16720 – TAs: Alisa Allaire, Kshitij Goel, Tarasha Khurana, Zhengyi Luo, Neehar Peri, Uksang Yoo Solved

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• Please pack your code and write-up into a single file <andrewid> hw2.zip, in accordance with the complete submission checklist at the end of this document.
• All tasks marked with a Q require a submission.
• Please stick to the provided function signatures, variable names, and file names.
• Verify your implementation as you proceed: otherwise you will risk having a huge mess of malfunctioning code that can go wrong anywhere.
• Code snippets: Please don’t forget to add code snippets for the questions that explicitly ask you to do so. For other questions, if you like you can add code snippets, but these are not required.
* * *
Both are available at:
https://www.ri.cmu.edu/pub_files/pub3/baker_simon_2002_3/baker_simon_2002_3.pdf. https://www.ri.cmu.edu/pub_files/pub3/baker_simon_2003_3/baker_simon_2003_3.pdf.
1 Lucas-Kanade Tracking
In this section, you will implement a simple Lucas & Kanade tracker with one single template. In the scenario of two-dimensional tracking with a pure translation warp function,
W(x;p) = x + p . (1)
The problem can be described as follows: starting with a rectangular neighborhood of pixels 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∗ = argminp (2)
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 the first-order Taylor expansion.
It+1(x′ + ∆p) ≈ It+1(x′) + ∂ I∂t+1x′T(x′) ∂W∂(pxT;p)∆p (4)
where ∆p = [∆px,∆py]T, is the template offset. Further, x′ = W(x;p) = x + p and is a vector of the x− and y− image gradients at pixel coordinate x′. Similarly to Equation 3, one can incorporate these linearized approximations into a vectorized form:
argmin (5)
such that p ← p + ∆p at each iteration.
• What is (should a 2×2 matrix)?
• What is A and b?
• What conditions must ATA meet so that a unique solution to ∆p can be found?
Q1.2 (15 points) Implement a function with the following signature
LucasKanade(It, It1, rect, p0 = np.zeros(2))
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, rect is the 4-by-1 vector that represents a rectangle describing all the pixel coordinates within N within the image frame It, and p0 is the initial parameter guess (δx,δy). 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., it includes all four corners. To deal with the fractional movement of the template, you will need to interpolate the image using the Scipy module ndimage.shift or something similar. You will also need to iterate the estimation until the change in is below a threshold. In order to perform interpolation, you might find RectBivariateSpline from the scipy.interpolate package. Read the documentation of defining the spline ( RectBivariateSpline) as well as evaluating the spline using RectBivariateSpline.ev carefully. Include a snippet of the code you wrote for this part in your write-up.
Q1.3 (10 points) Write a script testCarSequence.py that loads the video frames from carseq.npy, and runs the Lucas-Kanade tracker that you have implemented in the previous task to track the car. carseq.npy 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[:, :, 0])). The rectangle in the first frame is [x1,y1,x2,y2]T = [59,116,145,151]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.npy, which contains a single n × 4 matrix, where each row stores the rect that you have obtained for each frame, and n is the total number of frames. You are encouraged to play with the parameters defined in the scripts and report the best results.
Similarly, write a script testGirlSequence.py that loads the video from girlseq.npy and runs your Lucas-Kanade tracker on it. The rectangle in the first frame is [x1,y1,x2,y2]T = [280,152,330,318]T. Report your tracking performance (image + bounding rectangle) at frames 1, 20, 40, 60 and 80 in a format similar to Figure 1. Also, create a file called girlseqrects.npy, which contains one single n×4 matrix, where each row stores the rect that you have obtained for each frame.
Code for saving the images and playing the tracking animation is provided in the starter code.

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 that 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 (refer to section 2.1 and implement equation-3 from the paper using strategy-3). Write two scripts with a similar functionality to Q1.3 but with a template correction routine incorporated: testCarSequenceWithTemplateCorrection.py and testGirlSequenceWithTemplateCorrection.py. Save the rects as carseqrects-wcrt.npy and girlseqrects-wcrt.npy, and also report the performance at those frames. An example is given in Figure 2. Again, you are encouraged to play with the parameters defined in the scripts to see how each parameter affects the tracking results.

Figure 2: Lucas-Kanade Tracking with Template Correction
Here the blue rectangles are created with the baseline tracker in Q1.3, the red ones with the tracker in Q1.4. The tracking performance has been improved non-trivially. Include a visual comparison between template drift correction and no correction in your write-up for the same frames as in Q1.3. Make sure that the performance improvement can be easily visually inspected.
2 Affine Motion Subtraction
In this section, you will implement a tracker to estimate 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.npy, which consists of aerial views of moving vehicles from a non-stationary camera.
2.1 Dominant Motion Estimation
In the first section of this homework, we assumed 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 the 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 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 . (6)
One can represent this affine warp in homogeneous coordinates as,
x˜′ = Mx˜ (7)
M . (8)
Here M represents W(x;p) in homogeneous coordinates as described in [1]. Also 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 compared to the size of the image, image It will almost always not be fully contained in the warped version It+1. Therefore, only pixels lying in the region common to It and the warped version of It+1 when forming the linear system at each iteration.
Q2.1 (15 points) Write a function with the following signature
LucasKanadeAffine(It, It1)
which returns the affine transformation matrix M, and It and It1 are It and It+1 respectively. LucasKanadeAffine should be relatively similar to LucasKanade from the first section (you will probably also find scipy.ndimage.affine transform helpful). Include a snippet of the code you wrote for this part in your write-up.
2.2 Moving Object Detection
Once you can compute the transformation matrix M relating an image pair It and It+1, a naive way to determine the 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 the locations of moving objects. To obtain better results, you can check out the following scipy.morphology functions: binary erosion, and binary dilation.
Q2.2 (10 points) Using the function you have developed for the estimation of dominant motion, write a function with the following signature
SubtractDominantMotion(image1, image2)
where image1 and image2 form the input image pair, and the return value 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. Include a snippet of the code you wrote for this part in your write-up.
Q2.3 (10 points) Write two scripts testAntSequence.py and testAerialSequence.py that load the image sequence from antseq.npy and aerialseq.npy and run the motion detection routine you have developed to detect moving objects. Try to implement

