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DCT Image Compression

D. BHAVSINGHD. BHAVSINGH

EC94001EC94001

M.TECH ,E.IM.TECH ,E.I

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What is Compression?

Compression is an agreement between sender and receiver to a

system for the compaction of source redundancy and/or removal of

irrelevancy.

lossless and

lossy image compression.

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THEDISCRETECOSINETRANSFORM

the Discrete Cosine Transform (DCT) attempts to decorrelate the

image data

decorrelation each transform coefficient can be encoded

independently without losing compression efficiency

important properties.

One-Dimensional DCT

Two-Dimensional DCT

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One-Dimensional DCT

The most common DCT definition of a 1-D sequence of length N is

Similarly, the inverse transformation is defined as

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Two-Dimensional DCT

The 2-D Discrete Cosine Transform is just a one dimensional DCT

applied twice, once in the x direction, and again in the y direction.

The DCT equation (Eq.1) computes the i, jth entry of the DCT of an

image

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Lossy Image Compression (JPEG)

Block-based Discrete Cosine Transform (DCT)

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THEDCTMATRIX

the matrix form of Equation (1), we will use the following equation

8x8 block it results in this matrix

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DCTON AN 8x8 BLOCK

the pixel values of a black-and-white image range from 0 to

255

Pure black is represented by 0,

Pure white by 255 This particular block was chosen from the very upper- left-hand

corner of an image

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the DCT is designed to work on pixel values ranging from -128 to

127 the original block is leveled off by subtracting 128 from each

entry.

This results in the following matrix

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The discrete Cosine Transform, is accomplished by matrix

multiplication

D = TMT-----(5)

This block matrix now consists of 64 DCT coefficients

c (i, j) i and j range from 0 to 7

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QUANTIZATION

8x8 block of DCT coefficients is compression by quantization

Its decide on quality levels ranging from 1 to 100,

1 gives the poorest image quality and highest compression, 100 gives the best quality and lowest compression

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Quantization is achieved by dividing each element in the

transformed image matrix D by corresponding element in the

quantization matrix

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CODING

Before storage, all coefficients of C are converted by an encoder to a

stream of binary data (01101011...).

JPEG takes advantage of this by encoding quantized coefficients in

the zig-zag sequence

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DECOMPRESSION:

Reconstruction of our image begins by decoding the bit stream

representing the Quantized matrix C.

R i, j =Q i, j C i, j

The IDCT is next applied to matrix R, which is rounded to the nearest

integer. Finally, 128 is added to each element of that result, giving us

the decompressed JPEG version N of our original 8x8 image block M

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Lossy DCT Image Compression

The lossy DCT method compression

method is widely used in current

standards. For example, JPEG images

and MPEG-1 and MPEG-2 (DVD)

videos.

As we can see here, heavily DCT-

compressed images contain blocking

artefacts. Ringing artefacts can also be

seen around edges.

LosslessLossless

LossyLossy

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Rate/Distortion

As we have seen, quality can fall rapidly

as shown by the steep slope of rate/

distortion graph.

DCT methods typically* work well up to

around 10:1 compression ratios and

then quality falls rapidly beyond this.

a) Original b) DCT : QF 3 : CR 8:1

c) DCT : QF 10 : CR 11.6:1 d) DCT : QF 20 : CR 13.6:1

e) DCT : QF 25 : CR 14.2:1 f) Difference a-e

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DCTCompression

DCT QF 3 image (CR=8:1)

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DCTCompression

DCT QF 15 image (CR=12:1)

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DCTCompression

DCT QF 25 image CR=14:1

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DCTCompression

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The philosophy behind DCT

image compression is that

the human eye is less

sensitive to higher-frequency

information and also more

sensitive to intensity than tocolour.

The examples shown here

are from Dr Flowers MPEG

slides showing the effects of

percentage reduction of colour.

Original (100%) 0.5 UV (25%) 0.25 UV (6.25%)

0.2 UV (4%) 0.1 UV (1%) 0.0 UV (0%)

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DCT Image CompressionThe Discrete Cosine Method uses continuous cosine waves,like cos(x) below, of increasing frequencies to represent theimage pixels.

The bases are the set of 64 frequencies that can be combined

to represent each block of 64 pixels.

Firstly, the image must be transformed into the frequencydomain. This is done in blocks across the whole image.

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The Discrete CosineTransform BasesLow frequency

High frequency

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The DCT method is an example of a transform method.Rather than simply trying to compress the pixel valuesdirectly, the image is first TRANSFORMED into the frequencydomain. Compression can now be achieved by more coarselyquantizing the large amount of high-frequency componentsusually present.

Firstly, the image must be transformed into the frequencydomain. This is done in blocks across the whole image.

The JPEG* standard algorithm for full-colour and grey-scaleimage compression is a DCT compression standard that uses8x8 blocks.

It was not designed for graphics or line drawings and is notsuited to these image types.

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The DCT itself does not achieve compression, butrather prepares the image for compression.

Once in the frequency domain the image's high-

frequency coefficients can be coarsely quantised sothat many of them (>50%) can be truncated to zero.

The coefficients can then be arranged so that thezeroes are clustered (zig-zag collection) and Run-LengthEncoded.

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THEDCTMATRIX

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Summary of DCT Stages

Blocking (8x8)

DCT (Discrete Cosine Transformation)

Quantization

Zigzag Scan

DPCM* on the dc value (the average value in the top left)

RLE on the ac values (all 63 values which arent the dc/ average)

Huffman Coding

* DPCM Differential Pulse Code Modulation Instead of

sending the value send the difference from the previous

value.

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The DCT

Take each 8x8 pixel block andrepresent it as amounts(coefficients) of the basis functions(the frequency set).

represent the 8x8 pixels asamounts of lowest frequency(the average or DC value)through to the highestfrequency

64 pixels values areTRANSFORMED into 64coefficients which represent theamount of each frequency.

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DCTMathematics

The Discrete Cosine Transform below takes the pixels(x,y) and

generates DCT(i,j) values.

The pixel values can be calculated as shown in the 2nd line, where

DCT(i,j) values are used to calculate pixel(x,y) values.

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The Baseline JPEGStandardQuantization Matrix -

determined by subjective testing -

16 11 10 16 24 40 51 61

12 12 14 19 26 58 60 55

14 13 16 24 40 57 69 56

14 17 22 29 51 87 80 62

18 22 37 56 68 109 103 77

24 35 55 64 81 104 113 92

49 64 78 87 103 121 120 101

72 92 95 98 112 100 103 99

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Nelsons Simpler LinearQuantizer

The Nelson DCT implementation (this is the DCT compressorused in the laboratory) uses a very simple linear quantizationstrategy.

Q= quality or quantization factor

The higher Qthe LOWER the image quality.

Where each DCT coefficient (i,j) is quantised as

For (i=0;i

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NelsonQuanitizer forQ=2

3 5 7 9 11 13 15 17

5 7 9 11 13 15 17 19

7 9 11 13 15 17 19 21

9 11 13 15 17 19 21 23

11 13 15 17 19 21 23 25

13 15 17 19 21 23 25 27

15 17 19 21 23 25 27 29

17 19 21 23 25 27 29 31

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Before and AfterQuantization

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PROPERTIES OF DCT

Decorrelation

Energy Compaction

Separability

Symmetry

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Thank

You

94001 DCT

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