Patent Publication Number: US-2022239874-A1

Title: Rendering wide color gamut two-dimensional (2d) images on three-dimensional (3d) capable displays

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. patent application Ser. No. 17/064,578, filed Oct. 6, 2020, which is a continuation of U.S. patent application Ser. No. 16/312,256, filed Dec. 20, 2018, now U.S. Pat. No. 10,798,352, issued Oct. 6, 2020, which is the United States national stage of International Patent Application No. PCT/US2017/038829, filed Jun. 22, 2017, which claims priority to U.S. Provisional Patent Application No. 62/353,325, filed Jun. 22, 2016, and U.S. Provisional Patent Application No. 62/402,296, filed Sep. 30, 2016, all of which are incorporated herein by reference in their entirety. 
    
    
     BACKGROUND 
     Technical Field 
     This invention relates generally to devices and displays for displaying image data, and more particularly to the display of 2D images on 3D capable displays. 
     Description of the Background Art 
     Displays exist that are capable of displaying three-dimensional (3D) images. For example, such displays display a left eye image and a right eye image, which, when viewed together, give the appearance of a 3D image. The left eye image can be produced using 3 primary-color light sources (e.g., red1, green1, and blue1), and the right eye image can be produced using 3 different primary-color light sources (e.g., red2, green2, and blue2), each having a slightly different wavelength than their counterpart of similar color. The viewer views the displayed images through eyewear that includes a left lens and a different right lens. The left lens passes the left eye image and blocks the right eye image, and the right lens passes the right eye image and blocks the left eye image. 
     Conventional two-dimensional images can be displayed by 3D displays, by driving each pair of primary light sources with the same data. For example, the 2D red data values are used to drive both the red1 and the red2 primaries. Similarly, the 2D green data values are used to drive both the green1 and the green2 primaries, and the 2D blue data values are used to drive both the blue1 and blue2 primaries. The system is calibrated with the effective combined primaries and accurate images can be produced. However, the resulting color gamut can be limited significantly with respect to a desired color gamut (e.g., the established Rec 2020 gamut). 
     SUMMARY 
     The present invention overcomes the problems associated with the prior art by providing an improved means for displaying two-dimensional (2D) video data on a three-dimension (3D) display. 
     An example method for displaying image data includes identifying an established color gamut defined by a predefined number of primary colors and identifying a number of primary display colors associated with a light source, wherein the number of primary display colors associated with the light source exceeds the number of primary colors defining the established color gamut. The example method further includes defining a first virtual color gamut based on a combination of the primary display colors associated with the light source to approximate the established color gamut. Video data is received, which includes intensity values corresponding to a number of colors (e.g., 3) fewer than the number of primary colors associated with the light source (e.g., 6). The example method further includes generating intensity values associated with the first virtual color gamut based on the video data, and generating intensity values associated with the primary display colors of the light source based on the generated intensity values associated with the first virtual color gamut. The intensity values associated with the primary display colors are then provided to a spatial light modulator. 
     A particular example method further includes defining a second virtual color gamut, generating intensity values associated with the second virtual color gamut based on the video data, and using the intensity values associated with the second virtual color gamut in the step of generating the intensity values associated with the primary display colors. The second virtual color gamut is defined based on residual power of the light source after defining the first virtual color gamut. In a particular example method, the video data has a format associated with the established color gamut. 
     In a particular example method, the number of primary display colors is twice as large as the number of colors corresponding to the intensity values of the video data. In a more particular example method, the number of primary display colors is 6 and the video data includes intensity values corresponding to no more than 3 colors. 
     In an example method, at least one of the steps of generating intensity values associated with the first virtual color gamut based on the video data and generating intensity values associated with the second virtual color gamut based on the video data includes determining an intensity level indicated by the video data (e.g., a separate intensity level for each color associated with the video data) and generating the intensity values associated with at least one of the first virtual color gamut and the second virtual color gamut based on the intensity level indicated by the video data. The method additionally includes determining whether the intensity level indicated by the video data exceeds a predetermined intensity level (e.g., a separate predetermined intensity level for each color associated with the video data). If the intensity level indicated by the video data does not exceed the predetermined intensity level, then the intensity values associated with the first virtual gamut are generated based on the video data and the intensity values associated with the second virtual gamut are set to zero. On the other hand, if the intensity level indicated by the video data does exceed the predetermined intensity level, then an intensity value associated with the first virtual gamut is generated based on the predetermined intensity level, and intensity value associated with the second virtual gamut is generated based on an amount that the intensity level indicated by the video data exceeds the predetermined intensity level. 
     In one example method, the second virtual gamut is scaled to fit intensity values of the video data within an achievable gamut volume of the light source. The scaling is achieved, for example, by compressing the second virtual gamut toward white. 
     Another method includes clipping intensity values of at least one of the first virtual gamut and the second virtual gamut to fit within the achievable gamut volume of the light source. Optionally, the intensity values of at least one of the first virtual gamut and the second virtual gamut are clipped toward white. As another option, the intensity values of at least one of the first virtual gamut and the second virtual gamut are clipped to the edge of the achievable gamut volume in the direction of the negative primary. 
