Description
Objective: Gain experience with (a) how to process images on a computer by using
basic intensity transformations, and (b) how spatial filtering works by implementing
two-dimensional convolution.
1 TDT 4195
Digital Image Processing Assignments Overview
The image processing assignments in this course are meant to supplement the lectures by
providing theoretical and practical experience with the material.
There are three assignments in total and together they are worth 15 % of your final
grade. The assignments aim to show you how to look at images from a computational
perspective, and introduce you to a variety of techniques to manipulate them. This
include extracting or highlighting specific regions of interest, or improving the quality of
the image overall.
The assignments can be quite difficult and may at times be programming and maths
heavy. For this reason, I recommend you to start working on the assignments early. Seek
out help whenever you get stuck by making use of the assignment forum and the computer
lab sessions.
Assignment 1 (worth 4 %) introduces you to intensity transformations and spatial
filtering.
Assignment 2 (worth 5 %) shifts the focus from spatial to frequency filtering using
the convolution theorem.
Assignment 3 (worth 6 %) gives an introduction to image segmentation and
mathematical morphology.
To make the transition to programming with images a bit easier, we have made
available a getting started guide for Python (with NumPy) and MATLAB. This guide
walks through the basics of how to create and manipulate tensors1
, with a focus on
vectors and matrices. Additionally, the guide demonstrates how to read and write images,
manipulate pixels, inspect image histograms, traverse images in scanline, and more.
A link to the guide can be found below:
https://github.com/vicolab/tdt4195-public
1For our purposes, tensors are arrays of numbers organised in a regular grid with an arbitrary number
of axes.
Task 1: Theory Questions [1.5 points]
a) [0.5 points] Histogram equalisation is an algorithm that creates a transformation
Theq by exploiting an image histogram. Explain in your own words:
i) How can we see that an image has low contrast when looking at its histogram?
ii) What effect does Theq have when applied on an image?
iii) What happens when histogram equalisation is applied multiple times on the
same image in sequence?
b) [0.3 points] Perform histogram equalisation by hand on the 4-bit (2
4 = 16 intensity
values) 3 × 4 image in Figure 1. Your report must include all the steps you did to
compute the transformation, the transformed image, and its histogram.
12 3 1 9
3 0 4 3
2 6 15 2
Figure 1: An image with intensities in the [0, 15] range (4-bit). Each square represents a
pixel, where the number within is the intensity value.
c) [0.2 points] The convolution operator is associative. What does this mean, and
how is this property beneficial for convolution?
Hint: The same property is convenient for affine transformations as well.
d) [0.5 points] An M × M kernel is separable if it can be decomposed into an M × 1
and 1 × M vector whose outer product ⊗ form the original kernel. To determine if
a kernel is separable we can compute its rank 2
. A separable kernel has rank equal
to 1. The rank of a matrix can be found by counting the number of nonzero rows
remaining after Gaussian elimination. Answer the following questions in your own
words. Please show the steps you took when calculating your answers.
i) Determine if the convolution kernel (a) and (b) in Figure 2 are separable by
computing their rank. If a kernel is separable, also provide the separated row
and column vector.
ii) Why is it helpful to have convolution kernels that are separable?
2The column (or row) rank of a matrix A is the maximal number of linearly independent columns (or
rows) of A.
3 TDT 4195
1
16
1 2 1
2 4 2
1 2 1
(a)
1 1 1
1 −8 1
1 1 1
(b)
Figure 2: Two convolution kernels: (a) is a Gaussian kernel, and (b) approximates the
Laplacian ∇2
I =
∂
2
I
∂x2 +
∂
2
I
∂y2 (includes diagonal entries).
Task 2: Greyscale Conversion [0.5 points]
Converting a colour image to a greyscale representation can be done by taking the average
of the red (R), green (G), and blue (B) channels, as seen on the left-hand side of Equation 1.
However, because different colours have different wavelengths it follows that each colour
has a slightly different contribution to the image. Thus, to preserve the luminance, or
lightness, of the original colour image a weighted average is typically used instead. One
such weighted average – used by the sRGB colour space – can be seen on the right-hand
side of Equation 1.3
greyi,j =
Ri,j + Gi,j + Bi,j
3
greyi,j = 0.2126Ri,j + 0.7152Gi,j + 0.0722Bi,j (1)
a) [0.3 points] Implement two functions that manually convert a RGB colour image
to a greyscale representation. The first function must implement the averaging
method, while the second must implement the luminance-preserving method (see
Equation 1).
b) [0.2 points] Apply your functions on a couple of colour images and show the result
in your report. Briefly discuss the perceived differences between the output of your
two functions.
