Homework 1

Homework 2

Homework 3

Homework 4

Homework 5

Homework 6

Homework 7

Homework 7
1.  Implement Radon Transform & then Filtered Back Projection to reconstruct back the imag
a) Take any image, f, first implement Radon Transform in the discrete domain. To do this accurately,
        First  compute 2D Fourier transform of  f, estimate approximate bandwidth in x and y directions, choose zeta_0 = maximum of x and y bandwidths
        d = 1/(2 zeta_0)
        Choose Delta = pi/180
        Compute Discrete Radon Transform  g(md, n Delta),  m goes from -M/2 to M/2-1,  M = 2 D d  where D is diameter of the object (discretize the integral given in equation 10.6)

b) Reconstruct back the image by implementing Filtered Back Projection (as I explained in class, also read the book) with g(md, n Delta) as input.
        Be very careful: you will have to implement an approximation of the high pass filter H(f) = |f|, some examples are given in the book

c) Compute the reconstruction errror. Increase zeta_0, and see the trend. Vary Delta and see the trend

2.  AK Jain Chapter 10, Problem 10.6
3.  Extra credit: AK Jain Chapter 10, Problem 10.16



Homework 6 (Gaussian Mixture Model: Alternating Maximization and EM Algorithm)
You are given N data points Y1, Y2,... YN which are samples from a Gaussian mixture with K components. All Yi's are assumed independent. Each Yi is a P-dimensional vector.  For e.g. each Yi  can be the R,G,B color intensity of an image pixel, in that case, P=3 and N = number of pixels in the image and K is the number of different regions in the image (if 2 objects and background, then K=3). Also, for simplicity, assume Sigma_c for c= 1,2,.. K are diagonal covariance matrices.

The above problem can be reformulated as follows (in order to use Alt Max or EM): There is hidden data Xi which denotes the class membership of Yi. Conditioned on Xi, Yi is Gaussian distributed with mean mu_{Xi} and covariance Sigma_{Xi}. Also the prior probability of Xi taking the value c is p_c, i.e.
p(Yi|Xi) = N ( Yi; mu_c, Sigma_c)  and p(Xi = c) = p_c   for c = 1,2,... K

Denote X = [X1, X2,... XN]  and denote Y = [Y1, Y2,... YN]  and denote the parameters as theta = { mu_c, Sigma_c, p_c, c = 1,2,...K}

A. Use alternating maximization to compute  max_{theta, X}  p_{theta}( Y, X ), i.e. derive the expressions for both steps of the alternating maximization.
I gave the main idea for both steps in class. When theta is held fixed, you just get a Maximum Likelihood classifier. When X (class membership) is  held fixed, you need to take first derivative and set it to zero to get the max.

B. Use EM algorithm to compute max_{theta} sum_{ X }  p_{theta}( Y, X ), i.e. derive the expressions for the E step and the M step.
The E step was done in class, for M step also I gave the main idea in class.

C. Extra:  Pick a  colored image of any object  (K=2), and apply both algorithms. Manually changed the object intensity so it is not too different from background intensity. Then apply both methods. You will see that EM will work better in that case (when clusters close to each other).

Note both algorithms only give local maximas



Homework 5 (Extract contours, estimate geometric distortion & correct for it)
First use the circle image (convert to greyscale). 
Apply a geometric distortion to the image. Try rotation by 40 degrees, try uniform scaling, and try any type of affine deformation (x' = ax + by+c, y'=dx + ey + f). For applying any of these, first shift the origin to the center of the image, e.g. a 21x21 image will have coordinates x,y belonging to [-10, 10] x [-10, 10]. Also, remember you need to interpolate grey scale values for the distorted image  (interp2 command can be used or write your own code).
Note rotation will have almost no effect on circle (except due to interpolation errors).

Extract contour of the object from the original image and extract contour of the distorted image. If you're using the circle image, just threshold it to get a binary image, followed by using either bwtraceboundary or use the contour command in Matlab (try both). Both perform boundary tracing, but contour gives the boundary in sub-pixel coordinates, while bwtraceboundary gives the innermost boundary pixel locations on the object.

In general, the two contours will have different number of points and will also be of different lengths (remember arclength is different from number of points for the contour, I will explain this in next class). Use interp1 to resample both contours to M points where M = round( (L1+ L2)/2) where L1 and L2 are the lengths of the two contours.  Once you have M points on either contour, estimate the geometric distortion as taught in class, and then correct for it as taught in class. Again remember to do grey scale interpolation (interp2 command can be used or write your own code).

Now, try the same thing for any real image. Here you may need to use edge detection followed by edge linking to get the contours or other segmentation methods. Or use feature matching (KLT tracker) to get corresponding points.

Extra: Understand/experiment with using  the Matlab command imtransform.


Homework 4 (Image Enhancement & Wiener filtering) (you can use all MATLAB functions in this homework)
Do at least 4 of the following 5 parts. You're welcome to do all!

By the way, there is a command called dicomread in Matlab - to read .dcm images

A. Demonstrate using MATLAB that the covariance matrix of a zero mean periodic stationary random signal is  circulant.

B. Histogram Specification & Equalization.
Take any image. First do a histogram specification to get the pixels to follow a discrete version of the truncated Gaussian PDF N(20,3) truncated between [10, 40]. For e.g. the probability that a pixel takes value 12 is the truncated Gaussian integrated between 11.5, 12.5. Matlab has standard functions to compute Gaussian CDFs which can be used to evaluate this.
Then do a histogram equalization on it, so that the final histogram approximates a uniform between 0,255.
Also try the same histogram equalization method on the following image:  brain MRI

C. Do edge enhancement (using either high pass filtering or unsharp masking or bandpass filtering) on the following image: brain MRI
Try the same operations in DFT domain.

