Put the theory into practice. Use the Colour Conversion Matrix of Equation 6 to convert RGB coordinates into Y, (R-Y), (B-Y) coordinates. Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

+1 Y’ -1 +1 Fig 7 R’-Y’ B’-Y’ -1 +1 Each of the next 9 Black (0,0,0) 1 +1 Y’ -1 +1 Fig 7 R’-Y’ B’-Y’ Each of the next 9 foils will map one point of the R’G’B’ unity colour space into the Luma and colour difference space. Just ‘play’ the foils by flicking through them with the ‘PageDn’ key on your keyboard and watch how the unity colour cube is translated into a Luma & colour difference space cube. -1 +1 Luma & Colour Difference colour space Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the WHITE Co-ordinate (1,1,1) R’G’B’ White = [1,1,1] +1 Insert the R’G’B’ unit colour space coordinates into the matrix transform, here…. ……. And then evaluate the matrix multiplication. Plot the new coordinates on the Luma and colour difference axis -1 +1 R’-Y’ B’-Y’ Fig 8 -1 +1 Y’601 0.299 0.587 0.114 R’ [ 0.299*R’ + 0.587*G’ + 0.114*B’] = Y’601 B’- Y’601 = - 0.299 -0.587 0.886 G’ = [ -0.299*R’ - 0.587*G’ + 0.886*B’] = B’- Y’601 R’- Y’601 0.701 -0.587 -0.114 B’ [ 0.701*R’ - 0.587*G’ - 0.114*B’] = R’- Y’601 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the WHITE Co-ordinate (1,1,1) R’G’B’ White = [1,1,1] +1 -1 +1 Fig 9 R’-Y’ B’-Y’ Plot these co-ordinates -1 +1 Y’601 0.299 0.587 0.114 1 [ 0.299 + 0.587 + 0.114] 1 B’- Y’601 = - 0.299 -0.587 0.886 1 = [ -0.299 - 0.587 + 0.886] = 0 R’- Y’601 0.701 -0.587 -0.114 1 [ 0.701 - 0.587 - 0.114] 0 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the Black Co-ordinate R’G’B’ Black = [0,0,0] +1 -1 +1 Fig 10 R’-Y’ B’-Y’ Plot these co-ordinates -1 +1 Y’601 0.299 0.587 0.114 0 [ 0 + 0 + 0 ] 0 B’- Y’601 = - 0.299 -0.587 0.886 0 = [ -0 + 0 + 0 ] = 0 R’- Y’601 0.701 -0.587 -0.114 0 [ 0 + 0 + 0 ] 0 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the YELLOW Co-ordinate Black (0,0,0) 0.886 R’G’B’ Yellow = [1,1,0] +1 -1 - 0.886 +1 R’-Y’ Fig 11 0.114 B’-Y’ Plot these co-ordinates -1 +1 Y’601 0.299 0.587 0.114 1 [ 0.299 + 0.587 + 0] 0.884 B’- Y’601 = - 0.299 -0.587 0.886 1 = [ -0.299 - 0.587 + 0 ] = - 0.886 R’- Y’601 0.701 -0.587 -0.114 0 [ 0.701 - 0.587 - 0] 0.114 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the GREEN Co-ordinate (0,1,0) R’G’B’ Green = [0,1,0] +1 0.587 -1 +1 - 0.587 Fig 12 R’-Y’ B’-Y’ Plot these co-ordinates -0.587 -1 +1 Y’601 0.299 0.587 0.114 0 [ 0 + 0.587 + 0] 0.587 B’- Y’601 = - 0.299 -0.587 0.886 1 = [ 0 - 0.587 + 0] = - 0.587 R’- Y’601 0.701 -0.587 -0.114 0 [ 0 - 0.587 + 0] - 0.587 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the CYAN Co-ordinate (0,1,1) R’G’B’ Cyan = [0,1,1] +1 -1 0.701 +1 Fig 13 R’-Y’ B’-Y’ Plot these co-ordinates -0.701 0.299 -1 +1 Y’601 0.299 0.587 0.114 0 [ 0 + 0.587 + 0.114] 0.701 B’- Y’601 = - 0.299 -0.587 0.886 1 = [ 0 - 0.587 + 0.886] = 0.299 R’- Y’601 0.701 -0.587 -0.114 1 [ 0 - 0.587 - 0.114] - 0.701 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the BLUE Co-ordinate R’G’B’ Blue = [0,0,1] +1 -1 +1 Fig 14 R’-Y’ -0.114 B’-Y’ Plot these co-ordinates 0.114 -1 0.886 +1 Y’601 0.299 0.587 0.114 0 [ 0 + 0 + 0.114] 0.114 B’- Y’601 = - 0.299 -0.587 0.886 0 = [ - 0 - 0 + 0.886] = 0 .886 R’- Y’601 0.701 -0.587 -0.114 1 [ 0 - 0 - 0.114] - 0.114 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the MAGENTA Co-ordinate (1,0,1) R’G’B’ Magenta = [1,0,1] +1 -1 0.413 +1 0.587 Fig 15 R’-Y’ B’-Y’ Plot these co-ordinates 0.587 -1 +1 Y’601 0.299 0.587 0.114 1 [ 0.299 + 0 + 0.114] 0.413 B’- Y’601 = - 0.299 -0.587 0.886 0 = [ - 0.299 - 0 + 0.886] = 0.587 R’- Y’601 0.701 -0.587 -0.114 1 [ 0.701 - 0 - 0.114] 0.587 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Map the RED Co-ordinate (1,0,0) R’G’B’ Red = [1,0,0] +1 0.299 -1 +1 0.701 Fig 16 R’-Y’ -0.299 B’-Y’ Plot these co-ordinates -1 +1 Y’601 0.299 0.587 0.114 1 [ 0.299 + 0 + 0] 0.299 B’- Y’601 = - 0.299 -0.587 0.886 0 = [ -0.299 - 0 + 0] = - 0.299 R’- Y’601 0.701 -0.587 -0.114 0 [ 0.701 - 0 - 0] 0.701 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