Hint: The data type of your variables greatly impact the result of arithmetic operations.
Task 3: Intensity Transformations [0.5 points]
A greyscale intensity, or pixel brightness, transformation T maps the intensity values
p ∈ [p0, pk] in the original image into a new intensity range q ∈ [q0, qk]:
q = T (p) ⇔ Ji,j = T (Ii,j ) (2)
a) [0.2 points] Implement a function that takes a greyscale image and applies the
following intensity transformation: T (p) = pk − p. Include the following in your
report:
3More information can be found here: https://en.wikipedia.org/wiki/Greyscale.
i) In your own words, explain what this transformation does. What would you
call it?
ii) Apply the function on an image of your choice and show the result.
Hint: Remember, pk is the highest value a pixel can take, e.g. 15 for a 4-bit image.
A basic definition of the gamma transformation can be seen in Equation 3. It adjusts
the overall intensity of an image by squeezing pixel intensities to either low or high
intensities.
T (p) = cpγ
, c > 0 ∧ γ > 0 (3)
b) [0.3 points] Implement a function that takes a greyscale image and applies the
gamma transformation from Equation 3. You can let c = 1. For this task, it is
important that you normalise the image, so that the intensity values lie ∈ [0, 1],
before applying the gamma transformation4
. Include the following in your report:
i) In your own words, explain what happens with the image intensity values when
γ > 1 and γ < 1?
ii) Test out different γ-values on a greyscale image and show the result.
Task 4: Spatial Convolution [1.5 points]
Convolution f ∗ h is an operation on two functions f and h. When f and h are defined on
spatial variables, the operation is called spatial convolution. Formally, for functions f(x, y)
and h(x, y) of two discrete variables x and y, two-dimensional convolution is defined as:
(f ∗ h)(x, y) = X∞
i=−∞
X∞
j=−∞
f(i, j)h(x − i, y − j) (4)
Convolution is commutative – that is, f ∗h = h∗ f – however, to simplify explanations
we will let f be the image and h be the convolution filter/kernel/mask. We can think of
two-dimensional spatial convolution as sliding a kernel across the width and height of
an image. For each pixel in the image, a linear combination between the kernel, centred
on the image pixel, and a local neighbourhood around the image pixel is evaluated. The
result is assigned to the output image.
a) [0.5 points] Implement a function that takes a greyscale image and an arbitrary
linear convolution kernel and performs two-dimensional spatial convolution. Assume
that the size of the convolution kernel is odd numbered, e.g. 3 × 3, 5 × 5, or 7 × 7.
You must implement the convolution procedure yourself from scratch. You are not
required to implement procedures for adding and removing padding. In your own
words, explain how you implemented convolution your report.
4Assuming we have an 8-bit image, with intensity values ∈ [0, 255], then all we have to do is change
the data type to float and divide by 255.
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Equation 5 shows two convolution kernels that are used for smoothing images. The
one on the left is a 3 × 3 averaging kernel, while the one on the right is a 5 × 5 Gaussian
kernel approximated using binomial coefficients.
1
9
1 1 1
1 1 1
1 1 1
1
256
1 4 6 4 1
4 16 24 16 4
6 24 36 24 6
4 16 24 16 4
1 4 6 4 1
(5)
b) [0.2 points] Try the convolution function you made in task (a) by displaying the
result of the following in your report:
i) Convolve a greyscale image with the smoothing kernels in Equation 5.
ii) Convolve a colour image with the smoothing kernels in Equation 5.
Hint: To run convolution on a colour image, convolve each channel separately and
layer them afterwards.
The Sobel operator consists of a pair of directional convolution kernels and is used
to measure the gradient of an image. The convolution kernels for the horizontal and
vertical direction can be seen in Equation 6. They can be applied separately on an image
I , producing Ix and Iy . However, they can also be combined to measure the magitude of
the gradient at each point: |∇| =
q
Ix
2 + Iy
2
.
−1 0 1
−2 0 2
−1 0 1
−1 −2 −1
0 0 0
1 2 1
(6)
c) [0.8 points] Using your convolution implementation from (a), carry out the following
and record the result in your report:
i) Use the convolution kernels in Equation 6 to approximate the horizontal and
vertical gradient of a greyscale image.
ii) Compute the magnitude of the gradients |∇|.
iii) In your own words, explain what |∇| tell us about the image.
Hint: Convolution with Sobel kernels may yield negative numbers. This means that
your implementation must output images with signed numbers.