     Another example method includes modeling an achievable gamut volume of the first virtual gamut and the second virtual gamut and determining whether the generated intensity values of the first virtual gamut and the second virtual gamut fit within the modeled gamut volume. Intensity values of the first virtual gamut and the second virtual gamut that fit within the modeled gamut volume are left unmodified. Intensity values of the first virtual gamut and the second virtual gamut that do not fit within the modeled gamut volume are modified to fit within the modeled gamut volume. One example method of modifying the intensity values of the first virtual gamut and the second virtual gamut includes preserving the color balance (chromaticity) of the modified intensity values and reducing the intensity (e.g., magnitude) of the modified intensity values to fit within the achievable gamut volume. Another example method of modifying the intensity values of the first virtual gamut and the second virtual gamut includes preserving the intensity (e.g., magnitude) of the modified intensity values and adjusting the chromaticity of the modified intensity values towards a white point to fit within the achievable gamut volume. Yet another example method of modifying the intensity values of the first virtual gamut and the second virtual gamut includes reducing the intensity (e.g., magnitude) of the modified intensity values and adjusting the chromaticity of the modified intensity values towards a white point, whereby the intensity values are adjusted toward a surface of the achievable gamut volume. 
     An example display includes a light source, a spatial light modulator, and a controller. The light source includes a number of primary display colors (e.g., 6), which exceeds a number of primary colors (e.g., 3) defining an established color gamut. The spatial light modulator is illuminated by the light source. The controller is operative to receive video data including intensity values associated with a number of colors less than the number of primary display colors. The controller can be configured to receive video data having a format associated with the established color gamut. The controller generates intensity values associated with a first virtual color gamut based on the video data. The first virtual color gamut is defined by a combination of the primary display colors to match the established color gamut. The controller also generates intensity values for each of the primary display colors based on the intensity values associated with the first virtual gamut, and provides the intensity values for each of the primary display colors to the spatial light modulator. 
     In an example display, the controller is further operative to generate intensity values associated with a second virtual color gamut based on the video data. The second virtual color gamut is defined based on residual power of the light source considering the first virtual color gamut. The controller is further operative to generate intensity values associated with the second virtual color gamut based on the video data, and use the intensity values associated with the second virtual color gamut to generate the intensity values associated with the primary display colors. 
     In a particular embodiment, the number of primary display colors is twice as large as the number of colors corresponding to the intensity values of the video data. In a more particular embodiment, the number of primary display colors is 6 and the video intensity values correspond to no more than 3 colors. 
     In an example display, the controller is further operative to determine an intensity level indicated by the video data, and generate the intensity values associated with at least one of the first virtual color gamut and the second virtual color gamut based on the intensity level indicated by the video data. In one embodiment, the controller determines whether the intensity level indicated by the video data exceeds a predetermined intensity level. If the intensity level indicated by the video data does not exceed the predetermined intensity level, then the controller generates the intensity values associated with the first virtual gamut based on the video data and sets the intensity values associated with the second virtual gamut to zero. If the intensity level indicated by the video data does exceed the predetermined intensity level, then the controller generates an intensity value associated with the first virtual gamut based on the predetermined intensity level, and generates an intensity value associated with the second virtual gamut based on an amount by which the intensity level indicated by the video data exceeds the predetermined intensity level. 
     Optionally, the controller is operative to scale the second virtual gamut to fit intensity values of the video data within an achievable gamut volume of the light source. The scaling can include compressing the second virtual gamut toward white. As another option, the controller is operative to clip intensity values of at least one of the first virtual gamut and the second virtual gamut to fit within the achievable gamut volume of the light source. The intensity values of at least one of the first virtual gamut and the second virtual gamut can be clipped toward white, or the intensity values of at least one of the first virtual gamut and the second virtual gamut can be clipped to the edge of the achievable gamut volume. 
     Another example display incudes a modeler, a comparator, and a modifier. The modeler is operative to model an achievable gamut volume of the first virtual gamut and the second virtual gamut, in view of the light source capabilities. The comparator determines whether the generated intensity values of the first virtual gamut and the second virtual gamut fit within the modeled gamut volume. The modifier leaves the intensity values of the first virtual gamut and the second virtual gamut that fit within the modeled gamut volume unmodified, but modifies intensity values of the first virtual gamut and the second virtual gamut that do not fit within the modeled gamut volume, so that the modified values will fit within the modeled gamut volume. In a particular example display, the modifier is operative to preserve the color balance of the modified intensity values and reduce the intensity (e.g., magnitude) of the intensity values to fit within the achievable gamut volume. In another particular example display, the modifier is operative to preserve the intensity (e.g., magnitude) of the modified intensity values and adjust the chromaticity of the modified intensity values towards a white point to fit within the achievable gamut volume. In yet another particular example display, the modifier is operative to reduce the intensity (e.g., magnitude) of the modified intensity values and adjust chromaticity of the modified intensity values towards a white point, whereby the intensity values are adjusted toward a surface of the achievable gamut volume. 
     Any of the methods disclosed herein can be implemented with a non-transitory, electronically-readable medium having code embodied therein for causing a display device to perform the methods. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention is described with reference to the following drawings, wherein like reference numbers denote substantially similar elements: 
         FIG. 1  is a block diagram of an example 3D display system; 
         FIG. 2  is a block diagram of the controller of the 3D display system of  FIG. 1 ; 
         FIG. 3  is a chromaticity chart showing color gamuts associated with multiple display primary light sources; 
         FIG. 4  is a chromaticity chart showing virtual color gamuts associated with multiple display primary light sources; 
         FIG. 5  is a flow chart summarizing an example method of displaying 2D images on a 3D display; 
         FIG. 6A  is a flow chart summarizing an example method of performing a step of converting video data to first and second virtual color gamut values; 
         FIG. 6B  is a flow chart summarizing another example method of performing a step of converting video data to first and second virtual color gamut values; and 
         FIG. 6C  is a flow chart summarizing yet another example method of performing a step of converting video data to first and second virtual color gamut values. 
     
