# CSCI 576 Assignment 2 SOLUTION

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CSCI 576 Assignment 2

Theory Questions: (50 points)
Question 1: Color Theory (15 points)
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 you have, answer the following.
• In the side image, find the dominant wavelength of
color D. Show this wavelength. (2 points)
• Do all colors have a dominant wavelength? Explain
• Find the color which is complimentary to the color
C. Show this color (2 points).
colors in the chromaticity space that have a specific amount
of R (2 + 2 + 5 points)
• Show the locus of all points or colors that have R=0
• Show the locus of all points or colors that have R=1
• Show the locus of all points or colors that have R=0.5
Question 2: Generic Compression (10 points)
Consider a source that emits an alphabet with three symbols A, B, C having probability
distributions as follows – P(A) = 0.625, P(B) = 0.125, P(C) = 0.25
• Construct a Huffman code and calculate its average code length. (2 points)
• For this three-symbol vocabulary, how many Huffman codes are possible. What are
they? (2 points)
• Is the code you have defined optimal – give reasons! If not, what can you do to
improve upon this. Show a solution by an example computing the avg. code length. (6
points)
D . C .
Question 3: Arithmetic Compression (10 points)
Consider two symbols, A and B, with the probability of occurrence of 0.8 and 0.2, respectively. The
coding efficiency can be improved by combining N symbols at a time (called “symbol blocking”). Say N
= 3, so you are grouping symbols of 3 and giving them a unique code. (Assume that each symbol
occurrence is independent of previous symbol occurrences).
• How many types of different outcomes are there and what are their probabilities? (2 points)
• Show the arrangement of symbols on the unit interval [0, 1] and determine the arithmetic
code for the three-symbol sequence. (6 points)
• What is the average code word length? (2 points)
Question 4: Image Dithering (15 points)
Let’s say that we have an original 12×8 image represented as 8 bits per pixel. In the normalized
representation shown below, assume that 0 corresponds to white and 9 corresponds to black.
Answer the following questions You may want to code/script a process to generate the final
outputs, but only final outputs are expected. Also, we have asked you to plot this 12×8 image
block as a gray color image, and its processed outputs as binary black/white images. For reasons
of visible clarity (because a 12×8 image block is very small) you may want to show a zoomed or
magnified picture.
• Plot the image as an 8-bit gray scale map, that is – create a 12×8 image to show the
original gray image block. (2 points)
• If you thresholded the above 12×8 image block such that all values below 4.5 were 0 and
above 4.5 were 9, how does your output image B/W block look? Plot an image (3 points)
• We can create a better binary image output by using a dithering matrix. Compute the
binary output of a dithering operation on the gray level 12×8 image using the dithering
matrix D given below. Assume that the image top left coordinate indexes are [0, 0].
Show a graphical binary image plot of the dithered output. (5 points)
D =
ú
ú
ú
û
ù
ê
ê
ê
ë
é
5 2 7
1 0 3
6 8 4
• What if the image block’s top left coordinate indexes start with [1, 1]. Show a graphical
binary image plot of the dithered output. (5 points)
Programming on DCT based streaming (150 points)
This programming assignment will help you to understand the working of DCT and how
it is used by standard compression algorithms like JPEG and MPEG. Specifically, you
will implement a DCT based coder-decoder for compressing an image and simulate
decoding using the baseline mode as well as progressive modes of data delivery. Your
program will take as input 4 parameters and be invoked as
myProgram InputImage quantizationLevel DeliveryMode Latency
where the parameters are defined as :
• InputImage – is the image to input to your coder-decoder (you may assume a fixed size
and format that will be described on the class website).
• QuantizationLevel – a factor that will decrease/increase compression as explained below.
This value will range from 0 to 7.
• DeliveryMode – an index ranging from 1, 2, 3. A 1 implies baseline delivery, a 2 implies
progressive delivery using spectral selection, a 3 implies progressive delivery using
successive bit approximation.
• Latency – a variable in milliseconds, which will give a suggestive “sleep” time between
data blocks during decoding. This parameter will be used to “simulate” decoding as data
arrives across low and high band width communication networks.
Your program should display two images – the original on the left and the decoded
version on the right. Here are some example input parameter invocations
1. myProgram Example.rgb 0 1 0
Here you are encoding Example.rgb and using a 0 quantization level, which means no
quantization. You are using the baseline mode and there is no latency so you should see
the whole output image almost instantaneously, taking into account computational time.
2. myProgram Example.rgb 3 1 100
Here you are encoding Example.rgb and using a 3 quantization level. You are using the
baseline mode and there is a latency while decoding and so you should see the output
data blocks appear as they get decoded.
3. myProgram Example.rgb 1 2 100
Here you are encoding Example.rgb and using a 1 quantization level. You are using the
progressive spectral selection mode and there is a latency while decoding and so you
should see the output appear in stages.
And now for the details of each part –
The Encoder
You will have to start, with an RGB image file (images will be kept on the class website).
Implement jpeg-like compressor. Here you will do almost all of the JPEG compression
steps except the entropy coding part (RLE for AC or DPCM for DC, entropy coding) and
producing the actual formatted bit stream. Also, the JPEG pipeline contains chroma
subsampling, but I am not insisting that you convert to YCrCb and subsample Cr,Cb. The main
objective here is to study the DCT and its use in encoding and decoding. So, start with the RGB
image.
• For each component (R, G and B) break it into 8×8 blocks
• For each block, do a DCT on the blocks to compute the DC and AC coefficients. You
will start with 64 f(x,y) pixels and obtain 64 F(u,v) frequency coefficients.
• Quantize the DC and AC coefficients with a uniform quantization table, all of whose
entries are 2N
, where N is the quantization level given as a parameter above. Each table
entry is the same. Quantization works by
F'[u, v] = round ( F[u, v] / 2N
).
An entry here specifies the range of each interval. So if N = 0, then 2N = 1 and hence
every interval has range 1, In this case F'[u, v] is the same as F[u, v] and there is no
quantization effect.
Now you have the DCT coefficients for all the blocks for all the components computed
and quantized. This is the output of the encoder.
The Decoder
The next step is to write a decoder. To decode each image block
• Dequantize all the coefficients. This is done as
F[u, v] = F'[u, v] * 2N
• Do an Inverse DCT on the de-quantized AC and DC coefficients to recover the image
block signal. The recovered image is of course with some loss depending on N.
Simulating various modes of encoding-decoding
The main modes that you will be simulating are baseline, progressive encoding using
spectral selection and progressive encoding using successive bit approximation explained
in class. To better understand the results of this step, you will need to use the last two
parameters – Delivery Mode and Latency. Latency simulates communication at different
bandwidths. You may assume that all the data is not available at once for decoding but
data is available in limited amounts for decoding and display depending on the latency
and the mode used.
For this step you need to implement a display loop, which displays the currently decoded
data. The latency parameter simulates communication for different bandwidths. This
parameter simply controls the time delay (in milliseconds) between packets arriving
during communication. You are not implementing any networked communication as yet,
so this simulation would amount to inserting a “sleep(time)” statement as you are
decoding your data blocks. This means you will have to decode data, display it, sleep for
required latency time and repeat these decode-display-sleep steps for every iteration.
Here is how the decoding should proceed depending on the mode used.
1. Sequential Mode
Each image block is encoded in a single left-to-right, top-to-bottom scan. You may
assume that each latency iteration pertains to ONE BLOCK. So the process progresses as
Decode data of first block and display …sleep
Decode data of second block and display …sleep

