## Description

1 Lucas-Kanade Tracking In this section you will be implementing a simple Lucas & Kanade tracker with one single template. In the scenario of two-dimensional tracking with a pure translation warp function, 1 W(x; p) = x + p . (1) The problem can be described as follows: starting with a rectangular neighborhood of pixels N ∈ {xd} D d=1 on frame It, the Lucas-Kanade tracker aims to move it by an offset p = [px, py] T to obtain another rectangle on frame It+1, so that the pixel squared difference in the two rectangles is minimized: p ∗ = arg minp = X x∈N ||It+1(x + p) − It(x)||2 2 (2) = It+1(x1 + p) . . . It+1(xD + p) − It(x1) . . . It(xD) 2 2 (3) Q1.1 (5 points) Starting with an initial guess of p (for instance, p = [0, 0]T ), we can compute the optimal p ∗ iteratively. In each iteration, the objective function is locally linearized by first-order Taylor expansion, It+1(x 0 + ∆p) ≈ It+1(x 0 ) + ∂It+1(x 0 ) ∂x0T ∂W(x; p) ∂pT ∆p (4) where ∆p = [∆px, ∆py] T , is the template offset. Further, x 0 = W(x; p) = x + p and ∂I(x 0 ) ∂x0T is a vector of the x− and y− image gradients at pixel coordinate x 0 . In a similar manner to Equation 3 one can incorporate these linearized approximations into a vectorized form such that, arg min ∆p ||A∆p − b||2 2 (5) such that p ← p + ∆p at each iteration. • What is ∂W(x;p) ∂pT ? • What is A and b? • What conditions must AT A meet so that a unique solution to ∆p can be found? Q1.2 (15 points) Implement a function with the following signature [dp x,dp y] = LucasKanade(It, It1, rect) that computes the optimal local motion from frame It to frame It+1 that minimizes Equation 3. Here It is the image frame It, It1 is the image frame It+1, and rect is the 4-by-1 vector that represents a rectangle describing all the pixel coordinates within N within the image frame It. The four components of the rectangle are [x1, y1, x2, y2] T , where [x1, y1] T is the top-left corner and [x2, y2] T is the bottom-right corner. The rectangle is inclusive, i.e., in includes all the four corners. To deal with fractional movement of the template, you will need to interpolate the image using the MATLAB function interp2. You will also need to iterate the estimation until the change in ||∆p||2 2 is below a threshold. Q1.3 (10 points) Write a script testCarSequence.m that loads the video frames from carseq.mat, and runs the Lucas-Kanade tracker that you have implemented in the previous 2 task to track the car. carseq.mat can be located in the data directory and it contains one single three-dimensional matrix: the first two dimensions correspond to the height and width of the frames respectively, and the third dimension contain the indices of the frames (that is, the first frame can be visualized with imshow(frames(:, :, 1))). The rectangle in the first frame is [x1, y1, x2, y2] T = [60, 117, 146, 152]T . Report your tracking performance (image + bounding rectangle) at frames 1, 100, 200, 300 and 400 in a format similar to Figure 1. Also, create a file called carseqrects.mat, which contains one single n×4 matrix rects, where each row stores the rect that you have obtained for each frame, and n is the total number of frames. Figure 1: Lucas-Kanade Tracking with One Single Template Q1.4 (20 points) As you might have noticed, the image content we are tracking in the first frame differs from the one in the last frame. This is understandable since we are updating the template after processing each frame and the error can be accumulating. This problem is known as template drifting. There are several ways to mitigate this problem. Iain Matthews et al. (2003, https://www.ri.cmu.edu/publication_view.html?pub_id=4433) suggested one possible approach. Write a script testCarSequenceWithTemplateCorrection.m with a similar functionality to Q1.3, but with a template correction routine incorporated. Save the resulting rects as carseqrects-wcrt.mat, and also report the performance at those frames. An example is given in Figure 2. Figure 2: Lucas-Kanade Tracking with Template Correction Here the green rectangles are created with the baseline tracker in Q1.3, the yellow ones with the tracker in Q1.4. The tracking performance has been improved non-trivially. Note that you do not necessarily have to draw two rectangles in each frame, but make sure that the performance improvement can be easily visually inspected. 2 Lucas-Kanade Tracking with Appearance Basis The tracker we have implemented in the first secion, with or without template drifting correction, may suffice if the object being tracked is not subject to drastic appearance variance. However, in real life, this can hardly be the case. We have prepared another sequence sylvseq.mat (the initial rectangle is [102, 62, 156, 108]), with exactly the same 3 format as carseq.mat, on which you can test the baseline implementation and see what would happen. In this section, you will implement a variant of the Lucas-Kanade tracker (see Section 3.4 in [2]), to model linear appearance variation in the tracking. 2.1 Appearance Basis One way to address this issue is to use eigen-space approach (aka, principal component analysis, or PCA). The idea is to analyze the historic data we have collected on the object, and produce a few bases, whose linear combination would most likely to constitute the appearance of the object in the new frame. This is actually similar to the idea of having a lot of templates, but looking for too many templates may be expensive, so we only worry about the principal templates. Mathematically, suppose we are given a set of k image bases {Bk} K k=1 of the same size. We can approximate the appearance variation of the new frame It+1 as a linear combination of the previous frame It and the bases weighted by w = [w1, . . . , wK] T , such that It+1(x) = It(x) +X K k=1 wkBk(x) (6) Q2.1 (5 points) Express w as a function of It+1, It, and {Bk} K k=1, given Equation 6. Note that since the Bk’s are orthobases, thay are orthogonal to each other. 