseudo-code in Table 1. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Two-step Search Algorithm 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 Divide color space range into K segments 
               
               
                 //First Step 
               
               
                 (a) For each segment k, compute the prediction error J k ({u c1 }) 
               
               
                 assuming the boundary point u c1  is located approximately in the 
               
               
                 middle of the k-th segment 
               
               
                 (b) Determine the segment, say k c , for which J k ({u c1 }) is minimum 
               
               
                 //Second step 
               
               
                 (a) Within the k c  segment, use full search or repeat this two-step algorithm 
               
               
                 to find u c1  that minimizes the prediction error 
               
               
                   
               
            
           
         
       
     
     This two-step search algorithm can easily be modified for alternative embodiments. For example, instead of assuming that the boundary point is located approximately in the middle of the k-th segment, one may assume that the boundary point is located at the beginning, the end, or any other position of the segment. 
     For color spaces with more than two segments, similar heuristic and iterative search techniques may also be applied. For example, for 8-bit SDR data, after identifying the first boundary point u c1  in the range (1, 255), one may try to identify two candidates for a second boundary point: one candidate in the sub-ranges (0, u c1 ) and the other in the sub-range (u c1 , 255). By computing the overall distortion J({u c1 }) for each of these two candidates, then one can define the second boundary point (u c2 ) as the one that yields the smallest prediction error (e.g., using equation (22)) among the two candidate solutions. 
     Since color grading of video frames is highly correlated, especially for all the frames within the same scene, the search of boundary points for the n-th frame may also take into consideration known results from prior frames within the same scene. Alternatively, boundary points may be computed only once for the whole scene. An example of a scene-based search algorithm is described in pseudo code in Table 2. In this embodiment, after identifying a boundary point for the first frame using the full dynamic range of a color channel, subsequent frames use it as a starting point to define a boundary point within a far smaller segment of the color space. 
     
       
         
           
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Scene-based Search Algorithm 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 For the first frame in one scene 
               
               
                 (1) perform a two-step algorithm to identify a boundary point within a 
               
               
                 color channel 
               
               
                 For the rest of the frames in the same scene 
               
               
                 (2) use the boundary point from the previous frame to define a segment 
               
               
                 to be used as the starting point of the second step in the two-step 
               
               
                 search (see Table 1). 
               
               
                   
               
            
           
         
       
     
     It should be appreciated that the steps of this algorithm may be implemented in a variety of alternative ways. For example, in the (1) step, instead of using a two-step search algorithm to identify a boundary point, one may use full-search, or any other type of search algorithm. As another example, in the (2) step, given a starting point, that starting point can be considered the approximately middle point of a segment of a predefined length. Alternatively, it can be considered the starting point of a segment, the end point of a segment, or any predefined position in a segment. 
     The methodology described herein may also be applied in deriving other PCCC models. For example, a first-order PCCC model can be derived by utilizing only the first three terms of equation (11), using equations
 
 s   c     i     u(1) =[1 s   c     i     u   sc   c     i     u ],  (24)
 
and
 
 {circumflex over (v)}   c     i     u   =s   c     i     u(1)   M   c   u .  (25)
 
     Similarly, the data vectors in equations (8)-(11) can be extended to define third-order or higher-order PCCC models. 
     Image Decoding 
     Embodiments of the present invention may be implemented either on an image encoder or an image decoder.  FIG. 4  shows an example implementation of decoder  150  according to an embodiment of this invention. 
     Decoding system  400  receives a coded bitstream that may combine a base layer  490 , an optional enhancement layer (or residual)  465 , and metadata  445 , which are extracted following decompression  430  and miscellaneous optional inverse transforms  440 . For example, in a VDR-SDR system, the base layer  490  may represent the SDR representation of the coded signal and the metadata  445  may include information about the PCCC prediction model that was used in the encoder predictor  250  and the corresponding prediction parameters. In one example implementation, when the encoder uses a PCCC predictor according to the methods of this invention, metadata may include the boundary values that identify each color segment within each color channel, identification of the model being used (e.g., first order PCCC, second order PCCC, and the like), and all coefficients of the prediction parameter matrix associated with that specific model. Given base layer  490  s and the prediction parameters extracted from the metadata  445 , predictor  450  can compute predicted {circumflex over (v)}  480  using any of the corresponding equations described herein (e.g., equation (14)). If there is no residual, or the residual is negligible, the predicted value  480  can be outputted directly as the final VDR image. Otherwise, in adder  460 , the output of the predictor ( 480 ) is added to the residual  465  to output VDR signal  470 . 
     Example Computer System Implementation 
     Embodiments of the present invention may be implemented with a computer system, systems configured in electronic circuitry and components, an integrated circuit (IC) device such as a microcontroller, a field programmable gate array (FPGA), or another configurable or programmable logic device (PLD), a discrete time or digital signal processor (DSP), an application specific IC (ASIC), and/or apparatus that includes one or more of such systems, devices or components. The computer and/or IC may perform, control or execute instructions relating to PCCC-based prediction, such as those described herein. The computer and/or IC may compute any of a variety of parameters or values that relate to the PCCC prediction as described herein. The image and video dynamic range extension embodiments may be implemented in hardware, software, firmware and various combinations thereof. 
     Certain implementations of the invention comprise computer processors which execute software instructions which cause the processors to perform a method of the invention. For example, one or more processors in a display, an encoder, a set top box, a transcoder or the like may implement PCCC-based prediction methods as described above by executing software instructions in a program memory accessible to the processors. The invention may also be provided in the form of a program product. The program product may comprise any medium which carries a set of computer-readable signals comprising instructions which, when executed by a data processor, cause the data processor to execute a method of the invention. Program products according to the invention may be in any of a wide variety of forms. The program product may comprise, for example, physical media such as magnetic data storage media including floppy diskettes, hard disk drives, optical data storage media including CD ROMs, DVDs, electronic data storage media including ROMs, flash RAM, or the like. The computer-readable signals on the program product may optionally be compressed or encrypted. 
     Where a component (e.g. a software module, processor, assembly, device, circuit, etc.) is referred to above, unless otherwise indicated, reference to that component (including a reference to a “means”) should be interpreted as including as equivalents of that component any component which performs the function of the described component (e.g., that is functionally equivalent), including components which are not structurally equivalent to the disclosed structure which performs the function in the illustrated example embodiments of the invention. 
     Equivalents, Extensions, Alternatives and Miscellaneous 
     Example embodiments that relate to applying PCCC prediction in coding VDR and SDR images are thus described. In the foregoing specification, embodiments of the present invention have been described with reference to numerous specific details that may vary from implementation to implementation. Thus, the sole and exclusive indicator of what is the invention, and is intended by the applicants to be the invention, is the set of claims that issue from this application, in the specific form in which such claims issue, including any subsequent correction. Any definitions expressly set forth herein for terms contained in such claims shall govern the meaning of such terms as used in the claims. Hence, no limitation, element, property, feature, advantage or attribute that is not expressly recited in a claim should limit the scope of such claim in any way. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense.