Image compression

transcript

DIGITAL IMAGE PROCESSING

IMAGE COMPRESSION

by Paresh Kamble

Introduction

Image Compression: It is the Art & Science of reducing the amount of data required to represent an image.

It is the most useful and commercially successful technologies in the field of Digital Image Processing.

The number of images compressed and decompressed daily is innumerable.

Introduction To understand the need for compact image representation,

consider the amount of data required to represent a 2 hour Standard Definition (SD) using 720 x 480 x 24 bit pixel arrays.

A video is a sequence of video frames where each frame is a full color still image.

Because video player must display the frames sequentially at rates near 30fps, SD video data must be accessed at

30fps x (720x480)ppf x 3bpp = 31,104,000 bps

fps – frames per second,ppf – pixels per frame,bpp – bytes per pixel & bps – bytes per second

IntroductionThus a 2 hour movie consists of

31,104,000 bps x (602) sph x 2 hrs ≈ 2.24 x 1011 bytes. OR

224GB of datasph = second per hour

Twenty seven 8.5GB dual layer DVDs are needed to store it.

To put a 2hr movie on a single DVD, each frame must be compressed by a factor of around 26.3.

The compression must be even higher for HD, where image resolution reach 1920 x 1080 x 24 bits/image.

Introduction Web page images & High-resolution digital camera photos

also are also compressed to save storage space & reduce transmission time.

Residential Internet connection delivers data at speeds ranging from 56kbps (conventional phone line) to more than 12mbps (broadband).

Time required to transmit a small 128 x 128 x 24 bit full color image over this range of speed is from 7.0 to 0.03 sec.

Compression can reduce the transmission time by a factor of around 2 to 10 or more.

Similarly, number of uncompressed full color images that an 8 Megapixel digital camera can store on a 1GB Memory card can be increased.

Introduction Along with these applications , image compression plays an

important role in many other areas including:

Fundamentals Data Compression: It refers to the process of reducing the

amount of data required to represent a given quantity of information.

Data Vs

Information Data and Information are not the same thing; data are the

means by which information is conveyed.

Because various amount of data can be used to represent the same amount of information, representations that contain irrelevant or repeated information are said to contain redundant data.

Fundamentals

Fundamentals Let b & b’ denote the number of bits in two representations of

the same information, the relative data redundancy R of the representation with b bits is

• R = 1 – (1/C); where, C commonly called the compression ratio, is

defined as• C = b / b’

If C = 10 (or 10:1), for larger representation has 10 bits of data for every 1 bit of data in smaller representation.

So, R = 0.9, indicating that 90% of its data is redundant.

Fundamentals 2D intensity arrays suffers from 3 principal types of data

redundancies:

1) Coding redundancy: A code is a system of symbols used to represent a body of information or sets of events.

Each piece of event is assigned a code word (code symbol). The number of symbols in each code word is its length.

The 8-bit codes that are used to represent the intensities in most 2D intensity arrays contain more bits than are needed to represent the intensities.

Fundamentals2) Spatial & Temporal redundancy:

Because the pixels of most 2D intensity arrays are correlated spatially, information is replicated unnecessarily.

In video sequence, temporally correlated pixels also duplicate information.

3) Irrelevant Information:

Most 2D intensity arrays contain information that is ignored by the human visual system. It is redundant in the sense that it is not used.

Fundamentals1) Coding Redundancy: Assume a discrete random variable rk in interval [0 – L-1] is

used to represent the intensities of an M x N image. Also the each rk occurs with probability pr (rk). pr(rk) = nk / MN k = 0, 1, 2, ………….L-1 -----(a) Where, L is no. of intensity values & nk is no. of times kth intensity appears in the image.

If no. of bits used to represent each value of rk is l(rk), then avg no of bits required to represent each pixel is

Lavg = Σ l(rk) pr(rk) k = 0

Thus, total no. of bits required to represent an MxN image is MNLavg.

