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CS4551 Spring 2020 HW #2

 CS4551 Spring 2020 HW #2

CS4551 Multimedia Software Systems
Homework2 (15%) – Vector Quantization and DCT Coding
• Due: Electronic submission via CSNS by Sunday, 04/19/2020.
• What to turn in:
o Submit source code with necessary files for “compile and run”.
o Do NOT submit data files.
o You MUST provide a readme.txt file containing all information to help with the grading process.
• If your program produces any compile errors, you will receive 0 automatically no matter how close your 
program is to the solution.
• Programming requirements:
o You are not allowed to use any Java built-in image class methods, library, or tools to complete this 
homework.
o Do not create one mega-size main class.
o Do not change any given methods of MImage class nor create a new class that duplicates MImage 
class. Treat MImage as a part of imported library.
o Test your program with all test data.
o If you do not meet any of the requirements above, you will receive a significant reduction.
0. What your program should do
Name your main application CS4551_[YourLastName].java (e.g. CS4551_Doe.java) and expand the given 
template program to perform the following tasks.
Receive the input file as command line arguments.
On Command Prompt
java CS4551_Doe Ducky.ppm 
Read a 24bit input PPM image and display the following main menu to the user and receive the user’s input.
Main Menu-----------------------------------
1. Vector Quantization
2. DCT-based Coding
3. Quit 
Please enter the task number [1-3]:
After performing a selected task, go back to display the menu again.
1. Task 1 – Vector Quantization (50 pts) 
Compress the 24 bits per pixel input image to 2 bits per pixel using Vector Quantization (VQ). Implement VQ
encoding/decoding using the requirements below.
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Encoding:
• Input vectors are formed by 2×2 blocks of RGB pixels. Each input vector 𝒙 consists of RGB values of 
FOUR pixels, P1, P2, P3, and P4, and therefore 𝒙 is 12 dimensional.
P1 P2
P3 P4
Diagram of a 2×2 pixel block of the input image
𝒙 = {𝑃1𝑅,𝑃1𝐺,𝑃1𝐵,𝑃2𝑅, 𝑃2𝐺,𝑃2𝐵,𝑃3𝑅,𝑃3𝐺,𝑃3𝐵, 𝑃4𝑅,𝑃4𝐺,𝑃4𝐵
}
• Codebook and codebook vectors: The 2-bits per pixel quantization is equivalent to using 8 bits per 4 
pixels. Therefore, the VQ should the vector space into 256 (=28
) cells and the codebooks should have 
256 entries that are centroids of the 256 cells. After the vector quantization, each vector 𝒙 belongs to one 
cell and each cell number is represented by 1 byte. In order words, after the quantization, each 2×2 block 
(4×3=12 bytes) is encoded by a 1-byte codebook index. So, the compression ratio is 12.
• Codebook generation: Use K-means clustering algorithm to generate codebook vectors (centroids of 
cells).
K-means Clustering Algorithm
Inputs: K, number of clusters and the data set (input vectors 𝒙)
K is 256 in our case.
Assume that 𝒄[𝑖] store the K centroids. 𝑖 = 0, 1, ⋯ , 255.
Each 𝒄[𝑖] is a 12-dimensional vector.
1. Assign randomly generated initial values for the 𝐾 centroids.
2. For each 𝒙,
For each 𝑖 = 0 to 255
If 𝒄[𝑖] is the closest cell (cluster) to 𝒙 based on the Euclidean distance between 𝒙 and 𝒄[𝑖],
assign 𝒙 to 𝒄[𝑖] cluster
3. Update the 𝐾 centroids.
4. Iterate 2 & 3 until the algorithm meets a stopping condition (i.e. no data points change clusters, the 
sum of the distance is minimized, or the maximum number of iterations is reached.)
• Display the codebook. This is equivalent to displaying 𝒄[𝑖], 𝑖 = 0, 1, ⋯ ,255.
