Tài liệu Cơ sở dữ liệu hình ảnh P8 ppt

Compression can be thought of as redundancy reduction; its goal is to eliminatethe redundancy in the data to provide an efficient representation that preservesonly the essential informat

Image Databases: Search and Retrieval of Digital Imagery

Edited by Vittorio Castelli, Lawrence D Bergman Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-32116-8 (Hardback); 0-471-22463-4 (Electronic)

be an “image” — a two-dimensional (2D) array of pixels, and this chapterreviews the fundamentals of image compression as a 2D signal with specificstatistical characteristics Application to multicomponent imagery is achieved byseparately compressing each component Compression of the higher-dimensionalmulticomponent data is possible, it is very uncommon, so we concentrate on2D image compression

Compression can be thought of as redundancy reduction; its goal is to eliminatethe redundancy in the data to provide an efficient representation that preservesonly the essential information Compression can be performed in one of thetwo regimes: lossless compression and lossy compression Lossless compressionpermits an exact recovery of the original signal and permits compression ratiosfor images of not more than approximately 4 : 1 In lossy compression, theoriginal signal cannot be recovered from the compressed representation Lossycompression can provide images that are visually equivalent to the original atcompression ratios that range from 8 : 1 to 20 : 1, depending on the imagecontent Incorporation of human visual system characteristics can be important inproviding high-quality lossy compression Higher compression ratios are possible,but they produce a visual difference between the original and the compressedimages

A block diagram of a typical generic image-compression system is shown

in Figure 8.1 and consists of three components: pixel-level redundancy

211

redundancy

reduction

Data discarding

x Bit-levelredundancy reduction

Figure 8.1 Three components of an image-compression system.

reduction, data discarding, and bit-level redundancy reduction A lossless compression system omits the data-discarding step, and as such, losslesscompression results from redundancy reduction alone A lossy algorithm usesall three blocks, although extremely efficient techniques can produce excellentresults even without the third block Although both compression types can beachieved using simpler block diagrams (e.g., omitting the first block), these threesteps are required to produce state-of-the-art lossy image compression Each ofthese blocks is described briefly

image-Pixel-level redundancy reduction performs an invertible mapping of the input

image into a different domain in which the output data w is less correlated than

the original pixels The most efficient and widely used mapping is a frequency

transformation (also called a transform code), which maps the spatial

informa-tion contained in the pixels into a frequency space, in which the image data ismore efficiently represented numerically and is well matched to the human visualsystem frequency response Data discarding provides the “loss” in lossy compres-

sion and is performed by quantization of w to form x Both statistical properties

of images and human visual system characteristics are used to determine how the

data w should be quantized while minimally impacting the fidelity of the images.

Fidelity can be easily measured numerically but such metrics do not necessarilymatch subjective judgments, making visually pleasing quantization of image data

an inexact science Finally, bit-level redundancy reduction removes or reduces

dependencies in the data x and is itself lossless.

Instead of studying the blocks sequentially, this chapter begins by describingbasic concepts in both lossless and lossy coding: entropy and rate-distortiontheory (RD theory) Entropy provides a computable bound on bit-levelredundancy reduction and hence lossless compression ratios for specific sources,whereas RD theory provides a theory of lossy compression bounds Useful forunderstanding limits of compression, neither the concept of entropy nor RDtheory tell us how these bounds may be achieved and whether the computed

or theorized bounds are absolute bounds themselves However, they suggestthat the desired lossless or lossy compression is indeed possible Next, a briefdescription of the human visual system is provided, giving an understanding

of the relative visual impact of image information This provides guidance

in matching pixel-level redundancy reduction techniques and data-discardingtechniques to human perception Pixel-level redundancy reduction is thendescribed, followed by quantization and bit-level redundancy reduction Finally,several standard and nonstandard state-of-the-art image compression techniquesare described