Fundamentals If the intensities are represented using a natural m-bits fixed

length code, the RHS reduces to m bits. i.e. Lavg = m where m is substituted for l(rk). Constant m can be taken out the summation leaving only

sum of pr(rk) for 0 ≤ k ≤ L-1, which = 1.

rk pr(rk) code 1 l1(rk) code 2 l2(rk)r87 = 87 0.25 01010111 8 01 2r128 = 128 0.47 10000000 8 1 1r186 = 186 0.25 11000100 8 000 3r255 = 255 0.03 11111111 8 001 3rk for k ≠ 87, 0 - 8 - 0 128, 186, 255

FundamentalsWith respect to the above table, If a natural 8-bit binary code is

used to represent its 4 possible intensities, Lavg = 8, coz l1(rk) = 8 bits for all rk.

On the other hand, If code 2 scheme is used, the avg length of encoded pixels is,

Lavg = 0.25(2) + 0.47(1) + 0.25(3) + 0.03(3) = 1.81 bits.Total no. of bits rqd to represent entire image = MNLavg = 256 x 256 x 1.81 = 118,621Resulting compression C = 256 x 256 x 8 / 118,621 = 8 / 1.81 ≈ 4.42Relative redundancyR = 1 – 1/4.42 = 0.774Thus, 77.4% of data in original 8-bit 2D intensity array is

redundant.

Fidelity CriteriaFidelity Criteria:• Removal of irrelevant visual information involves a loss of

real or quantitative image information.

• Since information is lost, a means of quantifying the nature of loss is needed.

Objective fidelity criteria Subjective fidelity criteria Objective fidelity criteria: When information loss can be

expressed as a mathematical function of input & output of a compression process. Eg RMS error between 2 images.

Error between two imagese(x, y) = f’(x, y) – f(x, y)So, total error between two images M-1 N-1

Σ Σ [f’(x, y) – f(x, y)] x=0 y = 0

Fidelity CriteriaRMS error is given by M-1 N-1

erms = [(1/MN)Σ Σ [f’(x, y) – f(x, y)]2]1/2

x=0 y = 0

If f’(x, y) is considered to be the sum of original image f(x, y) & an error or noise signal e(x, y), the Mean Square SNR of output image denoted by SNRrms can be defined as

M-1 N-1

Σ Σ f’(x, y)2

SNRrms = x=0 y = 0

M-1 N-1

Σ Σ [f’(x, y) – f(x, y)]2

x=0 y = 0

Fidelity Criteria Subjective fidelity criteria:• A Decompressed image is presented to a cross section of

viewers and averaging their evaluations.

• It can be done by using an absolute rating scaleOr

• By means of side by side comparisons of f(x, y) & f’(x, y).

• Side by Side comparison can be done with a scale such as {-3, -2, -1, 0, 1, 2, 3} to represent the subjective valuations {much worse, worse, slightly worse, the same, slightly better,

better, much better} respectively.

Image Compression Models• The image compression system is composed of 2 distinct

functional component: an encoder & a decoder.

• Encoder performs Compression while• Decoder performs Decompression.

• Both operations can be performed in Software, as in case of Web browsers & many commercial image editing programs.

• Or in a combination of hardware & firmware, as in DVD Players.

• A codec is a device which performs coding & decoding.

Image Compression Models• Input image f(x,…..) is fed into the encoder, which creates a

compressed representation of input.

• It is stored for future for later use or transmitted for storage and use at a remote location.

• When the compressed image is given to decoder, a reconstructed output image f’(x,…..) is generated.

• In still image applications, the encoded input and decoder output are f(x, y) & f’(x, y) resp.

• In video applications, they are f(x, y, t) & f’(x, y, t) where t is time.

• If both functions are equal then the system is called lossless, error free. If not then it s referred to as lossy.