    
    
     DETAILED DESCRIPTION 
     The present invention overcomes the problems associated with the prior art, by providing a display system and display for displaying video data defining a first number of primary colors using an illumination source defining a second, greater number of primary light source colors. In the following description, numerous specific details are set forth (e.g., projector environment) in order to provide a thorough understanding of the invention. Those skilled in the art will recognize, however, that the invention may be practiced apart from these specific details. In other instances, details of well-known video processing practices and components have been omitted, so as not to unnecessarily obscure the present invention. 
       FIG. 1  is a block diagram of an example 3D display system  100  capable of displaying 2D video data with an improved color gamut. In this example embodiment, display system  100  is a projector including a light source  102  with 6 primary colors (e.g., laser light sources), illumination optics  104 , a separator  106 , one or more modulators  108 , a combiner  110 , projection optics  112 , and a controller  114 . Light source  102  produces an illumination beam including the six primary colors and directs the illumination beam through illumination optics  104  and into color separator  106 . Color separator  106  separates the multicolored beam into the 6 primary beams and directs each primary beam to an associated one of spatial light modulators  108 . After the primary illumination beams are modulated, projection optics  112  focuses the modulated beam to form an imaging beam, which is projected onto a viewing surface (not shown). 
     In this example embodiment, there is an individual modulator for each primary color. However, the number of modulators can be reduced using a field sequential modulation scheme. In another example embodiment, the light sources, the modulators, and other components of the projector can be separated into two separate, but coordinated projectors. In yet another embodiment, the modulators can include a plurality of modulators for each primary color, as, for example, in a dual modulation projector. 
       FIG. 2  is a block diagram of an example controller  114  of the 3D display system of  FIG. 1 . Controller  114  includes an Identification of Primaries Module for receiving and/or storing information (e.g., calibration data) about the primary colors of light source  102 . A Color Space Transformation module  204  transforms incoming 2D video data into an established tri-stimulus color space (e.g., Rec2020), if necessary. A Gamut Redefinition module  206  defines one or more virtual gamuts based on the target established color gamut and the primaries of light source  102 . An Established Tri-Stimulus To Virtual Gamut Transformation module  208  converts video data into intensity values associated with the virtual gamut(s). Then, a Virtual Gamuts To Primaries Transformation module  210  converts the intensity values of the virtual gamut(s) into intensity values corresponding to the primaries of the light source and provides the resulting intensity values to the modulators. 
       FIG. 3  is a CIE 1931 chromaticity chart showing color gamuts associated with the multiple display primary light sources and the target established color gamut (Rec2020). Gamut  302  is the Rec2020 gamut. Gamut  304  is the gamut defined by the first  3  light source primaries (R L , G L , and B L ), which provide illumination for a right eye image when the projector is operating in 3D mode. Gamut  306  is the gamut defined by the second 3 light source primaries (R S , G S , and B S ), which provide illumination for a left eye image when the projector is operating in 3D mode. Gamut  308  is the gamut defined by driving the left eye primaries and the right eye primaries with the same values. As shown, gamut  308  significantly differs from Rec2020 gamut  302 . 
       FIG. 4  is a chromaticity chart showing virtual color gamuts generated from the multiple display primary light sources. Gamuts  302 ,  306 , and  304  of  FIG. 3  are also shown for comparison. Virtual gamut  402  (Gamut A), is defined as a combination of the 6 primary light sources to closely approximate Rec2020 gamut  302 . Virtual gamut  404  (Gamut B) is defined by the residual power of the 6 primary light sources. In other words, Gamut B is defined by the left over light output of the 6 primaries after the light required for Gamut A has been subtracted. As shown, virtual gamut  402  (Gamut A) matches the Rec2020 gamut  302  much more closely than the simple summation gamut  308  ( FIG. 3 ). 
     The disclosed system optimizes the gamut volume for 2D images by driving the left and right eye signals differently (even though the content is only 2D and no glasses are worn). The gamut volume is divided into the two virtual gamuts: Gamut A and Gamut B. Each gamut uses virtual primaries that are specific blends of the original 6P primaries. The ‘A’ gamut is optimized to be as close to possible as Rec2020. The ‘B’ gamut uses the residual energy from the 6 primaries. The ‘A’ and ‘B’ gamuts are show in  FIG. 4 . Note that the ‘A’ gamut is very close to Rec2020. 
     Values in the ‘A’ and ‘B’ gamuts can be converted to the 6P primaries (R L , G L , B L , R S , G S , B S ) using “blend” matrices as follows: 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       R 
                       L 
                     
