CS485/685 Computer Vision Spring 2012 Dr. George Bebis Programming Assignment 2 Due Date: 3/27/2012

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1 CS8/68 Computer Vision Spring 0 Dr. George Bebis Programming Assignment Due Date: /7/0 In this assignment, you will implement an algorithm or normalizing ace image using SVD. Face normalization is a required preprocessing step or many ace recognition algorithms to account or variations with respect to ace location, orientation (in-plane), and scale. To address these issues, you will implement an iterative algorithm based on aine transormations. Figure. An example showing acial eatures o interest or normalization. The main idea is using an aine transormation to map certain acial eatures to predetermined locations in a ixed window (see Figure ). Figure shows examples o un-normalized and normalized aces; normalization was perormed by mapping the let eye center, right eye center, and mouth center to predetermined locations in a ixed 0 x 0 window. Figure. Examples o ace images beore (top) and ater (bottom) normalization. The ace images to be used in your experiments are available rom the course s webpage. ou will use the ollowing ive acial eatures or normalization: () let eye center, () right eye center, () tip o nose, () mouth center, and () tip o chin. The coordinates o these eatures have been extracted manually and are available rom the course s webpage. In practice, however, these eatures should be extracted automatically.

2 Fixed window ou will be using a ixed window o size 8 x 0. Note that the original images have size x 9, so this choice maintains the aspect ratio o the ace images. ou can download the coordinates o the acial eatures within the ixed window rom the course s webpage. Algorithm ou need to implement the algorithm described below to compute the parameters o an aine transormation that maps the acial eatures o each image to the predetermined acial eature locations in the 8 x 0 window. This can be done in dierent ways. We describe below an iterative algorithm (see page rom reerence []) which has shown to work well: Step: Use a vector F to store the average locations o each acial eature over all ace images (i.e., (i.e., F =( P ) where P and P correspond to the average positions o the eye centers corresponds to the average position o the nose s tip corresponds to the average position o the mouth center, and P corresponds to the average position o the chin s tip); For simplicity, initialize F with the eature locations o just the irst ace image F (i.e., F = F ). Step : Using SVD, compute the aine transormation that aligns F with the predetermined positions F =( P, ) in the 8 x 0 window. An aine transormation can be used in this step. Since each pair o corresponding eatures must satisy the aine transormation equations, we have the ollowing equations: P where P =A P P =A P P =A P P =A P P =A P Since there are 0 equations (i.e., two or each eature) in 6 unknowns, the system is overdetermined and can be solved using SVD. By separating the x-coordinates rom the y- coordinates, the above equations can be rewritten as ollows:

3 where P= X X X X X X X p x = X X X p y = Compute the aligned eatures F =A F and update F by setting F = F. Step : For every ace image F i, use SVD to compute the aine transormation (A i,b i ) that aligns the acial eatures o F i with the average acial eatures F ; let s call the aligned eatures F i where F i =A i F i i. Step : Update F by averaging the aligned eature locations F i or each ace image i. Step : I F (t) - F (t-) is less than a threshold, then stop; otherwise, go to Step. The above algorithm typically converges within -7 iterations depending on the threshold used. For each ace image, it yields an aine transormation that maps the ace image to the 8 x 0 window. ou should veriy that the computed aine transormation aligns the acial eatures with the ixed acial eatures. Computing the aine transormation using SVD In Step (and ), you will need to use SVD to ind c and c. The unction svdcmp(), provided on the course s webpage, computes the SVD o an (m x n) matrix A while the unction svbksb() uses the results o svdcmp() to solve Ax=b. I have provided an example (i.e., solve_system.c) to demonstrate how to use these unctions. I would recommend that you irst test these unctions by solving an over-determined system o questions whose solution you know. Both svdcmp() and svdcmp() are rom the book Numerical Recipes in C which is reely available on-line (

4 Computing the normalized images To avoid gaps when computing the normalized images, each point in the normalized images should be determined by applying the inverse aine transormation equations as shown in Figure. The inverse aine equations are given below. So, you should iterate through every point in the output image, ind the corresponding point in the input image, and copy it over in the output image. Make sure that you understand how to implement the equation shown below correctly (i.e., multiply [X ] with every column o the inverse matrix and divide the results by the quantity shown). a b -a b a b -a b a a -a a Figure. Inverse aine mapping. Graduate Students Only Once a ace image has been normalized with respect to position, scale, and orientation, some light correction can be applied to account or non-uniorm illumination (see pages 9-0, rom reerence []). First, a linear (or higher order) model is it to the intensity o the image; or example, the ollowing model can be used: (x,y)=axy+cxy+d Each pixel (x,y) in the input image must satisy the above equation where (x,y) is the intensity value o the input image at location (x,y). Using SVD, we can ind the "best" coeicients a, b, c, and d (i.e., in a "least-squares" sense). For an N x M image, this yields NM equations with our unknowns (i.e., over-determined system). Figure shows an example; the irst row shows some input images while the second row shows the linear model it to each o these images. To correct or lighting, simply subtract the linear model rom the original image. The last row o Figure shows some results. our task in this part o the assignment will be to apply light normalization on the normalized ace images.

5 Figure. Original images ( st row), model it ( nd row) light-corrected images (rd row). What to turn in ou are to turn in a report including a print-out o your source code. our report should include the ollowing: a description o the experiments, results (i.e., include graphic output o your results), discussion o results and comparisons, and a brie summary o what you have learned. The report is very important in determining your grade or the programming assignment. Be well organized, type your reports, and include igure captions with a brie description or all the igures included in your report. Motivation and initiative are greatly encouraged and will earn extra points. Reerences [] H. Rowley, S. Baluja, and T. Kanade, Neural Network-Based Face Detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 0, No., January, 998, pp. -8. [] G. Bebis, S. Uthiram, and M. Georgiopoulos, "Face Detection and Veriication Using Genetic Search", International Journal on Artiicial Intelligence Tools, vol 9, no, pp. -6, 000.