Assignment 2 Solution

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Theory Questions: (50 points)

Question 1: Color Theory (15 points)

In a three dimensional color space such as XYZ, any color C with coordinates (X, Y, Z) can be expressed as a linear combination of the primaries P1, P2, P3 with coordinates (X1, Y1, Z1), (X2, Y2, Z2) and (X3, Y3, Z3) respectively. This may be expressed as

  1. (X, Y, Z) = α1*P1 (X1, Y1, Z1) + α2*P2 (X2, Y2, Z2) + α3*P3(X3, Y3, Z3)

In this question you are asked to show that similarly, the normalized chromaticity coordinates of C can also be expressed as a linear combination of the normalized chromaticity coordinates of P1, P2, P3. Proceed by answering the following:

Find the normalized chromaticity coordinates of P1, P2, and P3 in terms of given known quantities (3 points)

Express the normalized chromaticity coordinates of the color C in terms of the chromaticity coordinates of P1, P2, and P3 (6 points)

Hence prove that the chromaticity coordinates of any color C (which is a linear combination of primaries P1, P2, and P3 in XYZ color space) can be represented also as a linear combination of the chromaticity coordinates of the respective primaries. (6 points)

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)

Question 3: Entropy Coding (10 points)

Consider a communication system that gives out only two symbols X and Y. Assume that the parameterization followed by the probabilities are P(X) = xk and P(Y) = (1-xk)

Write down the entropy function and plot it as a function of x for k=2. (1 points)

From your plot, for what value of x with k=2 does H become a minimum? (1 points) Your plot visually gives you the minimum value of x for k=2, find out a generalized

formula for x in terms of k for which H is a minimum (3 points).

From your plot, for what value of x with k=2 does H become a maximum? (1 points) Your plot visually gives you the maximum value of x for k=2, find out a generalized

formula for x in terms of k for which H is a maximum (4 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)

6 8 4

D =

1 0 3

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)

Firstly, you need to be able to display images in the RGB format that we will give to you for testing. We have provided a Microsoft Visual Studio C++ and java projects that read a given image and displays it correctly. This source has been provided as a reference for students who may not know how to read and display images. You are free to use this as a start or write your own. For this assignment you are required to use a non-scriptable language (such as C/C++ or java, no python, no matlab) where you will implement operations and not refer to libraries.

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 …

  1. 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

  1. 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

What should you submit for part 2 ?

Your source code, and your project file or makefile, if any, using the submit program. Please do not submit any binaries or images. We will compile your program and execute our tests accordingly.

If you need to, please include a readme.txt file with any special instructions on compilation for compilation.

Here is the dataflow pipeline of the encoding and decoding,

1.Read Input

Image

        1. Break each channel into 8×8 blocks

  1. Discrete Cosine Transform

    1. Quantize based on table

          1. Dequantize based on the same table

      1. Inverse DCT

and decode

based on

delivery mode

Display Input

This code is already provided to you,

Image

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

Quantize DC and ACs based on uniform

quantization table

Dequantize DC and ACs based on

uniform quantization table

Display Do Inverse DCT and display image

Output Image based on mode parameter


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