Report of Project 3

Name: Lijie Heng
Date: Oct 23th, 2007

Goal

This project is to implement a system to construct a height field from a series of images of a diffuse object under different point light sources, including calibrating the lighting directions, finding the best fit normal and albedo at each pixel,and then reconstucting the 3D modelbased on the depth at each pixel.

Results

Implementation of Each Step

Calibration

To determine the directions of point light sources, a shiny chrome sphere in the same location as all the other objects are used. Since we know the shape of this object, we can determine the normal at any given pointon its surface, and therefore we can also compute the reflection direction for the brightest spot on the surface.

To determine the direction vector L:

Centroid of the sphere
Using the chrome.mask, find out the average of the non-zero pixels location(xc, yc)as the center of the ball.
Radius of the sphere
Find out the maximal distance between the boundary pixels to the center(xc, yc) as the radius r.
Centroid of the highlight pixels
Find out the average of the non-zero pixels from each of the 12 image, as the hightlight point(xh, yh).
Light direction For each of the 12 image, computer the normal n of the highlight point:
n(x) = (xh-xc)/r;
n(y) = (yh-yc)/r;
n(z) = sqrt(r^2-n(x)^2-n(y)^2);

After we get the normal n for each pixel, using L = 2(N * R)N-R, R=(0, 0, 1) to get the light direction L. The light directions for 12 images in each sample dataset is here.

Solve for Normals

Through minimizing the following weighted linear least square problem, we get image normal for each pixels
Q = sum{(I(i)^2-I(i)*L(i)*g')^2}
g = kd*n;
In order to minimize Q, we make the derivtive of Q about g equal zero. Thus, foreach pixel, we get three equations for each g:
A = [sum(I.^2*L(:,1).^2), sum(I.^2*L(:,2)*L(:,1)), sum(I.^2*L(:,3)*L(:,1));
sum(I.^2*L(:,1)*L(:,2)), sum(I.^2*L(:,2).^2), sum(I.^2*L(:,3)*L(:,2));
sum(I.^2*L(:,1)*L(:,3)), sum(I.^2*L(:,2)*L(:,3)), sum(I.^2*L(:,3).^2))];
b = [sum(I.^3*L(:,1)); sum(I.^3*L(:,2)); sum(I.^3*L(:,3))];
g = A\b;

Surface Fitting

For each pixel, we compute the depth z based on the following two equations:
n(z)z(i,j)-n(z)z(i+1,j)=n(x);
n(z)z(i,j)-n(z)z(i,j+1)=n(y);
It can be written as the matix equation Mz = v. Using sparse matrix in matlab to solve it, we get the depth z for each value.

To Run

Summary