                   
                 
                 
                   
                     
                       G 
                       L 
                     
                   
                 
                 
                   
                     
                       B 
                       L 
                     
                   
                 
               
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             = 
             
               
                 
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                       ⁢ 
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                       ⁢ 
                       L 
                     
                   
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                           R 
                           B 
                         
                       
                     
                     
                       
                         
                           G 
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                           B 
                         
                       
                     
                   
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               and 
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                           B 
                           B 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Typical values for the blend matrices [B AL ], [B AS ], [B BL ], and [B BS ] are: 
     
       
         
           
               
               
             
               
                   
                   
               
             
            
               
                   
                 float B AL [3][3]= 
               
               
                   
                 { 
               
            
           
           
               
               
            
               
                   
                 { 0.0000f, 0.0000f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.2859f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.0000f, 0.9578f } 
               
            
           
           
               
               
            
               
                   
                 }; 
               
               
                   
                 float B AS [3][3] = 
               
               
                   
                 { 
               
            
           
           
               
               
            
               
                   
                 { 0.9903f, 0.0000f, 0.0000f }, 
               
               
                   
                 { 0.0097f, 0.7141f, 0.0422f }, 
               
               
                   
                 { 0.0000f, 0.0000f, 0.0000f } 
               
            
           
           
               
               
            
               
                   
                 }; 
               
               
                   
                 float B BL [3][3] = 
               
               
                   
                 { 
               
            
           
           
               
               
            
               
                   
                 { 1.0000f, 0.0000f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.7141f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.0000f, 0.0422f } 
               
            
           
           
               
               
            
               
                   
                 }; 
               
               
                   
                 float B BS [3][3] = 
               
               
                   
                 { 
               
            
           
           
               
               
            
               
                   
                 { 0.0097f, 0.0000f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.2340f, 0.0000f }, 
               
               
                   
                 { 0.0000f, 0.0000f, 1.0000f } 
               
            
           
           
               
               
            
               
                   
                 }; 
               
               
                   
                   
               
            
           
         
       
     
     The ‘A’ gamut is used for lower luminance levels and the ‘B’ gamut is added, where necessary, to achieve higher luminance levels. The ‘A’ gamut services approximately 50% of the luminance range, above which the less-favorable ‘B’ gamut is added. 50% of the linear luminance range represents all but the last stop of the perceptual range and, therefore, most of perceptual range can be handled with the ‘A’ gamut. Above that range the ‘B’ gamut is added and the gamut volume tapers towards the top. Hence, the vast majority of the perceptual range can be achieved with approximately Rec2020 gamut, and yet the full luminance range can still be achieved. 
     Although the ‘B’ gamut can represent chromaticities that are outside the ‘A’ gamut, it would be counter-intuitive for a colorist to utilize chromaticities that are only available at higher luminance values. Therefore the ‘B’ gamut would be limited to the intersection of the achievable ‘B’ gamut and the ‘A’ gamut. 
     There is a desire in the industry not to utilize colors outside the Rec2020 gamut. This is easily achieved by representing the source images in RGB with Rec2020 primaries and disallowing negative values. Content color-graded this way will never produce values outside Rec2020. When the content is packaged for distribution, it is transformed into a DCI-compliant package represented by CIE 1931 XYZ tri-stimulus values. Although XYZ can represent colors outside Rec2020, so long as the source was constrained to Rec2020, excursions in XYZ will never exceed Rec2020. 
     The methods in this disclosure cover various means for implementing this. The first two methods are computationally efficient and could easily be implemented in a simple filed-programmable gate array (FPGA) circuit. The third method is more computationally intensive and would typically be implemented in a graphics processor unit (GPU). 
     Method 1—Gamut scaling 
     In this method two functions are defined: 
         f   l ( C )=if ( C&lt; 0.5) C , else 0.5 
         f   u ( C )=if ( C&lt; 0.5)0, else  C− 0.5 
     
       
      
       C∈R 
       2020 
       ,G 
       2020 
       ,B 
       2020  
      
     
     For all incoming tri-stimulus pixel values (denoted R 2020 ), the ‘A’ and ‘B’ gamut signals are derived as follows: 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       R 
                       A 
                     