Figure 3: Lucas-Kanade Tracking with Motion Detection
testAntSequence.py first as it involves little camera movement and can help you debug your mask generation procedure. Report the image results at frames 30, 60, 90 and 120 with the corresponding binary masks superimposed, as exemplified in Figure 3. 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.
3 Efficient Tracking
3.1 Inverse Composition
The inverse compositional extension of the Lucas-Kanade algorithm (see [1]) has been used in literature to great effect for the task of efficient tracking. When utilized within tracking it attempts to linearize the current frame as:
∂It(x) ∂W(x;0)
It(W(x;0 + ∆p) ≈ It(x) + T ∂pT ∆p . (9) ∂x
In a similar manner to the conventional Lucas-Kanade algorithm one can incorporate these linearized approximations into a vectorized form such that,
argmin (10)
for the specific case of an affine warp where p ← M and ∆p ← ∆M this results in the update M = M(∆M)−1. Just to clarify, the notation M(∆M)−1 corresponds to W(W(x;∆p)−1;p) in Section 2.2 from [2].
Q3.1 (15 points) Reimplement the function LucasKanadeAffine(It,It1) as
InverseCompositionAffine(It,It1) using the inverse compositional method. Report the results to the query frames (ref. Q2.3) for both aerial and ant sequences. Also, report the runtime performance gain that you observe. In your own words, please describe why the inverse compositional approach is more computationally efficient than the classical approach.
4 Extra Credit (20 points)
Find a 10s video clip of your choice and run Lucas-Kanade tracking on a salient foreground object in this video. Needless to say, the object you track should undergo considerable amount of motion in the scene and should not be static. It is even better if this object undergoes an occlusion and your algorithm is able to track it across this occlusion. To report your results in the write-up, capture 6 frames of the tracked video at 0s, 2s, 4s, 6s, 8s, 10s and include these. Explain what changes you had to make in the algorithm to get it to work on this video.
5 Deliverables
The assignment (code and writeup) should be submitted to Gradescope. All parts that should be inluced in the writeup is highlighted in blue. The write-up should be submitted to Gradescope named <AndrewId> hw2.pdf. The code should be submitted as a zip named <AndrewId> hw2.zip to Gradescope. The zip when uncompressed should produce the following files.
• LucasKanade.py
• LucasKanadeAffine.py
• SubtractDominantMotion.py
• InverseCompositionAffine.py
• testCarSequence.py
• testCarSequenceWithTemplateCorrection.py
• testGirlSequence.py
• testGirlSequenceWithTemplateCorrection.py
• testAntSequence.py
• testAerialSequence.py
• carseqrects.npy
• carseqrects-wcrt.npy
• girlseqrects.npy
• girlseqrects-wcrt.npy
*Do not include the data directory in your submission.
6 Frequently Asked Questions (FAQs)
Q1: Why do we need to use ndimage.shift or RectBivariateSpline for moving the rectangle template?
A1: When moving the rectangle template with ∆p, you can either move the points inside the template or move the image in the opposite direction. If you choose to move the points, the new points can have fractional coordinates, so you need to use RectBivariateSpline to sample the image intensity at those fractional coordinates. If you instead choose to move the image with ndimage.shift, you don’t need to move the points and you can sample the image intensity at those points directly. The first approach could be faster since it does not require moving the entire image.
Q2: What’s the right way of computing the image gradients Ix(x) and Iy(x). Should I first sample the image intensities I(x) at x and then compute the image gradients Ix(x) and Iy(x) with I(x)? Or should I first compute the entire image gradients Ix and Iy and sample them at x?
A2: The second approach is the correct one.
Q3: Can I use pseudo-inverse the least-squared problem argmin ?
A3: Yes, the pseudo-inverse solution of A∆p = b is also ∆p = (ATA)−1ATb when A has full column ranks, i.e., the linear system is overdetermined.
Q4: For inverse compositional Lucas Kanade, how to deal with points outside out the image?
A4: Since the Hessian in inverse compositional Lucas Kanade is precomputed, we cannot simply remove points when they are outside the image since it can result in dimension mismatch. However, we can set the error It+1(W(x;p)) − It(x) to 0 for x outside the image.
Q5: How to find pixels common to both It1 and It?
A5: If the coordinates of warped It1 is within the range of It.shape, then we consider the pixel lies in the common region.
Note for Q1.4: Please be wary that we define p in the writeup to be the shift from the template rectangle. However, in the code for your LK function p0 is defined as the shift from the passed in variable rect. Therefore, when pass in a rectangle that isn’t the template rectangle like the initial frame rectangle in the 2nd LK stage, then you should be passing in p plus a shift that accounts for the difference in rectangle coordinates.
Note for Q2.2: When warping using the Affine Transformation function (cv2.warpAffine or scipy.ndimage.affine transform), read the function documentation carefully on whether M or M−1 needs to be passed to the function.
Note for Q3.1:


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