Activity 17 Photometric Stereo

As from our previous activities, we know that the brightness of an object highly depends on the amount of light it receives from the source and the reflectance of the object's surface.

Consider the image shown above. The variable P resembles the point at the surface, p(P) is the reflectance, S(P) is the vector from P to the source, n(P) is the normal vector at point P, and r(P) is the distance from P to the source. For a point source, the brightness decreases to 1/r^2 and this proportionality is more clearly seen for a nearby point source, wherein the brightness is given by:

If the light source is at infinity, the light wave approaching the object appears similar to a plane wave, and the brightness becomes:

where S(P) becomes equal for all direction. Finally, for a line source, the brightness decreases proportional to 1/r.

Given a light source shone to an object at different positions, we can reconstruct the 3D shape of the surface of the object. The technique employed for this problem is Photometric Stereo. In this method, we capture multiple images of the surface at different locations and use these images to generate the vectors normal to the surface.


In this activity, we reconstruct the 3D shape of an object which is illuminated by a source from a very far location. We assume that the brightness of the object is proportional to the intensity I captured by the camera at different positions V which are represented by the different images and matrix shown below.

We utilize the I and V values to the equations:

where g corresponds to the product of the reflectance and the normal vector. We solve for this variable in order to determine the surface normal vector n. The next two equation are used for this step.

Since the surface normals are related to the partial derivative of the elevation of the surface f, we can integrate the derivative to finally compute the surface elevation. The equation given below represents the elevation at any point (u,v) .

Following the steps above, f(u,v) was calculated and plotted to arrive at the 3D shape of the object displayed below.

The main objective of this activity which is the reconstruction of the 3D plot of the object was fulfilled so I am giving myself a grade of 10.


Reference:
"Photometric Stereo",Activity 17 Manual in AP 186.