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Problem 2 Spatially Variant Filtering (Matlab)Problem 4 Focus of expansionProblem 5 Center of rotationProblem 6 Stereo and Motion estimationProblem 7 Bayes’ TheoremProblem 8 Minimum-risk decision criterion6.801/6.866 Machine Vision Problem set 3 Assigned: 10/24/02 Due: 11/05/02 at 2:30pm Problem 1 Lucas-Kanade tracker (Matlab) Lucas & Kanade’s optic flow technique proposes to solve the brightness constant constraint equation (BCCE) by assuming constant flow over a patch: −=∑∑∑∑∑∑tytxyyxyxxIIIIvuIIIIII22 (1) a) Implement a single-iteration Lucas-Kanade tracker. Your function should have the following syntax: function [dx,dy] = lucaskanade(I1, I2, xl, yt, xr, yb, wsize)where I1 and I2 are the images, (xl,yt) and (xr,yb) are the top-left and bottom-right corners of the region where you want the optical flow computed, and wsize is the size of the squared image patch used to solve equation (1). Test your implementation using LK-0001.bmp and LK-0002.bmp with different window sizes: 5, 9, 13 and 17. - For each window size, plot the optical flow for the region (xl,yt);(xr,yb)=(80,170);(150,220) using the Matlab function quiver. - Is the estimated optical flow constant over the whole region? Should the true optical flow differ from your estimate? Why or why not? - Does the window size change the estimated optical flow? Why or why not? You should submit your file lucaskanade_yourlastname.m and print a copy of your code. Hint: To compute the image gradient, you should first blur image I1 and then use the Matlab function gradient. The temporal gradient can be approximated by the difference between blurred versions of I1 and I2. b) Derive the Lucas-Kanade solution (Eq. 1) from the BCCE by minimizing the following objective function: ()∑Ω∈++=yxtyxIvyxIuyxIvuE,2),(),(),( (2) Hint: Rewrite the objective function as the least-squares solution of a linear system of equations in matrix form. Problem 2 Spatially Variant Filtering (Matlab) The next two problems use pyramid image processing. Download the matlabPyrTools from http://www.cns.nyu.edu/~eero/software.html. When forming pyramid decompositions for these problems, you may always use the default decomposition filters. For both problems, you should submit your Matlab code and include a printout. Subjectively, our visual world appears to us to be high resolution everywhere. However, we have much higher spatial resolution in the center of our field of view than in the periphery. In this problem, we will synthesize an image approximating our visual resolution as a function of eccentricity.The figure shows a plot of the minimum angle resolvable as a function of the visual eccentricity. The visual eccentricity is measured in degrees away from the center of fixation. (From Rodieck, "The First Steps in Seeing", Sinauer, 1998). Approximate acuity, a, in minutes of arc (60 minutes to a degree) as a function of eccentricity, e, in degrees, by the expression, a = 0.23 e + 0.7 We will create an image with the effective spacing of the pixels equal to the angular size of the acuity limit. In the figure, that limit is defined as the white space between two ends of a circle. Adjacent black, white, black pixels could approximately represent that circle opening if the pixel spacing were equal to the angular size of the acuity limit. Assume that the image (or monitor) is square, and that you view it from a distance of three times the length of one side of the image. Where convenient, you may assume angles are small enough so thatθθ≈)tan( . a) How many evenly spaced pixels per side does the image need to have in order that the highest resolution part of the image has one pixel per length of finest acuity? Assume that some image point lies at zero eccentricity. b) Let the upper left corner of the image be (0,0), and the right and bottom edges of the picture be at a distance 1 from this corner. Assume that the upper left corner is the center of fixation. What effective pixel spacing, as a function of these units, causes the pixel spacing to equal the spatial acuity for the corresponding eccentricity? c) We can approximate images of this resolution by using a Gaussian pyramid, which generates images at different numbers of pixel samples, dividing the number of pixels by two at each level of the Gaussian pyramid. Start from an image at the full resolution of part (a). Each pyramid level increases the effective size of its pixels by a factor of two in each dimension. As a function of the coordinate system used in (b), by how many factors of two should the resolution of the original image be reduced as a function of position in the image in order tosimulate the human visual acuity, assuming the viewer stares at the upper left corner of the image? d) The expression in (c) involves fractional pyramid levels. We can visually approximate images at those intermediate resolution levels by linearly interpolating between our Gaussian pyramid levels. On the class web site is a 2000x2000 image, which should be more than enough pixels for you. Crop that image to the desired resolution such that the upper left corner will be at the maximum visual acuity, when viewed from 3 picture lengths away. Use the Gaussian pyramid to create an image that simulates the fall-off in visual acuity, assume the fixation point is at the upper left corner. At any given pixel, determine the coefficients for interpolating between images by linearly interpolating the corresponding pixel dimensions. Hint: You will want to use the upBlur function to transform the Gaussian pyramid levels to all have the same number of pixels. Assume that a pyramid level after upBlur has effectively the same number of pixels (in terms of picture content) as the original pyramid band before the upBlur operation. That is a reasonable approximation (take 6.341 for the details that we're glossing over here). Problem 3 Noise Removal (Matlab) We study noise removal using a wavelet/QMF pyramid. a) Download the three "training set" images from the web site. Convert each to a QMF pyramid. For the subband selecting vertical oriented structure, at the second highest resolution level, plot the histogram of the coefficient values, for each of the 3 training set images. Do these images share the subband histograms described in class?


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MIT 6 801 - Study Guide

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