ENTROPY — A BOUND ON LOSSLESS COMPRESSION 213

Entropy provides a computable bound by which a source with a known probabilitymass function can be losslessly compressed For example, the “source” could bethe data entering Block 3 in Figure 8.1; as such, entropy does not suggest howthis source has been generated from the original data Redundancy reduction prior

to entropy computation can reduce the entropy of the processed data below that

of the original data

The concept of entropy is required for variable-rate coding, in which a code

can adjust its own bit rate to better match the local behavior of a source Forexample, if English text is to be encoded with a fixed-length binary code, each codeword requires
log227 = 5 bits/symbol (assuming only the alphabet and a spacesymbol) However, letters such as “s” and “e” appear far more frequently than doletters such as “x” and “j.” A more efficient code would assign shorter code words

to more frequently occurring symbols and longer code words to less frequentlyoccurring symbols, resulting in a lower average number of bits/symbol to encodethe source In fact, the entropy of English has been estimated at 1.34 bits/letter [1],indicating that substantial savings are possible over using a fixed-length code

8.2.1 Entropy Definition

For a discrete source X with a finite alphabet of N symbols (x0, , x N−1) and

a probability mass function of p(x), the entropy of the source in bits/symbol is

It is easy to show (using the method of Lagrange multipliers) that the

uniform distribution achieves maximum entropy, given by H (X)= log2N Auniformly distributed source can be considered to have maximum randomnesswhen compared with sources having other distributions — each alphabet value

is no more likely than any other Combining this with the intuitive English textexample mentioned previously, it is apparent that entropy provides a measure ofthe compressibility of a source High entropy indicates more randomness; hencethe source requires more bits on average to describe a symbol

8.2.2 Calculating Entropy — An Example

An example illustrates the computation of entropy the difficulty in determiningthe entropy of a fixed-length signal Consider the four-point signal [3/4 1/4 0 0].There are three distinct values (or symbols) in this signal, with probabilities 1/4,

1/4, and 1/2 for the symbols 3/4, 1/4, and 0, respectively The entropy of thesignal is then computed as

is [10 11 0] for the symbols [3/4 1/4 0]

Now consider taking the Walsh-Hadamard transform of this signal based transforms are described in more detail in Section 8.5.2) This is an invert-ible transform, so the original data can be uniquely recovered from the trans-formed data The forward transform is given by

(block-12

00

Although the entropy example calculations given earlier are simple to compute,the results and the broader definition of entropy as the “minimum number of bitsrequired to describe a source” suggests that defining the entropy of an image isnot as trivial as it may seem An appropriate pixel-level redundancy reductionsuch as a transform can reduce entropy Such redundancy reduction techniquesfor images are discussed later in the chapter; however, it should be mentionedthat pixel transformation into the “right” domain can reduce the required bit rate

to describe the image

8.2.3 Entropy Coding Techniques

Entropy coding techniques, also known as noiseless coding, lossless coding,

or data compaction coding, are variable-rate coding techniques that provide

compression at rates close to the source entropy Although the source entropyprovides a lower bound, several of these techniques can approach this boundarbitrarily closely Three specific techniques are described

Huffman coding achieves variable-length coding by assigning code words of

differing lengths to different source symbols The code word length is directlyproportional to − log(f (x)), where f (x) is the frequency of occurrence of the symbol x, and a simple algorithm exists to design a Huffman code when the

RATE-DISTORTION THEORY 215

source symbol probabilities are known [2] If the probabilities are all powers

of (1/2), then the entropy bound can be achieved exactly by a binary Huffmancode Because a code word is assigned explicitly to each alphabet symbol, theminimum number of bits required to code a single source symbol is 1 Theexample variable-length code given in the previous section is a Huffman code

