Please note that for these operations to be correct, we really must operate on linear brightness values. If the input image is in a non-linear brightness space RGB colors must be transformed into a linear space before these matrix operations are used.
xformrgb(mat,r,g,b,tr,tg,tb) float mat[4][4]; float r,g,b; float *tr,*tg,*tb; { *tr = r*mat[0][0] + g*mat[1][0] + b*mat[2][0] + mat[3][0]; *tg = r*mat[0][1] + g*mat[1][1] + b*mat[2][1] + mat[3][1]; *tb = r*mat[0][2] + g*mat[1][2] + b*mat[2][2] + mat[3][2]; }
float mat[4][4] = { 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, };Transforming colors by the identity matrix will leave them unchanged.
float mat[4][4] = { rscale, 0.0, 0.0, 0.0, 0.0, gscale, 0.0, 0.0, 0.0, 0.0, bscale, 0.0, 0.0, 0.0, 0.0, 1.0, };Where rscale, gscale, and bscale specify how much to scale the r, g, and b components of colors. This can be used to alter the color balance of an image.
In effect, this calculates:
tr = r*rscale; tg = g*gscale; tb = b*bscale;
float mat[4][4] = { rwgt, rwgt, rwgt, 0.0, gwgt, gwgt, gwgt, 0.0, bwgt, bwgt, bwgt, 0.0, 0.0, 0.0, 0.0, 1.0, };Where rwgt is 0.3086, gwgt is 0.6094, and bwgt is 0.0820. This is the luminance vector. Notice here that we do not use the standard NTSC weights of 0.299, 0.587, and 0.114. The NTSC weights are only applicable to RGB colors in a gamma 2.2 color space. For linear RGB colors the values above are better.
In effect, this calculates:
tr = r*rwgt + g*gwgt + b*bwgt; tg = r*rwgt + g*gwgt + b*bwgt; tb = r*rwgt + g*gwgt + b*bwgt;
float mat[4][4] = { a, b, c, 0.0, d, e, f, 0.0, g, h, i, 0.0, 0.0, 0.0, 0.0, 1.0, };Where the constants are derived from the saturation value s as shown below:
a = (1.0-s)*rwgt + s; b = (1.0-s)*rwgt; c = (1.0-s)*rwgt; d = (1.0-s)*gwgt; e = (1.0-s)*gwgt + s; f = (1.0-s)*gwgt; g = (1.0-s)*bwgt; h = (1.0-s)*bwgt; i = (1.0-s)*bwgt + s;One nice property of this saturation matrix is that the luminance of input RGB colors is maintained. This matrix can also be used to complement the colors in an image by specifying a saturation value of -1.0.
Notice that when s
is set to 0.0, the matrix is exactly
the "convert to luminance" matrix described above. When s
is set to 1.0 the matrix becomes the identity. All saturation matrices
can be derived by interpolating between or extrapolating beyond these
two matrices.
This is discussed in more detail in the note on Image Processing By Interpolation and Extrapolation.
float mat[4][4] = { 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, roffset,goffset,boffset,1.0, };This can be used along with color scaling to alter the contrast of RGB images.
If we have functions:
identmat(mat), which creates an identity matrix. xrotatemat(mat,rsin,rcos), which multiplies a matrix that rotates about the x (red) axis. yrotatemat(mat,rsin,rcos), which multiplies a matrix that rotates about the y (green) axis. zrotatemat(mat,rsin,rcos), which multiplies a matrix that rotates about the z (blue) axis.Then a matrix that rotates about the 1.0,1.0,1.0 diagonal can be constructed like this: We make an identity matrix
identmat(mat);Rotate the grey vector into positive Z
mag = sqrt(2.0); xrs = 1.0/mag; xrc = 1.0/mag; xrotatemat(mat,xrs,xrc); mag = sqrt(3.0); yrs = -1.0/mag; yrc = sqrt(2.0)/mag; yrotatemat(mat,yrs,yrc);Rotate the hue
zrs = sin(rot*PI/180.0); zrc = cos(rot*PI/180.0); zrotatemat(mat,zrs,zrc);Rotate the grey vector back into place
yrotatemat(mat,-yrs,yrc); xrotatemat(mat,-xrs,xrc);The resulting matrix will rotate the hue of the input RGB colors. A rotation of 120.0 degrees will exactly map Red into Green, Green into Blue and Blue into Red. This transformation has one problem, however, the luminance of the input colors is not preserved. This can be fixed with the following refinement:
identmat(mmat);Rotate the grey vector into positive Z
mag = sqrt(2.0); xrs = 1.0/mag; xrc = 1.0/mag; xrotatemat(mmat,xrs,xrc); mag = sqrt(3.0); yrs = -1.0/mag; yrc = sqrt(2.0)/mag; yrotatemat(mmat,yrs,yrc); matrixmult(mmat,mat,mat);Shear the space to make the luminance plane horizontal
xformrgb(mmat,rwgt,gwgt,bwgt,&lx,&ly,&lz); zsx = lx/lz; zsy = ly/lz; zshearmat(mat,zsx,zsy);Rotate the hue
zrs = sin(rot*PI/180.0); zrc = cos(rot*PI/180.0); zrotatemat(mat,zrs,zrc);Unshear the space to put the luminance plane back
zshearmat(mat,-zsx,-zsy);Rotate the grey vector back into place
yrotatemat(mat,-yrs,yrc); xrotatemat(mat,-xrs,xrc);
Example C code that demonstrates these concepts is provided for your enjoyment.
These transformations allow us to adjust image contrast, brightness, hue and saturation individually. In addition, color matrix transformations concatenate in a way similar to geometric transformations. Any sequence of operations can be combined into a single matrix using matrix multiplication.