                   
                 
                 
                   
                     
                       G 
                       A 
                     
                   
                 
                 
                   
                     
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                       A 
                     
                   
                 
               
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                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
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     The signals representing the ‘A’ gamut (R A , G A , B A ) and the signals representing the ‘B’ gamut (R B , G B , B B ) then drive the 6P primaries according to the blend matrixes [B AL ], [B AS ], [B BL ], and [B BS ] described above. 
     In this method, for luminance values above 50%, all RGB values are moved towards white to fit within the available gamut volume. 
     Method 2—Gamut Clipping 
     In this method two functions are defined: 
     
       
         
           
             
               
                 
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                 MaxRGB 
               
             
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               else 
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     For all incoming tri-stimulus pixel values (denoted R 2020 ), the full unclipped ‘A’ and ‘B’ gamut signals are derived as follows: 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       R 
                       Af 
                     
                   
                 
                 
                   
                     
                       G 
                       Af 
                     
                   
                 
                 
                   
                     
                       B 
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                             2020 
                           
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                             2020 
                           
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                 and 
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                         Bf 
                       
                     
                   
                   
                     
                       
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                                 2 
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                                 ⁢ 
                                 2 
                                 ⁢ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             f 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               G 
                               
                                 2 
                                 ⁢ 
                                 0 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             f 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               B 
                               
                                 2 
                                 ⁢ 
                                 0 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 0 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             ; 
           
         
       
     
     where [C] A  and [C] B  are derived as will be described hereinafter. 
     These are then clipped as follows: 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       R 
                       A 
                     
                   
                 
                 
                   
                     
                       G 
                       A 
                     
                   
                 
                 
                   
                     
                       B 
                       A 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           max 
                           ⁡ 
                           
                             ( 
                             
                               
                                 R 
                                 Af 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           max 
                           ⁡ 
                           
                             ( 
                             
                               
                                 G 
                                 Af 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           max 
                           ⁡ 
                           
                             ( 
                             
                               
                                 B 
                                 Af 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   
 
                 
                 [ 
                 
                   
                     
                       
                         R 
                         B 
                       
                     
                   
                   
                     
                       
                         G 
                         B 
                       
                     
                   
                   
                     
                       
                         B 
                         B 
                       
                     
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         max 
                         ⁡ 
                         
                           ( 
                           
                             
                               R 
                               Bf 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         max 
                         ⁡ 
                         
                           ( 
                           
                             
                               G 
                               Bf 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         max 
                         ⁡ 
                         
                           ( 
                           
                             
                               B 
                               Bf 
                             
                             , 
                             0 
                           
                           ) 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     In this method, values within the achievable gamut volume are left untouched and values outside (with negative RGB values) are clipped to the edge of the gamut in the direction of the negative primary. 
     An alternative method instead clips towards white: 
     
       
         
           
             
               
                 min 
                 A 
               
               ⁢ 
               
                 = 
                 
                   min 
                   ⁡ 
                   
                     ( 
                     
                       
                         R 
                         Af 
                       
                       , 
                       
                         G 
                         Af 
                       
                       , 
                       
                         B 
                         Af 
                       
                     
                     ) 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               if 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   
                     min 
                     A 
                   
                   ⁢ 
                   
                     &lt; 
                     0 
                   
                 
                 ) 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 then 
                 ⁢ 
                 
                   
 
                 
                 [ 
                 
                   
                     
                       
                         R 
                         A 
                       
                     
                   
                   
                     
                       
                         G 
                         A 
                       
                     
                   
                   
                     
                       
                         B 
                         A 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         R 
                         Af 
                       
                       - 
                       
                         min 
                         A 
                       
                     
                   
                 
                 
                   
                     
                       
                         G 
                         Af 
                       
                       - 
                       
                         min 
                         A 
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         Af 
                       
                       - 
                       
                         min 
                         A 
                       
                     
                   
                 
               
               ] 
             
           
         
       
       
         
           
             
               
                 min 
                 B 
               
               ⁢ 
               
                 = 
                 
                   min 
                   ⁡ 
                   
                     ( 
                     
                       
                         R 
                         Bf 
                       
                       , 
                       
                         G 
                         Bf 
                       
                       , 
                       
                         B 
                         Bf 
                       
                     
                     ) 
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
               if 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 
                   minB 
                   &lt; 
                   0 
                 
                 ) 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 then 
                 ⁢ 
                 
                   
 
                 
                 [ 
                 
                   
                     
                       
                         R 
                         B 
                       
                     
                   
                   
                     
                       
                         G 
                         B 
                       
                     
                   
                   
                     
                       
                         B 
                         B 
                       
                     
                   
                 
                 ] 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         R 
                         Bf 
                       
                       - 
                       
                         min 
                         B 
                       
                     
                   
                 
                 
                   
                     
                       
                         G 
                         Bf 
                       
                       - 
                       
                         min 
                         B 
                       
                     
                   
                 
                 
                   
                     
