# Color Processing

## Color Space RGB to XYZ

### simple Method

The two transformation matrix RGB to XYZ and XYZ to RGB are:

$T=\left(\begin{array}{ccc}0.412453& 0.357580& 0.180423\\ 0.212671& 0.715160& 0.072169\\ 0.019334& 0.119193& 0.950227\end{array}\right)$
${T}^{-1}=\left(\begin{array}{ccc}3.240479& -1.537150& -0.498535\\ -0.969256& 1.875992& 0.041556\\ 0.055648& -0.204043& 1.057311\end{array}\right)$

then we applied the linear transformation on the RGB color space to convert.

$\left(\begin{array}{c}R\\ G\\ B\end{array}\right)=T\cdot \left(\begin{array}{c}X\\ Y\\ Z\end{array}\right)$ and $\left(\begin{array}{c}X\\ Y\\ Z\end{array}\right)={T}^{-1}\cdot \left(\begin{array}{c}R\\ G\\ B\end{array}\right)$

Here the program

```clear all; close all; [filename, pathname] = uigetfile({'*.jpg;*.tif;*.png;*.gif;*.bmp','All Image Files';...           '*.*','All Files' },'mytitle',...           'C:\Work\myfile.jpg'); img = imread(filename); [ligne colonne dimension]=size(img); img=double(img); R=img(:,:,1); G=img(:,:,2); B=img(:,:,3); % rgb to XYZ methode 1 X = 0.412453*R + 0.357580*G + 0.180423*B; Y = 0.212671*R + 0.715160*G + 0.072169*B; Z = 0.019334*R + 0.119193*G + 0.950227*B; Rp =  3.240479*X - 1.537150*Y - 0.498535*Z; Gp = -0.969256*X + 1.875992*Y + 0.041556*Z; Bp =  0.055648*X - 0.204043*Y + 1.057311*Z; figure(  'Name','RGB --> XYZ --> RGB methode 1',...             'NumberTitle','off',...             'color',[0.3137 0.3137 0.5098]); vide=zeros(ligne,colonne); subplot(421) imshow(uint8(cat(3,R,G,B))) title('Image origine') subplot(423) imshow(uint8(cat(3,R,vide,vide))) title('Image R origine') subplot(425) imshow(uint8(cat(3,vide,G,vide))) title('Image G origine') subplot(427) imshow(uint8(cat(3,vide,vide,B))) title('Image B origine')  subplot(422) imshow(uint8(cat(3,Rp,Gp,Bp))) title('Image reconstuite')  subplot(424) imshow(uint8(cat(3,Rp,vide,vide))) title('Image R reconstuite') subplot(426) imshow(uint8(cat(3,vide,Gp,vide))) title('Image G reconstuite') subplot(428) imshow(uint8(cat(3,vide,vide,Bp))) title('Image B reconstuite')```

and an example

### Method with gamma correctio

In this case, we apply a gamma correction on the RGB color. This correction is performed thanks to:

$R\text{'}=f\left(R\right),$ $G\text{'}=f\left(G\right),$ $B\text{'}=f\left(B\right)$

with
where s=0.03928. After the gamma correction, we apply the RGB to XYZ conversion above. To reverse the transformation we reverse the gamma correction too. We use the next equations:
$R=f\text{'}\left(R\text{'}\right),$ $G=f\text{'}\left(G\text{'}\right),$ $B=f\text{'}\left(B\text{'}\right)$

Where s = 0.0031308.
Here the program:

