Assignment 2

1. Questions (30%)

1. Detecting and computing symmetries of 3D objects has many applications in shape analysis, shape matching, shape representation, object detection, object recognition, etc. Assume you have been given an oriented point cloud: a dense set of 3D points (x,y,z) and corresponding surface normals (n_x,n_y,n_z), evenly distributed over the surface of some object. Using the Hough transform, develop an algorithm for detecting the planes of symmetry for this object. Hint: use simple examples and consider this problem in 2D before generalizing your approach to arbitrary shapes in 3D.
   
   
2. Imagine you are given two vectors, a "signal" S and a "template" T. Assume T is shorter than S. Now, you want to find the position within S at which T is the best match according to the sum of squared differences (SSD) criterion. That is, given an offset k at which you are looking for T within S, you want to find the k that minimizes:

   

   Show how you can do this without an explicit loop over k by computing two convolutions:
   1. S convolved with some variant of T
      Hint: this won't necessarily be T itself, but some simple transformation. Look at the equation above, and compare with the definition of discrete convolution.
   2. The vector consisting of the square of each element of S, convolved with a vector the same length as T but consisting of all ones.
   You will need this result for the face detection portion of the assignment below.
   
   
3. Suppose you only need to determine the most significant eigenvector (i.e., that whose corresponding eigenvalue has the largest magnitude) of an arbitrary NxN matrix A. Although computing the full SVD of A will certainly do the job, this approach is rather inefficient since it requires computing a complete decomposition of the matrix. This question asks you to devise a more efficient strategy.
   Assume A is not singular and has eigenvalues and corresponding eigenvectors . First, convince yourself that we can write any arbitrary N-element vector as a linear combination of these eigenvectors:

   

   Next, note what happens when we multiply v by A:

   

   Use this result to devise a simple iterative algorithm for approximating the most significant eigenvector of A. Discuss the expected running time of your algorithm as it relates to the eigenvalues of A. There is a situation (property of A) in which you will not be able to identify the eigenvector with the largest eigenvalue using this trick. What is it? What is a real-world application of this algorithm? Lastly, describe how to use this same property to find the second most significant eigenvector, the third, etc.

2. Aligned database of faces (20%)

Our first task is to assemble a database of aligned and cropped images of the faces of people who have take this class over the years. You should have received an email that contains directions for downloading a collection of face images (if this is not the case then please e-mail the instructor). There should be two sets of images (neutral_XX.jpg and smile_XX.jpg).

Write a simple program in MATLAB that:

• Loads and displays each image in turn
• Lets the user click on the centers of the eyes in each image and stores these coordinates (use the getpts function).
• Warps the images so that the eye points are mapped to fixed locations, 100 pixels apart horizontally (use the imtransform and cp2tform functions).
• Crops out an appropriate section of the image (e.g., 300 pixels tall by 200 pixels wide). Note the easiest way to do this might be to use the XData and YData options to imtransform
• Saves the results so you don't have to repeat the clicking multiple times.

At this point you should have a collection of equal-sized images with the person's eyes in roughly the same position.

Next compute an intensity image of the "average face" in which each pixel is set to the average intensity of that pixel across your set of aligned and cropped images. Note the quality of this average face (i.e., how well it resembles an actual face with distinguishable eyes, mouth, etc.) will depend on the quality of your alignment and is a great way to test that you got the first part correct. Submit this image as part of your write-up. Although not required for this assignment, You may want to also consider using more than just the eyes to perform the alignment (e.g., have the user click on the tip of nose, corners of the mouth, etc.).

3. Face detection (40%)

Because these images are large it is important to use the FFT to efficiently perform the convolutions required to compute the SSD between the template and target image (recall that you derived the necessary relationship between SSD and convolution in part 1 of this assignment).

Build a simple face detector that reports the presence of a face when the SSD is below a threshold. Play around with the value of this threshold and show images that indicate its relationship to the sensitivity of your detector. Extend this basic face detector in the following two ways:

• Perform non-maximum suppression whereby you return only the strongest match within local image regions (similar to what you did for your edge and corner detectors).
• Compute the SSD image for the target image at different scales (consult help imresize in MATLAB). Look for faces at 100%, 75%, 50% and 25% of the original size.

• How well does your detector work? How would you evaluate its performance in a principled way (i.e., how would you scientifically compare multiple face detectors)?
• Is there significant difference in its accuracy when applied to images of people not in our class and thus not included in the construction of the template?
• What types of false positives does it return (i.e. what image regions tend to resemble your face template)?
• How might you automate the alignment/cropping stage from before for a huge collection of face images using this detector? How would you create your template?

4. Eigenfaces (10%)

Apply SVD to this matrix of face images (consult help svd in MATLAB, especially the svd(X,O) syntax).

Include in your write-up a plot of the singular values and show the five most significant eigenfaces as images. Use a visualization of your choice (simple mapping to grayscale is fine).

Project each face image onto the FIRST TWO PRINCIPAL COMPONENTS. Hint: these are stored in V^T, but can also be computed as the dot product of a face image organized into a vector and each principal component. Now you have two coordinates associated with each face image in your database. Create a scatter plot of these 2-D points (consult help scatter in MATLAB). Answer the following questions:

• The eigenfaces encode the principal sources of variation in your dataset (e.g., absence/presence of facial hair, skin tone, glasses, etc.). What do the eigenfaces you computed correspond to?
• Using the plot of singular values as a guide, choose a number of principal components that you would use to form a compact and accurate representation of your face database. Justify/explain your decision. What is the total approximation error introduced by using this number of components in place of the original images? What is the total amount of space required to store this compressed representation? What is the compression ratio (i.e., ratio of compressed size to original size). Show your work for these calculations.

Submitting

This assignment is due Tuesday, October 16, 2007 at 11:59 PM. Please see the general notes on submitting your assignments, as well as the late policy and the collaboration policy.

Please submit:

• Your write-up as an HTML file, with links to your code and input/output images.

Note that programming in Matlab is not an excuse to write unreadable code. You are expected to use good programming style, including meaningful variable names, a comment or three describing what the code is doing, etc. Also, all images you submit should have been saved with the "imwrite" function - do not submit screen captures of the image window.