D. Compare Low Pass Filtering (say averaging or Gaussian filter) with Median filtering & Directional smoothing for the following images: 
(i) brain MRI , (ii) same image with salt and pepper noise added, (iii) same image with Gaussian noise added.
See the benefit of Directional Smoothing in this image which is already pretty blurred, so if you do smoothing in all directions, edges will get even more blurred.

E. Perform FIR Wiener smoothing (assume no blurring) for   all the three images used in part D. Compare with algorithms of part D.
I think you should write your own code for this one. Start with designing a 5x5 Wiener smoother for every  16x16 block in the image. Increase/decrease Wiener smoother size or block size as needed.
Try changing window size to keep output SNR constant.  Compare performance using MSE or visual inspection.



Homework 3 (postponed, due Feb 16)
Find an application of DFT in your research area (or in any image processing application) and implement it. Of course it doesn't necessarily have to be a "new" one.  Some examples are:

• use of DFT to detect regions and times where there is fast motion: apply DFT to image differences and look for high frequency regions
• use DFT (or DCT) to replace eigenvectors of coviance matrix (e.g. PCA) in any application. Can you define something like LDA but using DFT or DCT
• study Fourier descriptors literature (for feature extraction, recognition, retrieval...). FD is the Fourier series of some feature of the boundary contour of an object as a function of contour arclength, e.g. tangent angle, x,y location (complex FD), ... F series (defined only for periodice signals) is the same as DFT of one period
  
• study the use of face asymmetry as a biometric for face recognition (a recent PAMI paper uses the energy in the imaginary part of the DFT coefficients of each row as a measure of asymmetry).
  
• directional Fourier transforms... connection with Radon transform
  

Extra (this may be part of HW 4): Demonstrate using MATLAB that the covariance matrix of a zero mean periodic stationary random signal is  circulant?
Extra: Find an application of circulant matrices outside of signal/image processing and implement it. One domain is graph theory.


Homework 2 (Due Friday Feb 2)

1. Problem 4.9 of AK Jain.
   
2. Problem 4.11, part (a). Also verify using MATLAB. A continuous function can be simulated by choosing x = [0:1/(100 zeta_s): 5/zeta_0]  (using a very small sampling interval).
   
3. Demonstrate Moire effect for a DC source (constant grey scale image) using MATLAB. Note what I said in class wasn't exactly correct. You will need to decimate the image first and then interpolate using different size averaging filters. 
4. For the chessboard image of homework 1 (or any one row of it), write out equations for every image you got after decimation and after interpolation, i.e. do the entire thing on paper. Some of you haven't done it in the submission. One row of the chessboard can be written out as a difference of two unit step functions: u(x) - 2u(x-20)
   

Homework 1 (Decimation & Interpolation of Images)  Try to finish by Jan 19 or 22.
Use as I(m,n) a chessboard image or a fence image.

A. Decimation. First decimate without applying the low pass filter (Decimate in two steps: first do zeroing in the original image and then downsample). See the effect of aliasing in the decimated image. Start with decimating by a factor of 2 and keep going until you "see" the effect of aliasing. Let the chosen decimation factor is D. Now apply a low pass filter (say a 3x3 or a 5x5 averaging filter) to I(m,n) and then decimate by D. Keep increasing size of the averaging filter until the effect of aliasing is "negligible".

B. DFT. Take a 16x16 piece of the image and take its DFT (or you may even just take a row of the image and do 1dim DFT). Do this for all the images you get in the above steps. See the aliasing in the frequency domain and see the aliasing getting reduced if you low pass filter first.
    Note DFT  should help you decide what factor to decimate by to "see" aliasing, and then also decide how big an averaging filter to apply in order that the effect of aliasing is "negligible"
Details for decimation by 2 (similarly do for other factors D)

1. Take the image I(m,n), Set every alternate row and column to zero. Call this I_Z(m,n). Now downsample to get I_D(m,n) which is half the size.
   
2. Compute DFT of I, I_Z and I_D . Either do 1dim DFT or do 2dim DFT of a 16x16 or 32x32 region where you can "see" aliasing most easily
3. Low pass filter the original image and repeat the above steps
4. Do the same thing for different decimation rates D and for different
   

C. Interpolation. Take the decimated image I_D(m,n), add D-1 zeros to along rows and columns to get an original size image I_U(m,n). Interpolate the upsampled image (try zero-order hold, linear or cubic interpolation) to get the reconstructed image I_hat(m,n). Do this both for the decimation of the original image and decimation of the LPF'ed image. Notice that the effect of aliasing will not go away by interpolation. To get a quantitative measure, you may want to look at the mean squared error between the reconstructed and original image in both cases. Note for the LPF'ed case, the original image will be the LPF'ed one.

D. DFT. Compute DFT (again you may choose to just do a 1dim DFT of a relevant row since its easier to plot and see magnitudes) of all images (I_U and I_hat in both cases) and connect with what you learn in class.

Submission: Send me via email (or put in your webspace and send me the link, need not create an explicit webpage) 
- Send MATLAB code.
- Send a short report "explaining" what you did and what you observed and why based on what you've learnt about decimation and interpolation in class.
- Can be a handwritten report if you like. You won't lose any points for that. But I'll suggest learning LaTex!