2. Progressive Mode – Spectral Selection
The DC coefficients of every image blocks is decoded first and displayed. Next the first
AC coefficients is added for all the blocks and decoded. This goes on till all the
coefficients are added to the decoding process. You may assume that each latency
iteration occurs after EVERY SPECIFIC DCT COFFICIENT for all blocks. So the
process progresses as
Decode all blocks using only DC coefficient (set rest to zero) …sleep
Decode all blocks using only DC, AC1 coefficient …. sleep
Decode all blocks using only DC, AC1, AC2 coefficient …. sleep

3. Progressive Mode – Successive Bit Approximation
All DC and AC coefficients of all image blocks are decoded first and displayed in a
successive-bit manner. So you will decode all blocks using the all the DC and AC
coefficients, but only using the first significant bit of all coefficients Next, you will
decode all DC and AC coefficients using the first two significant bits of all coefficients
and so on. You may assume that each latency iteration occurs at EACH SIGNIFICANT
BIT usage. So the process progresses as
Decode all blocks using 1st significant bit of all coefficients …sleep
Decode all blocks using 1st , 2nd significant bit of all coefficients …. sleep
Decode all blocks using 1st , 2nd , 3rd significant bit of all coefficients …. sleep
Here is the dataflow pipeline of the encoding and decoding,
Decod
Image
Display Input
Image
2. Break each
channel into 8×8
blocks
3. Discrete Cosine
Transform
4. Quantize based
on table
5. Dequantize
based on the same
table
Display
Output Image
This code is already provided to you,
if you choose to make use of it
Before DCT, each channel of image
should be divided into 8×8-sized blocks
You need to use DCT function in the
Lecture notes
6. Inverse DCT
and decode
based on
delivery mode
Quantize DC and ACs based on uniform
quantization table
Dequantize DC and ACs based on
uniform quantization table
Encod
Do Inverse DCT and display image
based on mode parameter

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