                       
                         B 
                         Bf 
                       
                       - 
                       
                         min 
                         B 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     Again, the signals representing the ‘A’ gamut (R A , G A , B A ) and the signals representing the ‘B’ gamut (R B , G B , B B ) then drive the 6P primaries according to the blend matrices described above. 
     [C] A  and [C] B  used above to convert tri-stimulus values to full, unclipped A and B gamut values are derived as follows. As indicated above, two sets of three primaries of the virtual A and B gamuts {R A , G A , B A } and {R B , G B , B B } are related to the long and short primaries of the display {R L , G L , B L } and {R S , G S , B S } according to the following blend matrices: 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               R 
                               L 
                             
                           
                         
                         
                           
                             
                               G 
                               L 
                             
                           
                         
                         
                           
                             
                               B 
                               L 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           [ 
                           
                             B 
                             
                               A 
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   R 
                                   A 
                                 
                               
                             
                             
                               
                                 
                                   G 
                                   A 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   A 
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           [ 
                           
                             B 
                             
                               B 
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   R 
                                   B 
                                 
                               
                             
                             
                               
                                 
                                   G 
                                   B 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   B 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   [ 
                   1 
                   ] 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             R 
                             S 
                           
                         
                       
                       
                         
                           
                             G 
                             S 
                           
                         
                       
                       
                         
                           
                             B 
                             S 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           B 
                           
                             A 
                             ⁢ 
                             S 
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 A 
                               
                             
                           
                           
                             
                               
                                 G 
                                 A 
                               
                             
                           
                           
                             
                               
                                 B 
                                 A 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           B 
                           
                             B 
                             ⁢ 
                             S 
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 B 
                               
                             
                           
                           
                             
                               
                                 G 
                                 B 
                               
                             
                           
                           
                             
                               
                                 B 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   [ 
                   2 
                   ] 
                 
               
             
           
         
       
     
     Knowing the Normalized Primary Matrixes for {R L , G L , B L } and {R S , G S , B S }, the XYZ value for arbitrary points in {R L , G L , B L } and {R S , G S , B S } is: 
     
       
         
           
             
               [ 
               
                 
                   
                     X 
                   
                 
                 
                   
                     Y 
                   
                 
                 
                   
                     Z 
                   
                 
               
               ] 
             
             = 
             
               
                 
                   
                     [ 
                     NPM 
                     ] 
                   
                   LLL 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           L 
                         
                       
                     
                     
                       
                         
                           G 
                           L 
                         
                       
                     
                     
                       
                         
                           B 
                           L 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   
                     [ 
                     NPM 
                     ] 
                   
                   SSS 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           S 
                         
                       
                     
                     
                       
                         
                           G 
                           S 
                         
                       
                     
                     
                       
                         
                           B 
                           S 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     Substituting [1], [2]: 
     
       
         
           
             
               [ 
               
                 
                   
                     X 
                   
                 
                 
                   
                     Y 
                   
                 
                 
                   
                     Z 
                   
                 
               
               ] 
             
             = 
             
               
                 
                   
                     
                       [ 
                       NPM 
                       ] 
                     
                     LLL 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           [ 
                           
                             B 
                             
                               A 
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           
                             
                               
                                 R 
                                 A 
                               
                             
                           
                           
                             
                               
                                 G 
                                 A 
                               
                             
                           
                           
                             
                               
                                 B 
                                 A 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                         
                           [ 
                           
                             B 
                             
                               B 
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                         ⁢ 
                         
                             
                         
                         [ 
                         
                           
                             
                               
                                 R 
                                 B 
                               
                             
                           
                           
                             
                               
                                 G 
                                 B 
                               
                             
                           
                           
                             
                               
                                 B 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
                 + 
                 
                   
                     
                       [ 
                       NPM 
                       ] 
                     
                     SSS 
                   
                   ⁢ 
                   
                     
                       { 
                       
                         
                           
                             [ 
                             
                               B 
                               
                                 A 
                                 ⁢ 
                                 S 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             
                               
                                 
                                   R 
                                   A 
                                 
                               
                             
                             
                               
                                 
                                   G 
                                   A 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   A 
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           
                             [ 
                             
                               B 
                               
                                 B 
                                 ⁢ 
                                 S 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                               
                           
                           [ 
                           
                             
                               
                                 
                                   R 
                                   B 
                                 
                               
                             
                             
                               
                                 
                                   G 
                                   B 
                                 
                               
                             
                             
                               
                                 
                                   B 
                                   B 
                                 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                     ⁢ 
                     
                       
 
                     
                     [ 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ] 
                   
                 
               
               = 
               
                 
                   
                     
                       
                         
                           [ 
                           NPM 
                           ] 
                         
                         LLL 
                       
                       ⁡ 
                       
                         [ 
                         
                           B 
                           
                             A 
                             ⁢ 
                             L 
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               R 
                               A 
                             
                           
                         
                         
                           
                             
                               G 
                               A 
                             
                           
                         
                         
                           