```% rgb to XYZ methode 2 Rn=R/255; Gn=G/255; Bn=B/255; seuil = 0.03928; Rp= (Rn>seuil)  .* (((Rn+0.055)./1.055).^2.4); Rp= Rp + (Rn<=seuil) .* (Rn/12.92); Gp= (Gn>seuil)  .* (((Gn+0.055)./1.055).^2.4); Gp= Gp + (Gn<=seuil) .* (Gn/12.92); Bp= (Bn>seuil)  .* (((Bn+0.055)./1.055).^2.4); Bp= Bp + (Bn<=seuil) .* (Bn/12.92); X = 0.412453*Rp + 0.357580*Gp + 0.180423*Bp; Y = 0.212671*Rp + 0.715160*Gp + 0.072169*Bp; Z = 0.019334*Rp + 0.119193*Gp + 0.950227*Bp; Rpp =  3.240479*X - 1.537150*Y - 0.498535*Z; Gpp = -0.969256*X + 1.875992*Y + 0.041556*Z; Bpp =  0.055648*X - 0.204043*Y + 1.057311*Z; seuil = 0.0031308; Rr= (Rpp>seuil)  .* (1.055*Rpp.^(1/2.4)-0.055); Rr= Rr + (Rpp<=seuil)  .* (Rpp*12.92); Gr= (Gpp>seuil)  .* (1.055*Gpp.^(1/2.4)-0.055); Gr= Gr + (Gpp<=seuil)  .* (Gpp*12.92); Br= (Bpp>seuil)  .* (1.055*Bpp.^(1/2.4)-0.055); Br= Br + (Bpp<=seuil)  .* (Bpp*12.92); Rr=Rr*255; Gr=Gr*255; Br=Br*255; figure(  'Name','RGB --> XYZ --> RGB methode 2',...             'NumberTitle','off',...             'color',[0.3137 0.3137 0.5098]); vide=zeros(ligne,colonne); subplot(4,5,1) imshow(uint8(cat(3,R,G,B))) title('Image origine') subplot(4,5,6) imshow(uint8(cat(3,R,vide,vide))) title('Image R origine') subplot(4,5,11) imshow(uint8(cat(3,vide,G,vide))) title('Image G origine') subplot(4,5,16) imshow(uint8(cat(3,vide,vide,B))) title('Image B origine') subplot(4,5,2) imshow(uint8(cat(3,Rp,Gp,Bp)*255)) title('Image RGB linéarisé gamma correction') subplot(4,5,7) imshow(uint8(cat(3,Rp,vide,vide)*255)) title('Image R linéarisé gamma correction') subplot(4,5,12) imshow(uint8(cat(3,vide,Gp,vide)*255)) title('Image G linéarisé gamma correction') subplot(4,5,17) imshow(uint8(cat(3,vide,vide,Bp)*255)) title('Image B linéarisé gamma correction') %subplot(4,5,3) subplot(4,5,8) imagesc(X,[min(min(X)) max(max(X))]) colormap('gray') axis image title('Image X') subplot(4,5,13) imagesc(Y,[min(min(Y)) max(max(Y))]) colormap('gray') axis image title('Image Y') subplot(4,5,18) imagesc(Z,[min(min(Z)) max(max(Z))]) colormap('gray') axis image title('Image Z') subplot(4,5,4) imshow(uint8(cat(3,Rpp,Gpp,Bpp)*255)) title('Image non linéarisé gamma correction') subplot(4,5,9) imshow(uint8(cat(3,Rpp,vide,vide)*255)) title('Image R reconstuite linéarisé gamma correction') subplot(4,5,14) imshow(uint8(cat(3,vide,Gpp,vide)*255)) title('Image G reconstuite linéarisé gamma correction') subplot(4,5,19) imshow(uint8(cat(3,vide,vide,Bpp)*255)) title('Image B reconstuite linéarisé gamma correction')  subplot(4,5,5) imshow(uint8(cat(3,Rr,Gr,Br))) title('Image reconstuite non linéarisé gamma correction')  subplot(4,5,10) imshow(uint8(cat(3,Rr,vide,vide))) title('Image R reconstuite non linéarisé gamma correction') subplot(4,5,15) imshow(uint8(cat(3,vide,Gr,vide))) title('Image G reconstuite non linéarisé gamma correction') subplot(4,5,20) imshow(uint8(cat(3,vide,vide,Br))) title('Image B reconstuite non linéarisé gamma correction')  ```

An example