2.2 Tracking Given K bases, {Bk} K k=1, our goal is then to simultaneously find the translation p = [px, py] T and the weights w = [w1, . . . , wK] T that minimizes the following objective function: min p,w = X x∈N ||It+1(x + p) − It(x) − X K k=1 wkBk(x)||2 2 . (7) Again, starting with an initial guess of p (for instance, p = [0, 0]T ), one can linearize It+1(x+ p + ∆p) with respect to ∆p. In a similar manner to Equation 5 one can incorporate these linearized approximations into a vectorized form such that, arg min ∆p,w ||A∆p − b − Bw||2 2 . (8) As discussed in Section 3.4 of [2] (ignore the inverse compositional discussion) this can be simplified down to arg min ∆p ||B ⊥(A∆p − b)||2 2 (9) where B⊥ spans the null space of B. Note that ||B⊥z||2 2 = ||z − BBT z||2 2 when B is an orthobasis. Q2.2 (15 points) Implement a function with the following signature [dp x,dp y] = LucasKanadeBasis(It, It1, rect, bases) 4 where bases is a three-dimensional matrix that contains the bases. It has the same format as frames as is described earlier and can be found in sylvbases.mat. Q2.3 (15 points) Write a script testSylvSequence.m that loads the video frames from sylvseq.mat and runs the new Lucas-Kanade tracker to track the sylv (the toy). The bases are available in sylvbases.mat in the data directory. The rectangle in the first frame is [x1, y1, x2, y2] T = [102, 62, 156, 108]T . Please report the performance of this tracker at frames 1, 200, 300, 350 and 400 (the frame + bounding box), in comparison to that of the tracker in the first section. That is, there should be two rectangles for each frame, as exemplified in Figure 3. Also, create a sylvseqrects.mat for all the rects you have obtained for each frame. It should contain one single N × 4 matrix named rects, where N is the number of frames, and each row contains [x1, y1, x2, y2] T , where [x1, y1]T is the coordinate of the top-left corner of the tracking box, and [x2, y2] T the bottom-right corner. Figure 3: Lucas-Kanade Tracking with Appearance Basis 3 Affine Motion Subtraction In this section, you will implement a tracker for estimating dominant affine motion in a sequence of images and subsequently identify pixels corresponding to moving objects in the scene. You will be using the images in the file aerialseq.mat, which consists aerial views of moving vehicles from a non-stationary camera. 3.1 Dominant Motion Estimation In the first section of this homework we assumed the the motion is limited to pure translation. In this section you shall implement a tracker for affine motion using a planar affine warp function. To estimate dominant motion, the entire image It will serve as the template to be tracked in image It+1, that is, It+1 is assumed to be approximately an affine warped version of It. This approach is reasonable under the assumption that a majority of the pixels correspond to the stationary objects in the scene whose depth variation is small relative to their distance from the camera. Using a planar affine warp function you can recover the vector ∆p = [p1, . . . , p6] T , x 0 = W(x; p) = 1 + p1 p2 p4 1 + p5 x y + p3 p6 . (10) One can represent this affine warp in homogeneous coordinates as, x˜ 0 = Mx˜ (11) 5 where, M = 1 + p1 p2 p3 p4 1 + p5 p6 0 0 1 . (12) Note that M will differ between successive image pairs. Starting with an initial guess of p = 0 (i.e. M = I) you will need to solve a sequence of least-squares problem to determine ∆p such that p → p + ∆p at each iteration. Note that unlike previous examples where the template to be tracked is usually small in comparison with the size of the image, image It will almost always not be contained fully in the warped version It+1. Hence, one must only consider pixels lying in the region common to It and the warped version of It+1 when forming the linear system at each iteration. Q3.1 (15 points) Write a function with the following signature M = LucasKanadeAffine(It, It1) where M is the affine transformation matrix, and It and It1 are It and It+1 respectively. LucasKanadeAffine should be relatively similar to LucasKanade from the first section. 3.2 Moving Object Detection Once you are able to compute the transformation matrix M relating an image pair It and It+1, a naive way for determining pixels lying on moving objects is as follows: warp the image It using M so that it is registered to It+1 and subtract it from It+1; the locations where the absolute difference exceeds a threshold can then be declared as corresponding to locations of moving objects. To obtain better results, you can check out the following MATLAB functions: bwselect, bwareaopen, imdilate, and imerode. Q3.2 (10 points) Using the function you have developed for dominant motion estimation, write a function with the following signature mask = SubtractDominantMotion(image1, image2) where image1 and image2 form the input image pair, and mask is a binary image of the same size that dictates which pixels are considered to be corresponding to moving objects. You should invoke LucasKanadeAffine in this function to derive the transformation matrix M, and produce the aforementioned binary mask accordingly. Q3.3 (10 points) Write a script testAerialSequence.m that loads the image sequence from aerialseq.mat and run the motion detection routine you have developed to detect the moving objects. Report the performance at frames 30, 60, 90 and 120 with the corresponding binary masks superimposed, as exemplified in Figure 4. Feel free to visualize the motion detection performance in a way that you would prefer, but please make sure it can be visually inspected without undue effort. The MATLAB function imfuse may be useful. 6 Figure 4: Lucas-Kanade Tracking with Appearance Basis 4 Deliverables The assignment should be submitted to canvas. The writeup should be submitted as a pdf named .pdf. The code should be submitted as a zip named .zip. The zip when uncompressed should prodcue the following files. • LucasKanade.m • LucasKanadeAffine.m • LucasKanadeBasis.m • SubtractDominantMotion.m • testCarSequence.m • testSylvSequence.m • testCarSequenceWithTemplateCorrection.m • testAerialSequence.m • carseqrects.mat • carseqrects-wcrt.mat • sylvseqrects.mat Do not include the data directory in your submission. 7