In arithmetic coding, a variable number of input symbols are required to

produce each code symbol A sequence of source symbols is represented by asubinterval of real numbers within the unit interval [0,1] Smaller intervals requiremore bits to specify them; larger intervals require fewer Longer sequences ofsource symbols require smaller intervals to uniquely specify them and hencerequire more bits than shorter sequences Successive symbols in the input datareduce the size of the current interval proportionally to their probabilities; moreprobable symbols reduce an interval by a smaller amount than less probablesymbols and hence add fewer bits to the message Arithmetic coding is morecomplex than Huffman coding; typically, it provides a gain of approximately

10 percent more compression than Huffman coding in imaging applications

Lempel-Ziv-Welch (LZW) coding is very different from both Huffman and

arithmetic coding in that it does not require the probability distribution of theinput Instead, LZW coding is dictionary-based: the code “builds itself” from theinput data, recursively parsing an input sequence into nonoverlapping blocks ofvariable size and constructing a dictionary of blocks seen thus far The dictio-nary is initialized with the symbols 0 and 1 In general, LZW works best onlarge inputs, in which the overhead involved in building the dictionary decreases

as the number of source symbols increases Because of its complexity and thepossibility to expand small data sets, LZW coding is not frequently used in image-

compression schemes (it is, however, the basis for the Unix utilities compress and gzip).

FOR LOSSY COMPRESSION

Lossy compression performance bounds are provided by rate-distortion theory(RD theory) and are more difficult to quantify and compute than the losslesscompression performance bound RD theory approaches the problem of maxi-mizing fidelity (or minimizing distortion) for a class of sources for a given bitrate This description immediately suggests the difficulties in applying such theory

to image compression First, fidelity must be defined Numerical metrics are easy

to calculate but an accepted numerical metric that corresponds to perceived visualquality has yet to be defined Secondly, an appropriate statistical description forimages is required Images are clearly complex and even sophisticated statisticalmodels for small subsets of image types fail to adequately describe the source for

RD theory purposes Nevertheless, the basic tenets of RD theory can be appliedoperationally to provide improved compression performance in a system Theaim of this section is to introduce readers to the concepts of RD theory and theirapplications in operational rate distortion

Rate, bits/pixel

Figure 8.2 A sample rate-distortion curve.

A representative RD curve is shown in Figure 8.2 — as the rate increases,distortion decreases, and vice versa RD theory provides two classes ofperformance bounds Shannon theory, introduced by Claude Shannon seminalworks [3,4], provides performance bounds as the data samples to be coded aregrouped into infinitely long blocks Alternatively, high-rate low-distortion theoryprovides bounds for fixed block size as the rate approaches infinity Although thesecond class of bounds is more realistic for use in image compression (an imagehas a finite number of pixels), both classes only provide existence proofs; theyare not constructive As such, performance bounds can be derived, no instruction

is provided on designing a system that can achieve them Hence how can RDtheory be applied to practical image compression?

First, consider a simple example Suppose that an image-compression algorithmcan select among one of several data samples to add to the compressed stream Datasample 1 requires three bits to code and reduces the mean-squared error (MSE) ofthe reconstructed image by 100; data sample 2 requires 7 bits to code and reducesthe MSE by 225 Which sample should be added to the data stream? A purely rate-based approach would select the first sample — it requires fewer bits to describe.Conversely, a purely distortion-based approach would select the second sample, as

it reduces the MSE by over twice the first sample An RD-based approach comparesthe trade-off between the two: the first sample produces an average decrease in MSE

per bit of 100/3 = 33.3 and the second produces an average decrease in MSE per bit of 225/7 = 321/7 The RD-based approach selects data sample 1 as maximizing

the decrease in MSE per bit In other words, the coefficient that yields a steeperslope for the RD curve is selected

The previous example demonstrates how RD theory is applied in practice; this

is generally referred to as determining operational (rather than theoretical) RD

curves When dealing with existing compression techniques (e.g., a particulartransform coder followed by a particular quantization strategy, such as JPEG

or a zerotree-based wavelet coder), RD theory is reduced to operational ratedistortion — for a given system (the compression algorithm) and source model(a statistical description of the image), which system parameters produce thebest RD performance? Furthermore, human visual system (HVS) characteristics

