Fall 2004 

A color image can be described by three matrixes R,G,B, where R_{ij},
G_{ij} and B_{ij} are the amount of red, green and blue in the picture
at position (i,j). It is custom to encode with an integer from 0 to 255, which can also
be seen as 1 byte = 8 bits because 2^{8}=256.
The picture to the left with width 2 and height 2 can be described by three 2x2 matrices:
 0 255   0 0   255 0  R =   G =   B =    0 255   255 255   0 0 The data can also be described by one matrix alone, where we give the (rgb) value at each point  [0 0 1] [1 0 0]   blue red  A =   =    [0 1 0] [1 1 0]   green yellow  
Evenso the range of the coordinates is limited and discretized, we can think of a picture
as an element in a linear space. We can scale pictures, project onto subspaces or make
linear combinations. To illustrate this, we use Mathematica, which allows very easily to
import and export pictures. The pictures are matrices which can be manipulated.
For example
A=Import["zermatt.jpg"];imports a picture of a mountain near Zermatt in Switzerland. 
Here is a Mathematica procedure, which takes such imported image data and
applies a filter
ApplyFilter[F_,A_]:=Graphics[Raster[F[A[[1,1]]],A[[1,2]],A[[1,3]],A[[1,4]]],A[[2]],A[[3]],A[[4]]];Mathematica keeps more than just the color matrix. Important to know is that A[[1,1]] is the matrix which contains the RGB values of each picture. For example A = Graphics[Raster[{{{0, 255, 0}, {255, 255, 0}}, {{0, 97, 255}, {255, 0, 0}}}, {{0, 0}, {2, 2}}, {0, 255}, ColorFunction > RGBColor], ImageSize > {2, 2}, PlotRange > {{0, 1}, {1, 2}}, AspectRatio > Automatic]is, what Mathematica has stored about the 2x2 pixel picture above. We access the RGB matrix with A[[1,1]]. 
Here is an example, which "scales" a picture:
A=Import["zermatt.jpg"]; Dim[x_]:=Round[x/2]; B1=ApplyFilter[Dim,A]; Brighten[x_]:=Round[120+2x/3]; B2=ApplyFilter[Brighten,A]; Display["dimmed.jpg",B1,"JPEG"]; Display["brightend.jpg",B2,"JPEG"]; 
Here is an example of a projection onto the red green or blue planes:
red[{r_,g_,b_}]:={r,0,0}; Red[S_]:=Table[red[S[[i,j]]],{i,Length[S]},{j,Length[S[[1]]]}]; green[{r_,g_,b_}]:={0,g,0}; Green[S_]:=Table[green[S[[i,j]]],{i,Length[S]},{j,Length[S[[1]]]}]; blue[{r_,g_,b_}]:={0,0,b}; Blue[S_]:=Table[blue[S[[i,j]]],{i,Length[S]},{j,Length[S[[1]]]}]; B1=ApplyFilter[Red,A]; B2=ApplyFilter[Green,A]; B3=ApplyFilter[Blue,A]; Display["red.jpg",B1,"JPEG"]; Display["green.jpg",B2,"JPEG"]; Display["blue.jpg",B3,"JPEG"]; 
Here is an example of a rotation or reflection in color space.
rotate[{r_,g_,b_}]:={b,r,g}; Rotate[S_]:=Table[rotate[S[[i,j]]],{i,Length[S]},{j,Length[S[[1]]]}]; reflect[{r_,g_,b_}]:={255r,255g,255b}; Reflect[S_]:=Table[reflect[S[[i,j]]],{i,Length[S]},{j,Length[S[[1]]]}]; B4=ApplyFilter[Rotate,A]; B5=ApplyFilter[Reflect,A]; Display["rotate.jpg",B4,"JPEG"]; Display["reflect.jpg",B5,"JPEG"]; 
The procedure
Combine[F_,U_,V_]:=Graphics[Raster[F[U[[1,1]],V[[1,1]]],U[[1,2]],U[[1,3]],U[[1,4]]],U[[2]],U[[3]],U[[4]]];allows to combine two pictures using a function f[x,y].  
Here is an example on how to take two pictures and make a
simple linear combination, where we take 1/3 of a mountain
and 2/3 of a person and mix it up.
A1=Import["zermatt.jpg"]; A2=Import["oliver.jpg"]; LinearCombination[a_,b_]:=Function[{x,y},a*x+b*y]; B=Combine[LinearCombination[1/3,2/3],A1,A2]; Display["mixture.jpg",B,"JPEG"];

Note that each picture is in a huge dimensional space. The pictures we dealt with here had size 500x375 which are 187'500 pixels. Each pixels is encoded by 3 bytes so that this picture weights 1/2 megabytes. Indeed, uncompressed, the pictures needs this amount of space to be stored. Here is an other example of a linear combination of two vectors rsp. pictures. The combination amoung depends now on the pixel position.  
F1[a_,b_]:=Function[{x,y}, Module[{}, m=Length[x[[1]]]; (* width *) n=Length[x]; (* height *) z=x; Do[ If[jj>a*m, K=(ja*m)/(b*ma*m)]; If[j>b*m, K=1]; z[[i,j]]=Round[K*x[[i,j]]+(1K)*y[[i,j]]], {i,n},{j,m} ]; z ] ] B=Combine[F1[1/3,2/3],A1,A2]; Display["leftrighttransition.jpg",B,"JPEG"]; 
F2[r1_,r2_]:=Function[{x,y}, Module[{}, m=Length[x[[1]]]; mc=m/2; n=Length[x]; nc=n/2; rr1=(Min[m,n]*r1)^2; rr2=(Min[m,n]*r2)^2; z=x; Do[ rr=(inc)^2+(jmc)^2; K=0; If[rr1<=rr<=rr2, K=(rrrr1)/(rr2rr1)]; If[rr>rr2, K=1]; z[[i,j]]=Round[K*x[[i,j]]+(1K)*y[[i,j]]], {i,n},{j,m} ]; z ] ] B=Combine[F2[1/10,1/2],A1,A2]; Show["circle.jpg",B,"JPEG"]; 
To try it out, donwload the two pictures as well as the code picturemerge.m
and run it. You don't want to run Mathematica in the kiddie mode (graphical
user interface) don't you?.
math < picturemerge.m >out &You have to wait until the newly produced files are in the directory. There are many advantages not to use the graphical Mathematica interface
To run Mathematica from the commandline in OSX, add something like alias math='/Applications/Mathematica\ 5.0.app/Contents/MacOS/MathKernel'to your shell. (more details). Nothing prevents you now to write your own image filters using a computer algebra system. 