IMAGE COMPRESSION IMAGE COMPRESSION USING WEDGELETSUSING WEDGELETS

Submitted bySubmitted by Meeramol T.K.Meeramol T.K.

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ABSTRACTABSTRACT

Edges are dominant features in images,with great importance both for perception and compression.Most wavelet-based image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelet is introduced as a geometric tool for image compression. Wedgelets offer a convenient parameterization for the edges in an image. Wedgelets offer piecewiselinear approximations of edge contours and can be efficiently encoded.

INTRODUCTIONINTRODUCTIONUncompressed multimedia data requires considerable storage capacity and transmission bandwidth. The recent growth of data intensive multimedia based web applications have not only sustained the need for more efficientways to encode signals and images but have made compression of such signals central to storage and communication technology. For still image compression, the JPEG standard has been established by ISO and IEC .The performance of these coders generally degrades at low bit-rates.A variety of powerful and sophisticated wavelet-based schemes for image compression, have been developed. Most wavelet-based image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelets offer a convenient parameterization for the edges in an image.

IMAGE COMPRESSIONIMAGE COMPRESSION

In most images, the neighboring pixels are correlated.The foremost task is to find less correlated representation of the image. Two fundamental components of compression are

– redundancy reduction and – irrelevancy reduction

In general, three types of redundancy can be identified:– Spatial Redundancy – Spectral Redundancy– Temporal Redundancy

Image compression research aims at reducing the number of bits needed to represent an image.

WAVELET CODER

What is a Wavelet Transform ?

functions defined over a finite interval and having an average value of zero.The basic idea is to represent any arbitrary function ƒ(t) as a superposition of a set of waveletsThese basis functions are obtained from a single prototype waveletDiscrete Wavelet Transform of a finite length signal x(n) having N components, is expressed by an N x N matrix.

NEED OF WAVELET-BASED COMPRESSION

Blocking artifacts of JPEG

Wavelet transformation has been widely accepted in image compression

There is no need to block the input image

Robust under transmission and decoding errors

Better matched to the HVS characteristics

Suitable for applications where scalability and tolerable degradation are important.

GEOMETRY BASED TECHNIQUE

Edges represent abrupt changes in intensity.Smooth regions are characterized by slowly varying intensitiesTextures contain a collection of localized intensity changes. Edges are of particular interest for compression.Wavelets are well-suited to represent smooth and textured regions of images, but waveletbased descriptions of edges are highly inefficient.a simple twofold approach to compression.

A geometry-based compression scheme to compresses edge informationWavelets to compress the smooth and textured regions.

Better compression performancePSNR

WEDGELETS

Wedgelets is a tool for compression of edge information. Wedgelets approximate curved contours using an adaptive piecewise-linear representation.Wedgelets were first introduced by Donoho.A wedgelet is a piecewise constant function on a dyadic square with a linear discontinuity. These dyadic blocks contains a single straight edge with arbitrary orientation.Each wedgelet by itself can represent a straight edge within a certain region of the image. Smooth contours can be represented by concatenating individual wedgelets from this decomposition.

WEDGELET DICTIONARY

A wedgelet is a square, dyadic block of pixels containing a picture of a single straight edge. Wedgelet is parameterized by five numbers:

d : edge locationθ : edge orientationm1, m2 : shading N: block size

wedgelet dictionary is the dyadically organized collection of all possible wedgelets.A compression scheme based on the wedgelet representation requires a model which captures the dependency among neighboring wedgelet fits; this can be referred as “geometric modeling”.

Wedgelet DecompositionWedgelet Decomposition

Approximate edge contours by partitioning dyadic blocks Approximate edge contours by partitioning dyadic blocks along linesalong lines

d q1

q2

• Project image onto wedgelet at orientation (q1,q2)– linear edge projection close to image

WEDGELET ESTIMATION

Requires a technique for estimating wedgelet parameters which fit the pixelized data. A standard criterion, is to seek the set of parameters which minimize the distance l2 from the wedgelet approximation to an N*N block of pixel data.The set of possible wedgelets forms a nonlinear four dimensional subspace Finding the best wedgelet fit reduces to projecting the data onto this subspace. Accurate estimates may be obtained through an analysis of the block’s Radon transform. By restricting the wedgelet dictionary to a carefully chosen discrete set of orientations and locations, the inner products of all wedgelets may be quickly computed.

COMPRESSION VIA EDGE CARTOON

Two stage schemeTwo stage scheme– The image = {edge cartoon} + {texture}– f(x,y) = c(x,y) + t(x,y)

The edge cartoon contains the dominant edges of the image

Two-stage scheme produces compressed images with clean, sharp edges at low bitrates.