HUMAN VISUAL SYSTEM CHARACTERISTICS 217

must be taken into account even when determining operational RD curves Blindapplication of minimizing the MSE subject to a rate constraint for image datafollowing a redundancy-reducing transform code suggests that the average errorintroduced into each quantized coefficient be equal, and that such a quantizationstrategy will indeed minimize the MSE over all other step size selections at thesame rate However, the human visual system is not equally sensitive to errors

at different frequencies, and HVS properties suggest that more quantization errorcan be tolerated at higher frequencies to produce the same visual quality Indeed,images compressed with HVS-motivated quantization step sizes are of visuallyhigher quality than those compressed to minimize the MSE

The operational RD curve consists of points that can be achieved using agiven compression system, whereas RD theory provides existence proofs only.The system defines the parameters that must be selected (e.g., quantizer stepsizes), and a constrained minimization then solves for the parameters that will

provide the best RD trade-offs Suppose there are N sources to be represent using a total of R bits/symbol; these N sources could represent the 64 discrete

cosine transform (DCT) coefficients in a joint photographic experts group (JPEG)compression system or the 10 subband coefficients in a three-level hierarchicallysubband-transformed image (both the DCT and subband transforms are described

in Section 8.5) For each source, there is, an individual RD curve, which mayitself be operationally determined or generated from a model, that indicates that

source i will incur a distortion of D ij when coded at a rate of R ijat operating

point j Then the operational RD curve is obtained by solving the following

constrained minimization:

For each source i, find the operating point y(i) that minimizes the distortion

D = f (D 1y(1) , D 2y(2) , , D Ny(N ) ) such that

N

i=1

R iy(i) ≤ R ( 8.4) Most commonly, the distortion is additive and D =  N

i=1D iy(i) This constrainedminimization can be solved using the method of Lagrange multipliers When

solved, the result is that to minimize the total distortion D, each source should

operate at a point on its RD curve such that the tangents to the curves are equal;that is, the operating points have equal slopes The minimization finds that slope.Both RD theory and operational RD curves provide a guiding tenet for lossyimage compression: maximize the quality for a given bit rate Although someimage-compression algorithms actually use RD theory to find operating param-eters, more commonly, the general tenet is used without explicitly invoking thetheory This yields theoretically suboptimal compression, but as Section 8.8 willshow, the performance of many compression algorithms is still an excellent eventwithout explicit RD optimization

Lossy image compression must discard information in the image that is not or

is only minimally visible, producing the smallest possible perceptible change

to the image To determine what information can be discarded, an elementaryunderstanding of the HVS is required Understanding the HVS has been a topic

of research for over a century, but here we review the points salient to providinghigh-quality lossy image compression

8.4.1 Vision and Luminance-Chrominance Representations

Light enters the eye through the pupil and strikes the retina at the back of the eye.The retina contains two types of light receptors: cones and rods Approximatelyeight million cones located in the central portion of the retina and sensitive tored, blue, or green light provide color vision under high-illumination conditions,such as in a well-lighted room Each cone is connected to its own nerve end,

providing high resolution in photopic (color) vision Approximately 120 million

rods are distributed over the entire retina and provide vision at low-illumination

levels, such as a moonlit night (scotopic vision) A single nerve end is connected

to multiple rods; as such, resolution is lower and rods provide a general picture

of the field of view without color information With midrange illumination, both

rods and cones are active to provide mesopic vision Because most digital images

are viewed on well-lit displays, the characteristics of photopic vision are mostapplicable to digital imaging and compression

