Instructions:
EE5806: Topics in Digital Imaging Processing 2020-21: Semester A Assignment 2
Students must submit the assignment through Canvas. Click on the item ¡°Assignment¡± on the left panel. You should see a row with title ¡°Assignment 2¡±. Click on the ¡°Assignment 2¡± label. You can then see detailed instructions for submission. Please submit a report in Microsoft word format containing printouts of any code and figures requested in each of the problems. The report should also contain answers to all questions. Please also save your code and submit them as separate text files with .py extension as requested in the questions. Your report and all the requested .py files should be zipped as a single zip file for submission. When you are ready to submit the assignment, click on the ¡°Submit Assignment¡± label on the right panel and upload the requested zip file.
Problem 1 (35 marks)
The following piece of code filtered an input image with the Gaussian kernel with standard deviation equals to 20 in u- and v-directions (i.e., = = 20). The image is inside the demo4.zip package available in Canvas. The output image has wraparound error. Modify the program to mitigate this error by padding the image to size 2 ¡Á 2 using ¡®reflect¡¯ padding (See the definition of the ¡®reflect¡¯ setting here: https://numpy.org/devdocs/reference/generated/numpy.pad.html. You are also encouraged to use the pad function for the padding.) The output image generated by your modified program must be the same as that generated by the program presented below, except that there should be no wraparound in the output image you generated (i.e., the relative size of the Gaussian kernel used in the original and modified code must be the same. Hints: How
should and parentheses):
(a) (b) (c) (d) (e)
be modified in the revised program?). Plot the following images (size in
Original image ( ¡Á )
Image generated by the original code reproduced below ( ¡Á ) Padded image (2 ¡Á 2 )
Filtered padded image (2 ¡Á 2 )
Image with no wraparound error ( ¡Á )
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Problem 2 (35 marks)
(a) (15 marks) The inverse Fourier Transform of an image ( , ) in the Cartesian and polar coordinate system is given by:
( , ) = (, )() = | | ( , !)”( #$% & %'( &) ! )
What is the relationship between (, ) and ( , !)?
(b) (20 marks) Create an image containing a white rectangle on the black background
shown below using the code:
image = np.zeros((100,100))
image[10:50, 20:25] = 255
Fig. 1 Synthetic image used in Problem 2
Compute its Radon transform (radon) with an angular step of ¦È degrees and show the sinogram. Each figure in your report should have a caption. Compute the filtered back-projection (iradon) with the ¡®hamming¡¯ filters. Also, compute the simple (non- filtered) back-projection and show the images. Repeat the procedure for ! =
{45, 10, 5, 2, 1}.
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Problem 3 (30 marks)
In this question, the target image involved in the landmark registration is a scaled version of the source image as shown in Fig. 2. The scaling factors in the vertical and horizontal dimensions are the same. (Hints: Thus, no translation and rotational operations are involved). N landmarks (n = 1, 2, …., N) in the source image and corresponding landmarks (n = 1, 2, …., N) in the target image have been identified (e.g., black ¡®+¡¯ landmarks in Fig. 2). The 2D coordinates of these two sets of landmarks are denoted by:
and
The goal of this problem is to find out the optimal scaling operation based on the landmarks to register the source image with the target image.
You can express your answers in terms of the following constants:
Fig. 2
a. (10 marks) The cost function C quantifies the mean-squared distance between the target landmarks and the transformed source landmarks. Express C for this problem in terms of the coordinates of target landmarks, , the source landmarks, , and the scaling factor required for the registration, denoted by 0.
b. (20 marks) Find 0 that minimizes C derived in Part a of this question.
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