A13 - Photometric Stereo

Consider an object illuminated by a source. We assume that the intensity captured by the camera at point (x,y) is given by I(x,y). The goal of this activity is to approximate the shape of the surface of the object given the intensity. Multiple images of the surfaces with the sources at different locations, which we denote as v(x,y), will give information about the shape of the surface. We start with the following expression:

I = vg

where I is the intensity, v is the source location, and g is to be computed. I and v are given so that we can express g as:

g = (inverse(v*v))v*I

What we need is the norm of g which is just given by:

n = g/abs(g)

We then let z = f(x,y) be the equation of the surface. If we equate the gradient of the surface to the norm of g, we get the following expression:

df/dx = -nx/nz

df/dy = -ny/nz

where (nx, ny, nz) is the norm of g.

Finally, integrating the RHS of the two equations above, we get an approximate of f(x,y)

The codes are given below:

//We first load the gathered intensity of the object which is captured by the camera
loadmatfile('Photos.mat',['I1','I2','I3', 'I4']);
//Defining the matrices to be used
v = [0.085832 0.17365 0.98106; 0.085832 -0.17365 0.98106; 0.17365 0 0.98481; 0.16318 -0.34202 0.92542];
Im1 = I1(:);
Im2 = I2(:);
Im3 = I3(:);
Im4 = I4(:);
I = [Im1';Im2';Im3';Im4'];
g = inv(v'*v)*v'*I;
//To get the norm of g
for i=1:size(g,2);
n1(i)=sqrt((g(1,i)**2)+(g(2,i)**2)+(g(3,i)**2));
n1 = n1 + 0.000000001;
end;
n(1,:) = g(1,:)./n1';
n(2,:) = g(2,:)./n1';
n(3,:) = g(3,:)./n1';
//Differentials
dfx= -n(1,:) ./(n(3,:)+0.000000001);
dfy= -n(2,:)./(n(3,:)+0.000000001);
//Resizing
Imx = matrix(dfx,128,128);
Imy = matrix(dfy,128,128);
//Integration
intx = cumsum(Imx,2);
inty = cumsum(Imy,1);
Image = intx + inty;
mesh(Image);

The result is:

April and Rica helped me in this activity.

I rate myself 10/10 for my effort in this activity.