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CS 576 – Assignment 2
Solutions due on Monday 10/09/2023 by midday 12 pm noon
Late Policy: None, unless prior arrangement has been made.
PART 1: Theory Questions for practice, no need to submit,
Question1: Color Theory
One of uses of chromaticity diagrams is to find the gamut of colors given the primaries. It
can also be used to find dominant and complementary colors –
Dominant color of a given color D (or dominant wavelength in a color D) is defined as the
spectral color which can be mixed with white light in order to reproduce the desired D
color. Complementary colors are those which when mixed in some proportion create the
color white. Using these definitions and the understanding of the chromaticity diagram that
• In the image alongside find
the dominant wavelength of
color D. Show this
wavelength.
• Do all colors have a
dominant wavelength?
• Find the color which is
complimentary to the color
C. Show this color
• What colors in RGB color space map
to the equiluminous point E upon projection
into the chromaticity space
Question 2: Color Theory
• The chromaticity diagram in (x, y) represents the normalized color matching
functions X, Y and Z. Prove that
Z = [ (1-x-y)/y ] Y
Here you are tasked with mapping the gamut of a printer to that of a color CRT
monitor. Assume that gamuts are not the same, that is, there are colors in the printer’s
gamut that do not appear in the monitor’s gamut and vice versa. So, in order to print a
color seen on the monitor you choose the nearest color in the gamut of the printer.
• Comment (giving reasons) whether this algorithm will work effectively?
• You have two images – a cartoon image with constant color tones and a real image
with varying color tones? Which image will this algorithm perform better – give
reasons?
• Can you suggest improvements rather than just choosing the nearest color?
Entropy Coding –
Consider a communication system that gives out only two symbols X and Y. Assume that
the parameterization followed by the probabilities are P(X) = x2
and P(Y) = (1-x
2
).
• Write down the entropy function and plot it as a function of x
• From your plot, for what value of x does the Entropy become a minimum? At what
values of x is the Entropy a maximum?
• Although the plot visually gives you the value of x for which the entropy in
maximum, can you now mathematically find out the value(s) for which the entropy
is a maximum?
• Can you do the same for the minimum, that is can you find mathematically prove
the value(s) of x for which the entropy is a minimum?
Generic Compression –
The following sequence of real numbers has been obtained sampling a signal:
5.8, 6.2, 6.2, 7.2, 7.3, 7.3, 6.5, 6.8, 6.8, 6.8, 5.5, 5.0, 5.2, 5.2, 5.8, 6.2, 6.2, 6.2, 5.9, 6.3, 5.2, 4.2, 2.8,
2.8, 2.3, 2.9, 1.8, 2.5, 2.5, 3.3, 4.1, 4.9
This signal is then quantized using the interval [0,8] and dividing it into 32 uniformly distributed
levels.
• What does the quantized sequence look like? For ease of computation, assume that you
placed the level 0 at 0.25, the level 1 at 0.5P, level 2 at 0.75, level 3 at 1.0 and so on. This
should simplify your calculations. Round off any fractional value to the nearest integral
levels
• How many bits do you need to transmit the entire signal?
• If you need to encode the quantized output using DPCM. Compute the successive
differences between the values – what is the maximum and minimum value for the
difference? Assuming that this is your range (ie, ignore first value), how many bits are
required to encode the sequence now?
• What is the compression ratio you have achieved (ignoring first value)?
• Instead of transmitting the differences, you use Huffman coded values for the differences.
How many bits do you need now to encode the sequence? Show all your work and how you
• What is the compression ratio you have achieved now (ignoring first value)?
Programming Part (140 points)
This assignment will help you gain a practical understanding of color representations, color
spaces and data structures normally used in color analysis. Detecting and localizing objects
in an image is a challenging computational task. The overall problem space is complex and
robust solutions in practical scenarios need many computational descriptors and vectors. In
this assignment we will explore using color as one such data point to accomplish this.
While color representation is only one vector (among others) to help address this problem,
we are hopeful that you get a good understanding of color representation and color spaces
which is the intent of this assignment.
You will be given a small library of object images where each image represents an object.
All these images will be in a standard format representing the picture of one colorful object
placed against a green background. You may process these images to make color
representations of objects. You will be given an input image which contains a combination
of objects some of which will be in your library. You need to display your processed output
where you need to outline using a bounding box around the area of pixels that match any
object detected from the library. You are free to extend the display code sample given of
assignment 1 to implement this project. No external libraries or scripting environments are
allowed.
You program will take a few parameters.
• InputImage.rgb – This is the input image that contains a one of more of colorful
objects some of which (or none of which) will be in your library.
• A list of objects object1.rgb object2.rgb …objectn.rgb which will represent your
library of objects.
All the images will be of the same size 640x480.
To invoke your program, we will compile it and run it at the command line as
YourProgram.exe InputImage.rgb object1.rgb object2.rgb …. objectn.rgb
where the number of objects (objectn.rgb) may vary between 1 to n. Here is an example
data set of the input image and library images with an expected output.
Object1.rgb Object2.rgb
Inputimage.rgb
Process and Execution:
The main data structure you will be using is a color histogram. A histogram is a
representation of the distribution of data and the frequency of each type of data. For your
understanding, given a gray scale single channel image, its associated histogram is
illustrated by the example below, where the X axis describes all possible values of a pixel
0-255 and the Y axis gives the number of times the pixel value occurs in the image.
Processed output by program
Object1.rgb
A color representation has three channels R, G and B. You may create three one
dimensional histograms, one for each channel, or alternatively one three-dimensional
histogram. Now that you have understood the histogram representation lets see how to
make use of the this for detecting and localizing objects.
1. Creating histograms for object images.
The input parameters to your program has a list of objects (from a library). All these object
images are given to you against a constant green chroma background. The green chroma
background is given so you can conveniently choose pixels that belong to the object. Create
multiple histograms for each object image given on the command line list.
2. Creating a histogram of the input image
Given the input image, you may assume that it contains object(s). Create a histogram of the
image. At this point you should make note of the fact that the position/orientation/scale of
an object in the input image will be very different from its proportions in the object image.
This will result in the number of pixels that represent the object in the two images be vastly
different, though there is a pattern to the distribution which will be similar in both
histograms.
3. Object detection using histogram matching
Given object histograms and the input image histogram, implement a method that will
allow you detect the presence of an object histogram in the input image histogram
comparing all the channels. Given that color is the main purpose of comparison, you may
want to implement this matching algorithm in a space that ignores luminance eg
• Choice 1 – Convent the image from RGB to YUV, and create histograms for U and
V. Here you may ignore Y and compare the U&V histograms between the input
image and object images. You can read about the color conversion matrix from the
class lecture slides.
• Choice 2 - Convent the image from RGB to HSV, and create histograms for H.
Here you may ignore S & V and compare the H (hue) histogram between the input
image and library image objects. You can read about the color conversion here:
https://en.wikipedia.org/wiki/HSL_and_HSV & https://www.cs.rit.edu/~ncs/color/
4. Object Localization
Once you have detected the presence of the object by comparing histograms, you need to
highlight the pixels in the image that correspond to the object either by highlighting them
or showing a bounding box around them. To achieve this you need to keep track of [x,y]
locations of pixels when forming the histogram.
What should you submit?
• Your source code files, and your project file or makefile, if any – all combined into
a zip archive with your name on the archive. Please do not submit any binaries or
images. We will compile your program and execute our tests accordingly.
• If you need to include a readme.txt file with any special instructions on compilation,
that is fine too.