The HVS processes color information by converting the red, green, and bluedata from the cones into a luminance-chrominance space, with the luminancechannel having approximately five times the bandwidth of the chrominancechannel Consequently, much more error in color (or chrominance) than in lumi-nance information can be tolerated in compressed images Color digital imagesare often represented in a luminance-chrominance color space (one luminancecomponent, and two chrominance components) The chrominance componentsare often reduced in size by a factor of 2 in each dimension through low-passfiltering followed by downsampling; these lower-resolution chrominance compo-nents are then compressed along with the full-size luminance component Indecompression, the chrominance components are upsampled and interpolated tofull size for display No noticeable effects are seen when this is applied to naturalimages, and it reduces the amount of chrominance information by a factor of 4even before the compression operation

Because understanding and representation of color is itself a well-developedscience, the remainder of this section will focus on the HVS characteristics formonochrome (gray scale) images on which most lossy image-compression algo-rithms rely The resulting characteristics are typically also applied to chrominancedata without modification

8.4.2 The Human Contrast Sensitivity Function and Visibility Thresholds

Studies on perception of visual stimuli indicate that many factors influence thevisibility of noise in a degraded image when compared with the original Thesefactors are functions of the image itself and include the average luminance ofthe image, the spatial frequencies present in the image, and the image content

HUMAN VISUAL SYSTEM CHARACTERISTICS 219

Because images can have widely varying average luminances and content, the firstand third factors are more difficult to include in an image-compression algorithmfor general use However, all images can be decomposed into their frequencycontent and the HVS sensitivities to different frequencies can be incorporatedinto an algorithm

The human contrast sensitivity function (CSF) is a well-accepted,

experi-mentally obtained description of spatial frequency perception and plots contrastsensitivity versus spatial frequency A common contrast measure is the Michelsoncontrast, given in terms of minimum and maximum luminances in the stimulus

lmin and lmax as C = (lmax− lmin)/(lmax+ lmin) The visibility threshold (VT) isdefined as the contrast at which the stimulus can be perceived, and the contrastsensitivity is defined as 1/VT The units of spatial frequency are cycles/degree,where a cycle refers to a full period of a sinusoid, and degrees are a measure

of visual range, where the visual field is described by 180◦ Spatial frequency is

a function of viewing distance, so the same sinusoid represents a higher spatialfrequency at a larger viewing distance The CSF has been determined experi-mentally with stimuli of sinusoidal gratings at differing frequencies Figure 8.3plots a representative CSF This function peaks around 5–10 cycles/degree andexhibits exponential falloff — humans are exponentially less sensitive to higherfrequencies

The CSF represents measured sensitivities to a simple stimuli that is singlefrequency Although images can be decomposed into individual frequencies, they

in general consist of many such frequencies Factors influencing the VTs for

a complex stimuli include luminance masking, in which VTs are affected bybackground luminance and contrast masking, in which VTs are affected for oneimage component in the presence of another Contrast masking is sometimes

informally referred to as texture masking and includes the interactions of different

frequencies The combination of luminance and contrast masking is referred to as

spatial masking When spatial masking is exploited, more compression artifacts

can be hidden in appropriate parts of the an image For example, an observer isless likely to see compression artifacts in a dark, textured region, when compared

Spatial frequency, cycles/degree

Figure 8.3 The human contrast sensitivity function.

with artifacts in a midgray flat region However, fully exploiting spatial masking

in compression is difficult because it is fairly image-dependent

The CSF is nonorientation-specific A HVS model that incorporates orientation

frequency is the multichannel model [5], which asserts that the visual cortex contains sets of neurons (called channels) tuned to different spatial frequencies

at different orientations Although the multichannel model is itself not typicallyapplied directly to obtaining VTs, it is used as an argument for using wavelettransforms (described in Section 8.5)

8.4.3 Application to Image Compression

Although the CSF can be mapped directly to a compression algorithm (see [6–8]for details), it is more common to experimentally measure the VTs for basisfunctions of the transform used in an image-compression algorithm These VTsare then translated to quantizer step sizes such that quantization-induced distortion

in image components will be below the measured VTs Such an applicationassumes that results from individual experimental stimuli such as a single basisfunction or band-pass noise add independently, so that the measured VTs areequally valid when all transform coefficients in an image are simultaneouslyquantized This approach is argued to be valid when all individual distortions are

subthreshold that is, they are all below the experimentally measured VTs Such an

application is image-independent — only measured VTs are used in determining aquantization strategy Quantization can be made image-dependent by modifyingthe VTs by incorporating various spatial masking models [9]

