# ELEC/COMP 447/546 Assignment 3 solution

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1.0 Optical Flow
In this problem, you will implement both the Lucas-Kanade and Horn-Schunck
algorithms. Your implementations should use a Gaussian pyramid to properly account
for large displacements. You can use your pyramid code from Homework 2, or you may
simply use Opencv’s pyrDown function to perform the blur+downsampling. You may
also use Opencv’s Sobel filter to obtain spatial (x,y) gradients of an image.
Implement the Lucas-Kanade algorithm, and demonstrate tracking points on this video.
1. Select corners from the first frame using the Harris corner detector. You can use
this command: corners = cv.cornerHarris(gray_img,2,3,0.04).
2. Track the points through the entire video by applying Lucas-Kanade between
each pair of successive frames. This will yield one ‘trajectory’ per point, with
length equal to the number of video frames.
3. Create a gif showing the tracked points overlaid as circles on the original frames.
You can draw a circle on an image using cv.circle. You can save a gif with this
code:
import imageio
imageio.mimsave(‘tracking.gif’, im_list, fps=10)
where im_list is a list of your output images. You can open this gif in your web
browser to play it as a video and visualize your results. Show a few frames of
a. Do you notice any inaccuracies in the point tracking? Where and why?
b. How does the tracking change when you change the local window size
1.2 Horn-Schunck (5 points)
Implement the Horn-Schunck algorithm. Display the flow fields for the ‘Army,’
‘Backyard,’ and ‘Mequon’ test cases from the Middlebury dataset, located here.
Consider ‘frame10.png’ as the first frame, and ‘frame11.png’ as the second frame for all
cases.
Use this code to display each predicted flow field as a colored image:
hsv = np.zeros(im.shape, dtype=np.uint8)
hsv[…, 1] = 255
mag, ang = cv.cartToPolar(flow_x, flow_y)
hsv[…, 0] = ang * 180 / np.pi / 2
hsv[…, 2] = cv.normalize(mag, None, 0, 255, cv.NORM_MINMAX)
out = cv.cvtColor(hsv, cv.COLOR_HSV2RGB)
1.3 ELEC/COMP 546 Only: Improving Horn-Schunck with superpixels (5 points)
Recall superpixels discussed in lecture and described further in this paper. How might
you use superpixels to improve the performance of Horn-Schunck? Can you incorporate
your idea into the smoothness + brightness constancy objection function? Define any
notation you wish to use in the equation. You don’t have to implement your idea in code
for this question.
2.0 Image Compression with PCA
In this problem, you will use PCA to compress images, by encoding small patches in
Test Image 1
Test Image 2
Do the following steps for each image separately.
2.1 Use PCA to model patches (5 points)
Randomly sample at least 1,000 16 x 16 patches from the image. Flatten those patches
into vectors (should be of size 16 x 16 x 3). Run PCA on these patches to obtain a set
of principal components. Please write your own code to perform PCA. You may use
numpy.linalg.eigh, or numpy.linalg.svd to obtain eigenvectors.
Display the first 36 principal components as 16 x 16 images, arranged in a 6 x 6 grid
(Note: remember to sort your eigenvalues and eigenvectors by decreasing eigenvalue
magnitude!). Also report the % of variance captured by all principal components (not
just the first 36) in a plot, with the x-axis being the component number, and y-axis being
the % of variance explained by that component.
2.2 Compress the image (5 points)
Show image reconstruction results using 1, 3, 10, 50, and 100 principal components. To
do this, divide the image into non-overlapping 16 x 16 patches, and reconstruct each
patch independently using the principal components. Answer the following questions:
1. Was one image easier to compress than another? If so, why do you think that is
the case?
2. What are some similarities and differences between the principal components for
the two images, and your interpretation for the reason behind them?
Submission Instructions
All code must be written using Google Colab (see course website). Every student must
submit a zip file for this assignment in Canvas with 2 items:
1. An organized report submitted as a PDF document. The report should contain all
image results (intermediate and final), and answer any questions asked in this
document. It should also contain any issues (problems encountered, surprises)
you may have found as you solved the problems. The heading of the PDF file
should contain:
a. Your name and Net ID.
b. Names of anyone you collaborated with on this assignment.
c. A link to your Colab notebook (remember to change permissions on your
notebook to allow viewers).
2. A pdf copy of your Colab notebook.
Collaboration Policy
I encourage collaboration both inside and outside class. You may talk to other students
for general ideas and concepts, but you should write your own code, answer questions
independently, and submit your own work.
Plagiarism
Plagiarism of any form will not be tolerated. You are expected to credit all sources
explicitly. If you have any doubts regarding what is and is not plagiarism, talk to me.