• Extra credit (10 pts) – Save the quantized image (i.e. indices of all 2×2 blocks) into a PPM file. Given 
𝑊 × 𝐻 input image, the quantized image resolution is 𝑊/2 × 𝐻/2. The quantized image is a grayscale 
image because each pixel is an 8-bit index.
Decoding:
• Given the quantized image and the codebook, for each pixel of the quantized image, recover RGB pixel 
values of 4 pixels.
• Save the decoded image so that you can compare the output with the input.
2. Task 2 – DCT-based Coding (50 pts)
Implement a DCT-based transform coding. Notice that this is different from the standard JPEG steps. The
encoder/decoder steps are shown below.
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Encoding Steps Decoding Steps
E1. Read and resize the input image
Read the input ppm file containing RGB pixel values 
for encoding. First, if the image size is not a multiple 
of 8 in each dimension, make (increase) it become a 
multiple of 8 and pad with zeros. For example, if your 
input image size is 21×14, make it become 24×16 and 
fill the extra pixels with zeros (black pixels).
D4. Remove Padding and Display the image
Display the decompressed image. Remember that you
padded with zeros if the input image size is not 
multiple of 8 in both dimensions (width and height). 
Restore the original input image size by removing 
extra padded rows and columns.
E1. Resize Input Image
E2. Color Conversion & 
Subsampling
E3. Forward DCT
E4. Quantization
D4. Restore Original Size
D3. Supersampling & Inverse 
Color Conversion
D2. Inverse DCT
D1. De-quantization
Input RGB Image (PPM) Decompressed RGB Image (PPM)
Compressed Image 01001…
Entropy Encoding/Decoding
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E2. Color space transformation and Subsampling
Transform each pixel from RGB to YCbCr using the
equation below:
(
𝑌
𝐶𝑏
𝐶𝑟
) = (
0.2990 0.5870 0.1140
−0.1687 −0.3313 0.5000
0.5000 −0.4187 −0.0813
)(
𝑅
𝐺
𝐵
)
Initially, RGB value ranges from 0 and 255. After 
color transformation, Y should range from 0 to 255, 
while Cb and Cr should range from -127.5 to 127.5. 
(Truncate if necessary.)
Subtract 128 from Y and 0.5 from Cb and Cr so that 
they span the same range of values [-128,127]
Subsample Cb and Cr using 4:2:0 (MPEG1) 
chrominance subsampling scheme. If Cb(Cr) is not 
divisible by 8, pad with zeros.
D3. Supersampling and Inverse Color space 
transformation
Supersample Cb and Cr so that each pixel has Cb and 
Cr.
Add 128 to the values of the Y component and 0.5 to 
the values of the Cb and Cr components.
If using a color image, transform from the YCbCr 
space to the RGB space according to the following 
equation:
(
𝑅
𝐺
𝐵
) = (
1.0000 0 1.4020
1.0000 −0.3441 −0.7141
1.0000 1.7720 0
) (
𝑌
𝐶𝑏
𝐶𝑟
)
Common mistake: After this step, you have to make 
sure that the resulting RGB values are in the range 
between 0 and 255. Truncate if necessary.
E3. Forward DCT
Perform the forward DCT for Y image using the 
following steps:
• Divide the image into 8×8 blocks. Scan each 
block in the image in raster order (left to right, 
top to bottom)
• For each 8×8 block, perform the DCT 
transform to get the values 𝐹𝑢𝑣 from the values 
𝑓𝑥𝑦. The elements 𝐹𝑢𝑣 range from −2
10 to 2
10
Check max and min and assign −2
10 or 2
10
for the values outside of the range so that the 
values range from −2
10 to 2
10
.
Perform the DCT for Cb and Cr images, too.
D2. Inverse DCT
Perform the inverse DCT to recover the values 𝑓𝑥𝑦
from the values 𝐹𝑢𝑣 and recover Y, Cb, Cr images.
