In this exercise you are introduced to Markov random fields and their application to image denoising. We solve

p(true|noisy) = p(T|N) = p(N|T)P(T) (1)

with gradient ascent method and compare our results to median filtering method. Finally, we have a look at different image priors p(T) and question the meaning of this priors as well as the independence assumption.

1 Evaluation

Before we start with image denoising, we need an evaluation framework that generates noisy images and calculates a performance measure to compare our algorithms.

Tasks:

• Generate artificial images with binary pixel values x_ij {0,255} that shows stripes (see left image), i.e.,

T = toy stripes(n, m, sSize),

and a checkerboard pattern (right image), i.e.,

T = toy checkerboard(n, m, cSize).

T is the true image of size n m with stripe width sSize or black square size cSize.

• Write a function that adds Gaussian noise to an existing image T:

N = add noise(T, sigma)

Also, write a function which adds salt and pepper noise (randomly occurring black and white pixels):

N = add sp noise(T, p)

• Write a function that calculates peak signal-to-noise ratio

(2)

with reconstruction error err = (T N)²/(nm).

psnr = calc psnr(T, N)

Hints:

• You can use matlab function random() with = 0 to add or substract noise values from each pixel. Be sure that your pixel values are still in the range of [0,255].
• You should fix the random seed in your functions with Matlab function RandStream. This makes all results comparable to each other.

2 Median Filtering

A simple image denoising technique is median filtering. As you will find out, it often leads to blurred images.

Tasks:

• Write a function T = median filter(N, nsize) that replaces each pixel in a noisy image N with the median of the pixel values in a window of size nsize nsize around it. Take care at the image borders.
• Evaluate this denoising procedure on our different artificial examples with 10% saltnpepper noise and with the image png downloadable from the course website after adding Gaussian noise with = 25 (10% of the range). Show images before and after denoising and document PSNR for these images. What do you observe?
• Vary the amount of noise for the image png.

3 MRF-based Denoising with Gradient Ascent

A better denoising technique is a MRF-based method with gradient ascent:

T^t+1 T^t + _T logp(T|N) (3)

with logp(T|N) = logp(N|T) + logp(T) + const. const is ignored in all further computations as it doesnt depend on T. We need the gradient of both the likelihood logp(N|T) and the prior logp(T). For simplicity, lets start with a Gaussian log-likelihood

, (4)

as well as a Gaussian log-prior
logp(T) = Xlog(fH(Ti,j,Ti+1,j)) + log(fV (Ti,j,Ti,j+1))i,jwith	(5)

(6)

for horizontal neighbors and logf_V analogously.

Tasks:

• Write a function lp = denoising lp(T, N, sigma) to compute the log-posterior,

g = denoising grad llh(T, N, sigma) to compute the gradient of the log likelihood logp(N|T), and g = mrf grad log gaussian prior(T, sigma) to calculate the gradient of the Gaussian prior logp(T).

• Finally, implement the gradient ascent to denoise the image

T = denoising grad ascent(N, sigma, eta).

You should initialize the gradient ascent with the noisy image N.

• Explain your parameter tuning. Which observation do you make for different and ? (See hints.)
• Evaluate your implementation with the noisy images from Task 2 for the same noise parameters. Check the PSNR and the increasing log-posterior curve. Compare your results with median filtering which algorithm shows better results and why?
• What happens if you initialize gradient ascent with the output of median filtering? Is there an improvement in performance or a faster convergence observable?

Hints:

• The gradient g should have the same size as the input image while the log-posterior lp is just a scalar.
• You may need many iterations to reach approximate convergence (> 1000). You can verify your algorithm by displaying the log-posterior curve.
• Start with small artificial images to see the correctness of your algorithm and to get a feeling for the parameters. Usually, it is recommended to test different powers of 10, i.e., , {,10¹,10⁰,10¹,}.

4 A Different Prior

As we know from the lecture, a Gaussian distribution does not match the statistics of a natural image very well. A more appropriate distribution is the Studentt distribution:

(7)

Tasks:

• Implement the gradient of this log-prior given above as

g = mrf grad log student prior(T, sigma, alpha).

Do not forget the log in your partial derivatives.

• Evaluate your denoising algorithm with this new prior, = 1 works fine. What do you observe?
• Display the gradient of the log-prior for the test images. Explain your results.

5 Independence Assumption

Finally, we question the assumption that the noise in each pixel is independent.

Tasks:

• Is this assumption reasonable?
• What happens if you add spatially dependent noise to your image, e.g., a noisy stripe like in old movies.
• Show results for both priors (Gaussian and Student-t distributions).

6 Bonus

So far, we have used MRF-based image denoising for gray images only. How would you denoise a color image?