Image Compression Models

Mapper Quantizer Symbol coder

Symbol Decoder

Inverse Mapper

Image Compression ModelsEncoding or Compression process:Encoder is used to remove the redundancies through a series of

3 independent operations.

Mapper: It transforms f(x,…) into a format designed to reduce spatial and temporal redundancies.

• It is reversible• It may / may not reduce the amount of data to represent

image.

Ex. Run Length coding

• In video applications, mapper uses previous frames to remove temporal redundancies.

Image Compression ModelsQuantizer: It keeps irrelevant information out of compressed

representations.

• This operation is irreversible.

• It must be omitted when error free compression is desired.

• In video applications, bit rate of encoded output is often measured and used to adjust the operation of the quantizer so that a predetermined average output is maintained.

• The visual quality of the output can vary from frame to frame as a function of image content.

Image Compression ModelsSymbol Encoder: Generates a fixed or variable length code to

represent the quantizer output and maps the output in accordance with the code.

• Shortest code words are assigned to the most frequently occurring quantizer output values. Thus minimizing coding redundancy.

• It is reversible.

• Upon its completion, the input image has been processed for the removal of all 3 redundancies.

Image Compression ModelsDecoding or Decompression process:

• Quantization results in irreversible loss, an inverse quantizer block is not included in the decoder block.

Some Basic Compression MethodsHuffman Coding:• Most popular technique for removing coding redundancies.

• It yields smallest possible code symbol per source symbol.

Original Source Source reductionSymbol Probability 1 2 3 4

a2 0.4 0.4 0.4 0.4 0.6 a6 0.3 0.3 0.3 0.3 0.4 a1 0.1 0.1 0.2 0.3 a4 0.1 0.1 0.1 a3 0.06 0.1 a5 0.04

Some Basic Compression MethodsHuffman Coding:

Original Source Source reductionSymbol Probability Code 1 2 3 4

a2 0.4 1 0.4 1 0.4 1 0.4 1 0.6 0 a6 0.3 00 0.3 00 0.3 00 0.3 00 0.4 1 a1 0.1 011 0.1 011 0.2 010 0.3 01 a4 0.1 0100 0.1 0100 0.1 011 a3 0.06 01010 0.1 0101 a5 0.04 01011

Lavg = (0.4)(1) + (0.3)(2) + (0.1)(3) + (0.06)(5) + (0.04)(5) = 2.2 bits / pixel.

Some Basic Compression MethodsHuffman Coding:• It is instantaneous. • Coz each code word in a string of code symbols ca be

decoded without referencing succeeding symbols.

• It is uniquely decodable.• Coz any string of code symbols can be decoded by examining

individual symbols of string from left to right.

Ex. 010100111100

Some Basic Compression MethodsHuffman Coding:• It is instantaneous. • Coz each code word in a string of code symbols ca be

decoded without referencing succeeding symbols.

• It is uniquely decodable.• Coz any string of code symbols can be decoded by examining

individual symbols of string from left to right.

Ex. 01010 011 1 1 00First valid code: 01010 – a3, 011 – a1, Thus, completely decoding the message, we get, a3a1a2a2a6

Some Basic Compression MethodsArithmetic coding: It generates non block codes.One to One correspondence between source symbols and code

words does not exist.Instead, an entire sequence of source symbols is assigned a

single arithmetic code.Code word defines an integer of real numbers between 0 & 1.

As No. of symbols in msg. interval to represent it no. of bits to represent info Each symbol of msg size of interval in accordance with its

probability of occurrence.

Some Basic Compression MethodsBasic Arithmetic coding process:5 symbol message, a1a2a3a3a4 from 4 symbol source is coded.

Source Symbol Probability Initial Subinterval

a1 0.2 [0.0, 0.2)

a2 0.2 [0.2, 0.4)

a3 0.4 [0.4, 0.8)

a4 0.2 [0.8, 1.0)

Some Basic Compression MethodsArithmetic coding: a1 a2 a3 a3 a4