DCT

Chapter 18Chapter 18

Discrete Cosine TransformDiscrete Cosine Transform

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004Chapter 18, Slide 2

Learning ObjectivesLearning Objectives

Introduction to the DCT and IDCT.Introduction to the DCT and IDCT. Decomposition of a 2-D DCT to two 1-D Decomposition of a 2-D DCT to two 1-D

DCTs.DCTs. Implementation of a 2-D DCT using a 1-Implementation of a 2-D DCT using a 1-

D DCT.D DCT.


IntroductionIntroduction

To perform the JPEG coding, an image (in colour or grey To perform the JPEG coding, an image (in colour or grey scales) is first subdivided into blocks of 8x8 pixels.scales) is first subdivided into blocks of 8x8 pixels.

The Discrete Cosine Transform (DCT) is then performed on The Discrete Cosine Transform (DCT) is then performed on each block.each block.

This generates 64 coefficients which are then quantised to This generates 64 coefficients which are then quantised to reduce their magnitude.reduce their magnitude.

DCT

IDCT

Quantiser

Dequantiser

EntropyEncoder

EntropyDecoder

Channelor

Storage

reversezigzag

zigzag

An8x8

block

Im age

8 pixels

8 p

ixels


IntroductionIntroduction

The coefficients are then reordered into a one-dimensional array The coefficients are then reordered into a one-dimensional array in a zigzag manner before further entropy encoding.in a zigzag manner before further entropy encoding.

The compression is achieved in two stages; the first is during The compression is achieved in two stages; the first is during quantisation and the second during the entropy coding process.quantisation and the second during the entropy coding process.

JPEG decoding is the reverse process of coding.JPEG decoding is the reverse process of coding.

DCT

IDCT

Quantiser

Dequantiser

EntropyEncoder

EntropyDecoder

Channelor

Storage

reversezigzag

zigzag

An8x8

block

Im age

8 pixels

8 p

ixels


Implementation of the DCTImplementation of the DCT

DCT-based codecs use a two-DCT-based codecs use a two-dimensional version of the transform.dimensional version of the transform.

The 2-D DCT and its inverse (IDCT) of The 2-D DCT and its inverse (IDCT) of an N x N block are shown below:an N x N block are shown below:

Note: The DCT is similar to the DFT Note: The DCT is similar to the DFT since it decomposes a signal into a series since it decomposes a signal into a series of harmonic cosine functions.of harmonic cosine functions.

2-D DCT:2-D DCT:

2-D IDCT:2-D IDCT:

1

0

1

0

]2

)12(cos[]

2

)12(cos[),()()(

2),(

N

y

N

x N

vy

N

uxyxfvCuC

NvuF

1

0

1

0

]2

)12(cos[]

2

)12(cos[),()()(

2),(

N

v

N

u N

vy

N

uxvuFvCuC

Nyxf


2-D DCT using a 1-D DCT Pair2-D DCT using a 1-D DCT Pair

One of the properties of the 2-D DCT is One of the properties of the 2-D DCT is that it is separable meaning that it can that it is separable meaning that it can be separated into a pair of 1-D DCTs.be separated into a pair of 1-D DCTs.

To obtain the 2-D DCT of a block a 1-D To obtain the 2-D DCT of a block a 1-D DCT is first performed on the rows of DCT is first performed on the rows of the block then a 1-D DCT is performed the block then a 1-D DCT is performed on the columns of the resulting block.on the columns of the resulting block.

The same applies to the IDCT.The same applies to the IDCT. This process is illustrated on the This process is illustrated on the

following slide.following slide.


2-D DCT using a 1-D DCT Pair2-D DCT using a 1-D DCT PairRaw B lock of P ixels

F inal result after DCT by colum ns Result after DCT by rows

128

128

125

127

127

130 128 134 130 128 128

128128130134128130

126 130 127 130 131 128 127

127127

131

131

131

129

129

128

130129

132

133131

129

126127

130

134131129

131

124 128

132

129 134134 136 136 137 134 132

132135140137138136135133

365

365

362

359

368

370

378

383

-1

-1

-3

-3

1

0

-3

-1

-4

-4

-4

-3

1

-3

-6

-6

1

1

0

0

-1

-3

0

2

-3

-3

-3

-3

1

1

-1

-1

-2

00-3

1

1

-3

-3

1

-1

-3

2

0

-2

-1

-1

3

3

1

1

-1

-2

2

-3

pixel[n]

An 8x8block

Part of a picture

Colum ns (256 pixels)

Row

s (

256 p

ixels

)

p ixel[256+n]

pixel[n+1]

1024

-18

11

0

-2

-4

0

2

-4

-1

-11

-1

1

-5

2 2

1 1

-3 -3

0 -1

0 1 0

-2

-1

1 0 -1 -1 -2

0 -1 -1 -1

011-1

-3 -1 0 -2

3-2-23-3

0 1

-1

-12

0

0 -4 -6 0 1

0

2 4-22


2-D DCT using a 1-D DCT Pair2-D DCT using a 1-D DCT Pair

1-D DCT:1-D DCT:

1-D IDCT:1-D IDCT:

]2

)12(cos[)()(

2)(

1

0

N

i N

kiixkC

NkX

]2

)12(cos[)()(

2)(

1

0

N

kN

kikXkC

Nix

k = 0, 1, 2, …, N-1.and i = 0, 1, 2, …, N-1.