                             
                               B 
                               A 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         
                           [ 
                           NPM 
                           ] 
                         
                         SSS 
                       
                       ⁡ 
                       
                         [ 
                         
                           B 
                           
                             A 
                             ⁢ 
                             S 
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               R 
                               A 
                             
                           
                         
                         
                           
                             
                               G 
                               A 
                             
                           
                         
                         
                           
                             
                               B 
                               A 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         
                           [ 
                           NPM 
                           ] 
                         
                         SSS 
                       
                       ⁡ 
                       
                         [ 
                         
                           B 
                           
                             B 
                             ⁢ 
                             S 
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               R 
                               B 
                             
                           
                         
                         
                           
                             
                               G 
                               B 
                             
                           
                         
                         
                           
                             
                               B 
                               B 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       
                         
                           
                             [ 
                             NPM 
                             ] 
                           
                           LLL 
                         
                         ⁡ 
                         
                           [ 
                           
                             B 
                             
                               B 
                               ⁢ 
                               L 
                             
                           
                           ] 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 R 
                                 B 
                               
                             
                           
                           
                             
                               
                                 G 
                                 B 
                               
                             
                           
                           
                             
                               
                                 B 
                                 B 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       
 
                     
                     [ 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                       
                         
                           Z 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   
                     
                       { 
                       
                         
                           
                             
                               [ 
                               NPM 
                               ] 
                             
                             LLL 
                           
                           ⁡ 
                           
                             [ 
                             
                               B 
                               
                                 A 
                                 ⁢ 
                                 L 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             
                               [ 
                               NPM 
                               ] 
                             
                             SSS 
                           
                           ⁡ 
                           
                             [ 
                             
                               B 
                               
                                 A 
                                 ⁢ 
                                 S 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               R 
                               A 
                             
                           
                         
                         
                           
                             
                               G 
                               A 
                             
                           
                         
                         
                           
                             
                               B 
                               A 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       { 
                       
                         
                           
                             
                               [ 
                               NPM 
                               ] 
                             
                             SSS 
                           
                           ⁡ 
                           
                             [ 
                             
                               B 
                               
                                 B 
                                 ⁢ 
                                 S 
                               
                             
                             ] 
                           
                         
                         + 
                         
                           
                             
                               [ 
                               NPM 
                               ] 
                             
                             LLL 
                           
                           ⁡ 
                           
                             [ 
                             
                               B 
                               
                                 B 
                                 ⁢ 
                                 L 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               R 
                               B 
                             
                           
                         
                         
                           
                             
                               G 
                               B 
                             
                           
                         
                         
                           
                             
                               B 
                               B 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Putting in terms of primary matrices (Note [PM] A  and [PM] B  are not normalized primary matrices. Although the middle row of [NPM] LLL  and [NPM] SSS  each sum to 1, [PM] A,B  take arbitrary blends of each column so the resulting middle row no longer sums to 1.): 
     
       
         
           
             
               [ 
               
                 
                   
                     X 
                   
                 
                 
                   
                     Y 
                   
                 
                 
                   
                     Z 
                   
                 
               
               ] 
             
             = 
             
               
                 
                   
                     [ 
                     PM 
                     ] 
                   
                   A 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           A 
                         
                       
                     
                     
                       
                         
                           G 
                           A 
                         
                       
                     
                     
                       
                         
                           B 
                           A 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 
                   
                     [ 
                     
                       P 
                       ⁢ 
                       M 
                     
                     ] 
                   
                   B 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           R 
                           B 
                         
                       
                     
                     
                       
                         
                           G 
                           B 
                         
                       
                     
                     
                       