Roughly speaking, then, a good image-compression algorithm will discardmore higher frequencies than lower frequencies, putting more compression arti-facts in the frequencies to which the eye is less sensitive Note that if thedata-discarding step is uniform quantization, this result is in conflict with thatfrom RD theory, which indicates that all frequencies should be quantized with thesame step size As such, a perceptually weighted distortion measure is required

to obtain the best-looking images when using an RD technique

In this section, techniques for redundancy reduction in images are examined.These include predictive coding and transform coding The high redundancypresent in image pixels can be quantified by the relatively high correlation coef-ficients and can be intuitively understood by considering predicting a single pixel.High correlation intuitively means that given a group of spatially close pixels and

an unknown pixel in that group, the unknown pixel can be predicted with verylittle error from the known pixels As such, most of the information required

to determine the unknown pixel is contained in the surrounding pixels and theunknown pixel itself contains relatively little information that is not represented inthe surrounding pixels Predictive coding exploits this redundancy by attempting

PIXEL-BASED REDUNDANCY REDUCTION 221

to predict the current data sample from surrounding data and can be appliedboth in the pixel domain and selectively on transform coded image data If theprediction is done well, the resulting coded data contains much less redundancy(i.e., has lower correlation) than the original data

Transformation from the pixel domain to a domain in which each data pointrepresents independent information can also eliminate this redundancy It is notcoincidental that state-of-the-art compression algorithms employ frequency trans-formations as a method of pixel-based redundancy reduction A transform can beconsidered to be a spectral decomposition, providing a twofold insight into theoperation of transform coding First of all, image data is separated into spatialfrequencies, which can be coded according to human perception Secondly, the

spectral information (or coefficient data) is significantly less correlated than the

corresponding pixel information and hence is easier to represent efficiently.This section describes two types of frequency transformations: block-basedtransforms and wavelet transforms Block-based transform coding has been verywidely used in compression techniques for images, due in great part to the use

of the DCT for the still-image coding standard JPEG [10] and in the video

coding standard motion pictures experts group (MPEG) [11] Wavelet transform

coding is an alternative to block-based transform coding and is the transformused in JPEG-2000 [12] Wavelet decompositions naturally lend themselves toprogressive transmission, in which more bits result in improvement either in aspatially scalable manner (in which the size of the decoded image increases)

or in a quality-scalable manner (in which the quality of the full-size decodedimage increases) It has been argued that representing images using their wavelettransforms is well suited to the HVS because wavelets provide a decompositionthat varies with both scale and orientation, which matches the HVS multichannelmodel

Many excellent texts exist on both transform coding techniques, and the ested reader is directed to those listed in the bibliography Ref [13]

inter-8.5.1 Predictive Coding

In predictive coding, rather than directly coding the data itself, the coded dataconsists of a difference signal formed by subtracting a prediction of the datafrom the data itself The prediction for the current sample is usually formedusing past data A predictive encoder and decoder are shown in Figure 8.4, with

the difference signal given by d If the internal loop states are initialized to the same values at the beginning of the signal, then y = x If the predictor

is ideal at removing redundancy, then the difference signal contains only the

“new” information at each time instant that is unrelated to previous data This

“new” information is sometimes referred to as the innovation, and d is called the

innovations process If predictive coding is used, an appropriate predictor must

be determined

Predictors can be nonlinear or linear A nonlinear predictor can in generaloutperform a linear predictor at removing redundancy because it is not restricted

in form, but can be difficult or impossible to implement The predictor thatminimizes the MSE between the current sample and its prediction is nonlinear

and is given by the conditional expectation operator: the MMSE predictor of x given z is E(x |z) However, this predictor is impossible to implement in practice.