Implementation IssuesImplementation Issues Precalculate the DCT coefficients and Precalculate the DCT coefficients and

scale them (see scale them (see \Links\DCT Coeffs.pdf).).

0.707

0.981

0.924

0.707

0.831

0.707 0.707 0.707 0.707 0.707 0.707

-.981-0.8310.556-0.1950.1950.556

0.383 -0.383 -.0924 -0.924 -0.383+0.38

30.924

-0.8310.195

0.556

0.707

0.981

-0.707

-0.195

-0.707

0.9810.556

0.707

-0.8310.831

0.707

-0.556-0.981

-.707

0.981-0.9810.556

0.707

0.831 -0.195

-0.707

0.383 0.924-0.924 0.383 -0.383 0.924 -0.924 0.383

-0.1950.556-0.8310.981-0.9810.8310.5560.195

1448

2008

1892

1448

1702

1448 1448 1448 1448 1448 1448

-2008-1702-1137-3993991137

783 -783 -1892 -1892 -783 783 1892

-1702399

-1137

1448

2008

-1448

-399

-1448

20081137

1448

-17021702

1448

-1137-2008

-1448

399-20081137

1448

1702 -399

-1448

783 1892-1892 -783 -783 1892 -1892 783

-3991137-17022008-20081702-1137399

Multiplied by 211

]16

)12(cos[),(*2

kikicoef

coef(i, k) in Q12 form at


Block-based DCT and IDCT in CBlock-based DCT and IDCT in C

The most straightforward way of The most straightforward way of implementing a DCT and an IDCT is to implementing a DCT and an IDCT is to use the 1-D DCT and IDCT.use the 1-D DCT and IDCT.

The following C code shows how to The following C code shows how to translate the 1-D equations for the DCT translate the 1-D equations for the DCT and IDCT into C code.and IDCT into C code.

The program also takes into account the The program also takes into account the numerical issues associated with fixed-numerical issues associated with fixed-point processors.point processors.


Implementation IssuesImplementation Issues(1)(1) The program runs on the DSK with the C6711 and does The program runs on the DSK with the C6711 and does

not use any communication with the host.not use any communication with the host.

(2)(2) The The scenary.hscenary.h file is quite large (256x256 8-bit pixels or file is quite large (256x256 8-bit pixels or 65536 bytes) and therefore needs to be located in a specific 65536 bytes) and therefore needs to be located in a specific memory location in the DSK. By using the memory location in the DSK. By using the #pragma #pragma DATA_SECTION (…)DATA_SECTION (…) directive the picture can be located directive the picture can be located in the DRAM (origin address 0x0800 0000).in the DRAM (origin address 0x0800 0000).

(3)(3) For example, For example, #pragma DATA_SECTION (image_in, #pragma DATA_SECTION (image_in, “ext_sdram”)“ext_sdram”) allows the image_in data array to be allows the image_in data array to be loaded into the section called “ext_sdram” which is loaded into the section called “ext_sdram” which is defined in the command file. The same applies to defined in the command file. The same applies to image_out.image_out.

(4) The directory:(4) The directory:

..\ Chapter 18 - Discrete Cosine transform\DCT_C_Fixed..\ Chapter 18 - Discrete Cosine transform\DCT_C_Fixed

contains the entire source and executable files required. contains the entire source and executable files required. The program can be loaded into the DSK and by using The program can be loaded into the DSK and by using the display feature of the CCS as shown below the the display feature of the CCS as shown below the contents of image_in and image_out can be displayed and contents of image_in and image_out can be displayed and compared.compared.


Implementation IssuesImplementation Issues

Graph properties for Graph properties for image_inimage_in

Graph properties for Graph properties for image_outimage_out

image_inimage_in

image_outimage_out


DCT Code and LinksDCT Code and Links

Code location:Code location: ……\Chapter 18 - Discrete Cosine transform\DCT_C_Fixed\Chapter 18 - Discrete Cosine transform\DCT_C_Fixed

Projects:Projects: Fixed Point in C language: Fixed Point in C language: dct_c_fixed.pjtdct_c_fixed.pjt

Links:Links: \Links\Imaging Kit.pdf

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