                         
                           B 
                           B 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     where: 
       [PM] A =[NPM] LLL [ B   AL ]+[NPM] SSS [ B   AS ] 
       Similarly: 
       [PM] B =[NPM] SSS [ B   BS ]+[NPM] LLL [ B   BL ] 
     Given the normalized primary matrix for Rec2020 
       [NPM] 2020 , 
       then: 
       [ C ] A =[PM] A   −1 [NPM] 2020 , and 
       [ C ] B =[PM] B   −1 [NPM] 2020 . 
     Method 3—Gamut Volume Modelling 
     In this method, the gamut volume achievable by utilizing the ‘A’ gamut for low luminance values and the ‘B’ gamut for higher luminance values, is modeled such that tri-stimulus values can be tested to determine whether they fall within the achievable gamut volume or not. 
     In one method, tri-stimulus values within the achievable gamut volume are unmodified. For values outside the gamut volume, the ratio of RGB values is preserved but the values are scaled to reduce luminance to fit within the available gamut volume. 
     In an alternative method, again the values within the achievable gamut volume are unmodified. For values outside the gamut volume, the luminance is preserved, but chromaticity is moved towards the mastering white point (e.g. D6500) to fit within the gamut volume. 
     In another method, tri-stimulus values within some threshold of the surface of the gamut volume would be dropped in luminance or moved towards the white point to achieve a “soft clip” to the surface of the gamut volume. This may reduce artifacts in cases where high luminance saturated colors are utilized heavily. 
     Flowcharts Summarizing Example Methods 
       FIG. 5  is a flow chart summarizing an example method  500  of displaying 2D images on a 3D display. In a first step  502 , an established color gamut defined by a number (e.g., 3) primary colors is identified. Then, in a second step  504  a different number (e.g., 6) of primary display colors associated with a light source are identified. Next, in a third step  506 , a first virtual gamut is defined based on a combination of the identified primary display colors to closely approximate the identified established gamut. In a fourth step  508 , a second virtual gamut is defined based on the residual light source power for the primary display colors. Then, in a fifth step  510 , video data is converted to intensity values of the first and second virtual gamuts. Next, in a sixth step  512 , the intensity values of the first and second virtual gamuts are converted to intensity values associated with the primary display colors. Then, in a seventh step  514 , the intensity values associated with the primary display colors are provided to one or more spatial light modulators. 
       FIG. 6A  is a flow chart summarizing an example method  600  of performing a step of converting video data to intensity values associated with the first and second virtual color gamuts. In a first step  602 , a luminance level (L) at or below which colors can be represented by the first virtual color gamut alone is defined. Then, in a second step  604 , the luminance level of video data is determined. Next, in a third step  606 , it is determined whether the luminance level of the video data is at or below the defined luminance level (L). If it is determined that the luminance level of the video data is at or below the defined luminance level (L), then, in a fourth step  608 , intensity values associated with the first virtual gamut are generated from the video data, and the intensity values associated with the second virtual gamut are set to zero. 
     If, in third step  606  it is determined that the luminance level of the video data is above the defined luminance level (L), then, in a fifth step  610 , intensity values associated with the first virtual gamut are generated corresponding to the luminance level (L). Next, in a sixth step  612 , the second gamut is scaled to be within an achievable gamut volume of the light source. Then, in a seventh step  614 , intensity levels associated with the second virtual gamut are generated corresponding to an amount that the luminance level of the video data exceeds the defined luminance level (L). 
       FIG. 6B  is a flow chart summarizing another example method  620  of performing a step of converting video data to intensity values associated with the first and second virtual color gamuts. In a first step  622 , a luminance level (L) at or below which colors can be represented solely by the first virtual color gamut is defined. Then, in a second step  624 , the luminance level of video data is determined. Next, in a third step  626 , it is determined whether the luminance level of the video data is at or below the defined luminance level (L). If it is determined that the luminance level of the video data is at or below the defined luminance level (L), then, in a fourth step  628 , intensity values associated with the first virtual color gamut are generated from the video data, and the intensity values associated with the second virtual color gamut are set to zero. 
     If, in third step  626  it is determined that the luminance level of the video data is above the defined luminance level (L), then, in a fifth step  630 , intensity values associated with the first virtual gamut are generated corresponding to the luminance level (L). Next, in a sixth step  632 , intensity levels associated with the second virtual gamut are generated corresponding to an amount that the luminance level of the video data exceeds the defined luminance level (L). Then, in a seventh step  634 , the intensity values associated with the first and second virtual gamuts are clipped to be within the achievable gamut volume. 
       FIG. 6C  is a flow chart summarizing yet another example method  640  of performing a step of converting video data to first and second virtual color gamut values. In a first step  642 , an achievable gamut volume of the light source is modeled using the first virtual gamut for luminance values at or below a predetermined luminance level (L). Then, in a second step  644 , video data is converted to intensity values associated with the first and second virtual color gamuts. Next, in a third step  646 , it is determined whether the intensity values associated with the first and second virtual color gamuts fit within the modeled gamut volume. It the intensity values associated with the first and second virtual color gamuts do fit within the modeled gamut volume, method  640  ends. Otherwise, the intensity values associated with the first and second virtual color gamuts are modified according to one of the following optional steps. In an optional fourth step  648 , the luminance of the intensity values associated with the first and second virtual color gamuts is reduced, while preserving the primary color balance of the data, until the intensity values fit within the modeled gamut volume. In an optional fifth step  650 , the luminance of the intensity values associated with the first and second virtual color gamuts is preserved, but chromaticity is adjusted towards a white point until the intensity values fit within the modeled gamut volume. In an optional sixth step  652 , the luminance of the intensity values associated with the first and second virtual color gamuts is reduced and the chromaticity is adjusted toward the white point until the intensity values associated with the first and second virtual color gamuts fit within the modeled gamut volume, thus achieving a “soft clip” to the surface of the gamut volume. 
     The description of particular embodiments of the present invention is now complete. Many of the described features may be substituted, altered or omitted without departing from the scope of the invention. For example, the example embodiment is a shown as a projector. However, the inventive methods and components can be employed in a color grading desk. As another option, the methods and components of this disclosure can be embodied in an intermediate device interposed between a color grading desk and a projector or other display. This and other deviations from the particular embodiments shown will be apparent to those skilled in the art, particularly in view of the foregoing disclosure.