Description
1 Information Theory
In this section, you will explore how information theory can be used to for the
analysis of medical images. The image for analysis, I, can be seen in Figure
1. For all the questions below, you are permitted to use in-built functions,
except when performing the information theoretic computations themselves (e.g.
entropy).
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Figure 1: Image of a retina. courtesy: Ajit Rajwade.
1.1 Entropy of an image
Let X be a random variable defining the intensity of a pixel in an image and
let {xi
|i = 1, 2, 3, . . . } be the set of discrete values that X can take on. The
entropy of X is given by:
H(X) = −
X
i
p(xi) log2
(p(xi))
1. Compute the entropy of I. (1 point)
2. Generate an image NA of the same size as I such that pixel intensities of
NA are drawn from U[−A, A](discrete uniform noise). Vary A from 0 to
200 in steps of 5 and generate instances of NA. Compute and plot in a
single graph the entropy of: I, NA and I + NA for multiple values of A.
Discuss the trend in the graph. (5 points)
1.2 Mutual Information and KL divergence
1. Generate the noise images, NA, by varying A from 0 to 200 in the steps
of 5 as described in the previous question. Compute and plot the mutual
information between the images I and I + NA as a function of A. Discuss
the trend in the graph. (5 points)
2. Generate a single noise image, N20, (i.e. NA with A = 20). Compute the
joint entropy of the image pair: H(I, I + N20). Also, verify numerically
that:
H(I; I + N20) = H(I) + H(I + N20) − MI(I; I + N20)
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(a) (b) (c) (d)
Figure 2: Images to be registered. Images are titled as (a) I1 1 (b) I1 2 (c) I2 1
(d) I2 2.
(Note: Reuse H(I), H(I + N20) and MI(I; I + N20) from the previous
questions.) (2 points)
3. KL divergence is defined as follows:
DKL(P||Q) = −
X
i
P(xi) log2
Q(xi)
P(xi)
Compute the forward and the backward KL divergences, i.e. DKL(P||Q)
and DKL(Q||P) between: (i) I and the noise image N of the same size as
I; where pixel intensities of N are drawn from U[0, 255] (ii) I and I + N20
from the previous question. Discuss the values. (Note: For (ii), use Parzen
window filtering on the histogram of I, and mention the parameters of the
window.) (4 points)
1.3 Registration
A set of brain images are shown in Figure 2. The images consist of 2D slices
of Magnetic Resonance Images (MRI) from patients with brain tumours. The
images are taken from the MICCAI 2017 BRATS Challenge dataset [1, 2].
1. You will register pairs of images from Figure 2 using a simple transformation space: x-y translations. Integer values are sufficient. Consider two
similarity metrics: mutual information and mean squared error. Report
the optimal translation computed by these two metrics.
Below are each of the image pairs to be registered. Consider the first of
the pair to be the moving image and the second to be the fixed image.
• I1 1 and I1 2 (4 points)
• I2 1 and I2 2 (4 points)
2. Discuss the above results in terms of the assumptions inherent to the metrics. Describe the context in which each metric should be used. Support
your arguments with an example or two. (3 points)
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2 Face Recognition
You will explore and compare eigenfaces to probabilistic face recognition, based
on face images from a subset of the publicly available Color FERET Database
[3] provided with the assignment. Whereas the full database contains images
from approximately 1000 different subjects, the given subset contains images
from 52 subjects. The subset is arranged into several folders each containing
images of a specific subject in different poses. The number of images per subject
varies from 32 to 96. Each image is 768×512 pixels and the files are in PPMformat. You may choose to convert images to gray-scale and/or down-sample
images by the scale which does not degrade the performance of the system.
You will design a face recognition system using this subset. Randomly separate the images into training and test sets as described below. A new random
selection should be made by your program every time the system is retrained.
• Train set. For each subject, randomly select 80% of the total images
given for that subject. This will be used as the training set. (1 point)
• Test set. All the remaining images from the given subset which are not
used in the training set will be used as test images. (1 point)
2.1 Principal Components Analysis
1. The training set can be placed into a matrix X = [x1, x2, . . . xD] of size
N ×D, N being the total number of pixels in an image x and D being the
total number of training images. Compute the principal components using
the snap-shot method. Display the mean face and the first 10 Eigenfaces.
(6 points)
2. Decide on the number of principal components Np required to represent
the data. You can use either of the two methods for dimensionality estimation discussed in the class (refer PCA slides 69 and 72). Justify your
choice and support it with an appropriate graph. (3 points)
2.2 Probabilistic Face Recognition
You will explore recognizing faces using Bayesian techniques. Let φ(x) be
eigenspace representation of x. Let y be the random variable describing class
label i.e. the subject face identity.
1. Using Bayes rule, derive an expression for the posterior density of a subject
label y for a given test image x∗. (1 point)
2. Find eigen representation φ(x) for each training image x i.e project the
image x on the first Np Eigenfaces and find corresponding Np coefficients.
(3 points)
3. Build total 52 Gaussian density functions for the likelihood, one for each
subject in the training set. (3 points)
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4. Do the following for each test image x∗.
• Find eigen representation φ(x∗) of x∗. (2 point)
• (Method 1) Assign the subject label to x∗ based on the MAP. Display
the posterior density function for one test image in the report. (4
points)
• (Method 2) Use nearest neighbor classifier to assign the subject label
to x∗. (2 points)
5. Calculate the recognition rate as the ratio of correctly classified images to
the total number of images for both the methods. Compare the results.
(3 points)
References
[1] Menze, Bjoern H., et al. ”The multimodal brain tumor image segmentation
benchmark (BRATS).” IEEE transactions on medical imaging 34.10 (2015):
1993.
[2] Bakas, Spyridon, et al. ”Advancing the cancer genome atlas glioma MRI
collections with expert segmentation labels and radiomic features.” Scientific
data 4 (2017): 170117.
[3] Phillips, P. Jonathon, et al. ”The FERET database and evaluation procedure
for face-recognition algorithms.” Image and vision computing 16.5 (1998):
295-306.
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