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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 consists of RGB values of
FOUR pixels, P1, P2, P3, and P4, and therefore is 12 dimensional.
P1 P2
P3 P4
Diagram of a 2×2 pixel block of the input image
= {1 , 1 , 1, 2 , 2 , 2, 3 , 3 , 3, 4 , 4 , 4}
• 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 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 )
K is 256 in our case.
Assume that [] store the K centroids. = 0, 1, ⋯ , 255.
Each [] is a 12-dimensional vector.
1. Assign randomly generated initial values for the centroids.
2. For each ,
For each = 0 to 255
If [] is the closest cell (cluster) to based on the Euclidean distance between and [],
assign to [] cluster
3. Update the 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 [], = 0, 1, ⋯ ,255.
• Extra credit (10 pts) – Save the quantized image (i.e. indices of all 2×2 blocks) into a PPM file. Given
× input image, the quantized image resolution is /2 × /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:
(



) = (
0.2990 0.5870 0.1140
−0.1687 −0.3313 0.5000
0.5000 −0.4187 −0.0813
) (



)
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:
(



) = (
1.0000 0 1.4020
1.0000 −0.3441 −0.7141
1.0000 1.7720 0
) (



)
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 from the values
. The elements range from −2
10 to 210
Check max and min and assign −210 or 210
for the values outside of the range so that the
values range from −210 to 210.
Perform the DCT for Cb and Cr images, too.
D2. Inverse DCT
Perform the inverse DCT to recover the values
from the values and recover Y, Cb, Cr images.
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Forward DCT Formula
=
1
4
∑ ∑ cos (
(2 + 1)
16
) cos (
(2 + 1)
16
)
7
=0
7
=0

= {
1/√2, = 0
1, otherwise

= {
1/√2, = 0
1, otherwise

is the -th row and -th column pixel of the 8×8 image block ( and range from 0 to 7); is the DCT
coefficient value in the -th row and -th column ( and range from 0 to 7).
Inverse DCT Formula

′ =
1
4
∑ ∑
′ cos (
(2 + 1)
16
) cos (
(2 + 1)
16
)
7
=0
7
=0

E4. Quantization
Given in an 8×8 DCT block, quantize using:
Quantized() = round (


)
The default intervals corresponding and 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
( 0 ≤ ≤ 5 ), which controls the quality of the
compression by changing the quantization intervals.
The actual quantization is done by
Quantized() = round (


′ )

′ = × 2

For example, if = 0,
′ is same as ; if = 1,

′ is double of , which will divide with
bigger values and result in more compression.
D1. De-quantization
Assume that the quantization tables (basis ones) and
the compression quality parameter are available for
decoding. Given the quantized value for DCT
coefficient , multiply it by the corresponding
quantization interval
′ .

′ = Quantized() ×

Notice that the recovered
′ will be different from
the original .
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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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