Mini project

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Mini project
Image Compression (using DCT transform)

• ASPI8 gr. 841

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Outline

• Transform coding
• 1D-DCT, 2D-DCT
• Quantization
• Uniform
• Codeword assignment

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Source Coder Block Diagram
Encoder
Codeword assignment
Transform (DCT)
Input data
Quantization
Coded bit-string
Decoder
Codeword decoder
Inverse Transform (IDCT)
Output data
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One-dimensional Discrete Cosine Transform (1D-DCT)
The one-dimensional forward Discrete Cosine
Transform (1-D DCT) of N samples is formulated by
for u 0, 1, . . . , N - 1, where The
function f(x) represents the value of the xth
sample of the input signal. F(u) represents a
Discrete Cosine Transformed coefficient for u
0, 1, , N 1 First of all we apply this
transformation to the rows, then to the columns
of image data matrix.
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One-dimensional Inverse Discrete Cosine Transform
(1D-IDCT)
The one-dimensional inverse Discrete Cosine
Transform (1-D IDCT) of N samples is formulated
by
for x 0, 1, . . . , N 1, where
The function f(x) represents the value of the xth
sample of the input signal. F(u) represents a
Discrete Cosine Transformed coefficient for u
0, 1, , N 1 For image decompression we use
this 1D-DCT
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DCT by Rows and Columns
Image Matrix
Transformed Matrix
pixel
DCT coefficient
Transformed Matrix
Transformed Matrix
Transformed Matrix
Transformed Matrix
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Two-dimensional Discrete Cosine Transform (2D-DCT)

• We divide image matrix 8x8 blocks and apply
  2D-DCT which is defined by

Inverse DCT
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Partitioning to 8x8 Blocks
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Applying 2D-DCT to 8x8 Block
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Methods Comparison (¼ of all DCT coefficients)
DCT by rows and columns
DCT by 8x8 blocks
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Methods Comparison (1/16 of all DCT coefficients)
DCT by rows and columns
DCT by 8x8 blocks
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Methods Comparison (1/64 of all DCT coefficients)
DCT by rows and columns
DCT by 8x8 blocks
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Methods Comparison
Comparison of Signal to Noise Ratio (SNR)
between different methods

• DCT by rows and columns

• DCT by 8x8 blocks

Number of coefficients SNR
1/4 54.2612
1/16 42.8034
1/64 35.1766
Number of coefficients SNR
1/4 52.6321
1/16 39.9717
1/64 30.9931
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Quantization

• Why?
• To reduce the number of different values in the
  transformed signal
• How?
• Transform data values to values from interval
  -128,127
• Round off values to the nearest integer.
• Note
• before quantization we take out some number of
  the most significant values, from the transformed
  matrix.

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Codeword assignment

• Divide quantized matrix into 8x8 blocks and find
  the biggest value of each block
• Identify the minimal number of bits necessary to
  each block.

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Results

• Original image 512x512 (257KB)
• DCT(rows/2columns/2) quantization -gt 64.2 KB
  (compression rate 4)
• Codeword assignment -gt 22.4 KB
• (compression rate 2.86)
• Finaly 257KB -gt 22.4KB
• compression rate 11.5

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Original Picture
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Decompressed Picture (compression rate 11.5 )