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Forward DCT Formula
𝐹𝑢𝑣 =
1
4
𝐶𝑢𝐶𝑣 ∑ ∑𝑓𝑥𝑦 cos (
(2𝑥 + 1)𝑢𝜋
16 )cos (
(2𝑦 + 1)𝑣𝜋
16 )
7
𝑦=0
7
𝑥=0
𝐶𝑢 = {
1/√2, 𝑢 = 0
1, otherwise
𝐶𝑣 = {
1/√2, 𝑣 = 0
1, otherwise
𝑓𝑥𝑦 is the 𝑥-th row and 𝑦-th column pixel of the 8×8 image block (𝑥 and 𝑦 range from 0 to 7); 𝐹𝑢𝑣 is the DCT 
coefficient value in the 𝑢-th row and 𝑣-th column (𝑢 and 𝑣 range from 0 to 7).
Inverse DCT Formula
𝑓𝑥𝑦
′ =
1
4
∑ ∑𝐶𝑢 𝐶𝑣 𝐹𝑢𝑣
cos (
(2𝑥 + 1)𝑢𝜋
16 ) cos (
(2𝑦 + 1)𝑣𝜋
16 )
7
𝑣=0
7
𝑢=0
E4. Quantization
Given 𝐹𝑢𝑣 in an 8×8 DCT block, quantize 𝐹𝑢𝑣 using:
Quantized(𝐹𝑢𝑣) = round (
𝐹𝑢𝑣
𝑄𝑢𝑣
)
The default intervals 𝑄𝑢𝑣 corresponding 𝑢 and 𝑣 are 
specified in Table 1 and Table 2.
In this homework, we want to provide a variety of
compression quality options (high compression or 
low compression). User shall specify one parameter 
𝑛 (0 ≤ 𝑛 ≤ 5 ), which controls the quality of the 
compression by changing the quantization intervals. 
The actual quantization is done by
Quantized(𝐹𝑢𝑣) = round (
𝐹𝑢𝑣
𝑄𝑢𝑣
)
𝑄𝑢𝑣
′ = 𝑄𝑢𝑣 × 2
𝑛
For example, if 𝑛 = 0, 𝑄𝑢𝑣
is same as 𝑄𝑢𝑣; if 𝑛 = 1, 
𝑄𝑢𝑣
is double of 𝑄𝑢𝑣 , which will divide 𝐹𝑢𝑣 with 
bigger values and result in more compression.
D1. De-quantization
Assume that the quantization tables (basis ones) and 
the compression quality parameter 𝑛 are available for 
decoding. Given the quantized value for DCT 
coefficient 𝐹𝑢𝑣 , multiply it by the corresponding
quantization interval 𝑄𝑢𝑣
.
𝐹𝑢𝑣
′ = Quantized(𝐹𝑢𝑣) × 𝑄𝑢𝑣
Notice that the recovered 𝐹𝑢𝑣
′ will be different from 
the original 𝐹𝑢𝑣.
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Default Quantization Tables
The following table gives the default quantization intervals for each element in the 8×8 DCT block for the 
luminance (Y) and chrominance (Cb and Cr).
Table 1. Luminance Y Quantization 
Table
Table 2. Chrominance Cb and Cr 
Quantization Table
4 4 4 8 8 16 16 32 8 8 8 16 16 32 32 64
4 4 8 8 16 16 32 32 8 8 16 16 32 32 64 64
4 8 8 16 16 32 32 32 8 16 16 32 32 64 64 64
8 8 16 16 32 32 32 32 16 16 32 32 64 64 64 64
8 16 16 32 32 32 32 48 16 32 32 64 64 64 64 96
16 16 32 32 32 32 48 48 32 32 64 64 64 64 96 96
16 32 32 32 32 48 48 48 32 64 64 64 64 96 96 96
32 32 32 32 48 48 48 48 64 64 64 64 96 96 96 96
• E1/D4 – 10 pts
• E2/D3 – 10 pts
• E3/D2 – 20 pts
• E4/D1 – 10 pts
An important requirement – After each encoding step, implement the corresponding decoding step 
immediately and check if your output is correct or not.
You will receive credits for each encoding step if only if you complete to implement the corresponding 
decoding step.
Sample results will be posted on CSNS.
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