The diagram below shows how the (R’G’B’) to (Y’, B’-Y’, R’-Y’) colour space transform re-maps the R’G’B’ unity cube to an area that is 25% of the volume allowed by the maximum excursion of the new axis’. +1 If the colour difference cube was mapped using 8 bit quantization for each axis, only 25% of the codes would be legal. -1 +1 Fig 17 R’-Y’ B’-Y’ -1 This introduces the concept of valid signals (i.e., luma and color difference signals are found on the Y and B-Y, R-Y axis, but they may not be legal because they do not sit inside the 25% of volume that represents the translated colour space.) +1 Colour legalizers and colour correctors are based on colour space convertors. Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

B A F C -1 D E R’-Y’ B’-Y’ G -1 Fig 18 +1 H …In fact, the remapped colour space occupies maximum excursions in all three axis that are not convenient for engineering in either analogue or digital systems. To understand this better, let’s project all the colour points down onto the R-Y and B-Y axis. A F C -1 D E R’-Y’ B’-Y’ G -1 Fig 18 +1 H Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

B A C -1 D R’-Y’ B’-Y’ -1 Fig 19 +1 View the cube via the plane ABCD Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

B A C -1 D R’-Y’ B’-Y’ -1 Fig 20 +1 See how the points project down onto the B-Y and R-Y axis A C -1 D R’-Y’ B’-Y’ -1 Fig 20 +1 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

B A C -1 D R’-Y’ B’-Y’ -1 Fig 21 +1 Now, take away the box and look at a top-level projection of the axis on the next foil. A C -1 D R’-Y’ B’-Y’ -1 Fig 21 +1 Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

A need to scale the colour difference axis +1 B C We can see that the maximum excursions of the colour difference axis are B’-Y’ = +/- 0.886 and R’-Y’ = +/- 0.701 These are inconvenient for transmission in practical systems, so the new colour space is re-scaled (or weighted), to give the colour difference axis an excursion of +/- 0.5. 0.701 +0.866 -1 B’-Y’ +1 -0.866 -0.701 Fig 22 R’-Y’ A -1 D Scaling the above B’-Y’ and R’-Y’ axis to give maximum excursions of +/- 0.5 causes some confusion because of the different nomenclatures used to represent the scaled axis. (Note that the luminance channel Y’ still has a max excursion of ‘1’ so no scaling is necessary and it is still named Y’). Scaled B’-Y’, R’-Y’ are sometimes referred to as “weighted colour difference channels” or “± 0.5V colour difference components” or “PB & PR” . Whilst there is plenty of scope for confusion in the actual nomenclature, PB & PR are the terms often used to represent the ± 0.5V analogue outputs on the rear panels of most studio broadcast equipment. Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Create Y’, PB PR From Y,B-Y,R-Y -0.5 0.5 +0.5 PR A B C D The vector scope image now looks more symmetrical after the colour difference channels have been scaled to ± 0.5V. The scaling/weighting factors used to normalise the colour difference channels to unity excursion are : PB PR = 0.5 ( R’- Y’601 ) = 0.71327 (R’- Y’601 ) 0.701 (Equation 7) PB = 0.5 (B’- Y’601 ) = 0.56433(B’- Y’601 ) 0.886 (Equation 8) Fig 23 The matrix multiplication below is the definition of Y PBPR. We can see that the matrix multiplication required to convert R’G’B’ signals into Y, PB, PR signals now has the scaling factors of 0.56433 and 0.71327 included to normalise the B-Y and R-Y axis to ± 0.5. Y’601 0.299 0.587 0.114 R’ PB = (-0.299 x 0.564) (-0.587 x 0.564) (0.886 x 0.564) G’ (Equation 9) PR (0.701 x 0.713) (-0.587 x 0.713) (-0.114x 0.713) B’ Andy Miller © Copyright 2000 Xilinx - All Rights Reserved

Re-Normalize B’-Y’ & R’-Y’ to Create Y’PBPR Evaluating the bracketed multiplications in the previous matrix equation for Y,PBPR will give numbers that are close to those below. Some rounding has taken place, but the matrix now represents the scaling factors that have to be applied to R’G’B’ data in order to get Y’ PB PR signals. Y’601 0.299 0.587 0.114 R’ PB = -0.168736 -0.331261 0.5 G’ (Equation 10) PR 0.5 -0.418688 -0.081312 B’ A quick digression…..The Matrix transform for Y’UV Another set of colour coordinates that are found within the video industry are “YUV.” These are legacy signals from the days of analogue processing and were introduced to ensure that the composite luma & modulated chroma signal were contained within the amplitude limits of the analogue signal processing and recording equipment. The colour difference channels (R-Y’ and B-Y’) were scaled/weighted by the following factors. U = 0.493 (B’-Y’601) V = 0.877 (R’-Y’601) The matrix for R’G’ B’ to YUV conversion is given by. (Equation 11) Which evaluates to the following coefficients. (Equation 12) Y’601 0.299 0.587 0.114 R’ U = (-0.299 x 0.493) (-0.587 x 0.493) (0.886 x 0.493) G’ V (0.701 x 0.877) (-0.587 x 0.877) (-0.114x 0.877) B’ Y’601 0.299 0.587 0.114 R’ U = -0.147407 -0.289391 0.436798 G’ V 0.614777 -0.514799 -0.099978 B’ (Press your “Page Down” key to end the presentation.)