An N -step linear predictor is given as

Pred{x(n)} =

N

i=1

where ρ i is the i-th-step correlation coefficient of x Linear predictors are easy

to implement as FIR filters A special case of linear prediction is one-stepprediction From Eq (8.6), the optimal one-step prediction coefficient is given

by α1 = ρ1 However, a more common predictor is to take α1 = 1, in whichcase the predictor is simply the previous data sample and can be easily imple-

mented with a delay element This is known as differential pulse code modulation

or DPCM

When predictive coding is combined with a data-discarding step (namely,quantization), care must be taken to ensure that the resulting error does notaccumulate in the decoded samples If a quantizer is placed between the encoding

and decoding operations shown in Figure 8.4, each decoded sample y(n) will contain not only the quantization error associated with quantizing d(n) but also the quantization error associated with quantizing all previous samples d(n−

1), d(n − 2), To avoid such error accumulation, the encoder must include the

quantized data in the predictor, as shown in Figure 8.5 The decoder is the same

PIXEL-BASED REDUNDANCY REDUCTION 223

In one-dimensional transform coding, an input signal x(n) is first segmented into blocks of length N , where N is the size of the transform Each block, represented as a column vector p, is then multiplied by an orthonormal transform matrix T of size N × N (T may be unitary, but most transforms used in image compression are real) The columns of T are the basis vectors of the particular transform, and the transform is applied by forming c = T p The vector c consists

of the transform coefficients, each of which is the result of an inner product of

a basis vector and the block p Because T is orthonormal, the data vector is

recovered from the transform coefficients by simply writing it as an expansion of

the corresponding basis vectors: p = T c If no lossy coding of the coefficients has taken place, t the transform provides exact invertibility, since T c = T T p = p.

Block-based transform coding easily extends to two dimensions by applyingthe transform independently in each dimension In the case of images, this corre-sponds to a horizontal and vertical application Hence, the data segment is now

a 2D matrix P of size N × N that is referred to as a pixel block The

corre-sponding transform coefficients represent the inner product of the 2D pixel blockwith the 2D basis functions, which are the outer products of the one-dimensionalbasis vectors with each other The coefficients are obtained by first applying the

transform to the columns of P by forming ˜ C = T P and then by applying the

transform to the resulting rows of ˜C by forming C = ˜CT = T CT The pixel

block is recovered from the coefficient block by pre- and postmultiplying the

coefficient block by T and T , respectively: P = T CT Again, the 2D transform

obviously provides exact invertibility

In the context of lossy data compression, a transform that minimizes theerror between the original and decompressed signals and still provides highcompression is desirable Under certain assumptions, the Karhunen-Loeve trans-form (KLT) approximately accomplishes this goal [13] The KLT diagonalizes

the data autocovariance matrix by performing an eigen-decomposition: the form matrix is the eigenvector matrix, and the transform coefficients are theeigenvalues As such, the KLT coefficients are completely uncorrelated.

trans-An example of a widely-used block-based transform is the DCT The DCTbasis functions are cosines Several versions of the DCT exist, but the version

used in JPEG is considered here The transform matrix T is given by

For small block sizes, the pixels in the block are highly correlated, and this

DCT is asymptotically equivalent to the KLT for fixed N as the one-step lation coefficient ρ approaches 1 Thus the DCT coefficients within a block are

corre-considered approximately uncorrelated

The basis vectors for the 2D DCT are the outer products of horizontal and

vertical cosine functions of frequencies ranging from DC to (N − 1)/2 The coefficient c 0,0 is called the DC coefficient, and the remaining coefficients c k,l are

called AC coefficients As the coefficient indices k and l in the coefficient block

increase, the corresponding vertical and horizontal spatial frequencies increase.The basis functions for an 8× 8 DCT are illustrated in Figure 8.6

Figure 8.6 DCT basis functions for an 8× 8 transform.