Description
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: (2 points)
• Generate artificial images with binary pixel values xij ∈ {0, 255} that
shows stripes (see left image), i.e.,
T = toy stripes(n, m, sSize),
1
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
P SNR = 20 log10
255
√
err
(2)
with reconstruction error err = (T − N)
2/(nm).
psnr = calc psnr(T, N)
Hints:
• You can use matlab function random() with µ = 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: (3 points)
• 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% salt’n’pepper noise and with the image la.png downloadable from
the course website after adding Gaussian noise with σ = 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 la.png.
2
3 MRF-based Denoising with Gradient Ascent
A better denoising technique is a MRF-based method with gradient ascent:
T
t+1 ← T
t + η∇T log p(T|N) (3)
with log p(T|N) = log p(N|T) + log p(T) + const. const is ignored in all
further computations as it doesn’t depend on T. We need the gradient of both
the likelihood log p(N|T) and the prior log p(T). For simplicity, let’s start with
a Gaussian log-likelihood
log p(N|T) = X
i,j
−
1
2σ
2
(Ni,j − Ti,j )
2
, (4)
as well as a Gaussian log-prior
log p(T) = X
i,j
log(fH(Ti,j , Ti+1,j )) + log(fV (Ti,j , Ti,j+1)) (5)
with
log fH = −
1
2σ
2
(Ti,j − Ti+1,j )
2
(6)
for horizontal neighbors and log fV analogously.
Tasks: (8 points)
• 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 log p(N|T), and
g = mrf grad log gaussian prior(T, sigma)
to calculate the gradient of the Gaussian prior log p(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 σ and η? (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?
3
• 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., σ, η ∈ {…, 10−1
, 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:
fH(Tij , Ti+1j ) =
1 +
1
2σ
2
(Ti,j − Ti+1,j )
2
−α
(7)
Tasks: (5 points)
• 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, α = 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: (2 points)
• 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).
4
6 Bonus
(2 points) So far, we have used MRF-based image denoising for gray images
only. How would you denoise a color image?
5