ESTIMATION AND COMPRESSION OF THE EDGE CARTOON

Wedgelet decomposition offers a piecewise-linear approximation to a contour.Resulting image resembles a “cartoon sketch” It contains approximations of the image’s dominant edges, and spaces between the edges are filled with constant values.The sizes of wedgelet blocks should be chosen intelligentlyBegin with a full dyadic tree of wedgelets.Each node n of the tree is associated with the wedgelet parameters which give the best l2 fit to the data in the corresponding image block.

WEDGELET QUADTREEWEDGELET QUADTREE

Wedgelets live on leaves of a Wedgelets live on leaves of a quadtreequadtree

– deep where curvature is highdeep where curvature is high

– shallow where curvature is lowshallow where curvature is low

E , C : leaf nodesE , C : leaf nodes

I : interior nodesI : interior nodes

We code the pruned wedgelet tree using a top-down predictive framework.

parameters of each node are transmitted to the decoder

Transmits all information in a single pass

ESTIMATION AND COMPRESSION OF THE EDGE CARTOON…

Three types of information must be sent: (1) a symbol from {E, I, C} (2) edge parameters (d, θ) (3) grayscale values (m) or (m1, m2)

For a given node, we predict its edge parameters and grayscale values based on the previously coded parameters of its parentWe make the prediction based on a simple spatial ntuition: The parent’s wedgelet is divided dyadically to predict the wedgelets of its four childrenAfter coding the pruned wedgelet tree, we translate it into the cartoon sketch

MULTISCALE WEDGELET PREDICION

a) Parent wedgelet b) Predicted hildren

IMPROVING THE COMPRESSION SCHEME

The wedgelet-based cartoon compression scheme, can be combined with the tapered masking scheme for wavelet compression of the residual image.geometric modeling to attain improvements in visual quality and PSNRThe wedgelet decomposition in Stage I has been optimized only locally. The consideration of placing wedgelets is made without knowledge of any residual compression scheme to follow. The resulting wedgelet placements often create residual artifacts. Wedgelets should be placed only when they actually improve the overall rate-distortion performance of the coderAchieving global rate-distortion optimality requires sharing information between the geometry-based coder and the residual coder.

W-SFQ: Geometric Modeling with Rate-Distortion

OptimizationGeometric modeling and compression of edge contours must be very effective.

A natural image coder should wisely apply its geometric techniques in a rate-distortion sense

Here introduces a method which uses a simple wedgelet-based geometric representation

Wedgelets are used only when they actually increase the final rate-distortion performance of the coder.

THE SFQ COMPRESSION FRAMEWORK

SFQ FUNDAMENTALS

zerotree quantization frameworkThe dyadic quadtree of wavelet coefficients is transmitted in a single pass from the top down, and each directional subband is treated independently. Each node includes a binary map symbol. A symbol indicates a zerotree: descendants are quantized to zero. A symbol indicates that the node’s four children are significant: their quantization bins are coded with an additional map symbol Thus, the quantization scheme for a given wavelet coefficient is actually specified by the map symbol of its parent Themap symbol transmitted at a given node refers only to the quantization of wavelet coefficients descending from that node. All significant wavelet coefficients are quantized uniformly by a common scalar quantizer; The quantization stepsize is optimized for the target bitrate.

SFQ FUNDAMENTALS……

A tree-pruning operation optimizes the placement of zerotree symbols by weighing the rate and distortion costsBottom-upIt is assumed that all coefficients are significant, and decisions must be made regarding whether to group them into zerotrees.convergence is guaranteed because the number of zerotrees can only increase in each iteration.At the beginning of each iteration, the coder estimates the probability density of the collection of significant coefficients;This yields an estimate of the entropy of each quantized coefficient.Adaptive arithmetic coding is used for transmission of these quantization bin indices. Wi be the wavelet coefficient at node ni , andŴi denote the coefficient quantized by stepsize q . Ci denote the set of the four children of node ni Ui denote the subtree of descendants of node

PHASE1PHASE1Those nodes currently labeled significant are examinedThe coder has two options at each such node:

create a zerotree (symbol 0) or maintain the significance (symbol 1) .

Each option requires a certain number of bits and results in a certain distortion relative to the true wavelet coefficients. The first option, zerotree quantization of the subtree beginning with node ni , Ri(0) = 0 requires bitsThis option results in distortionDi (0) = Σ Wj

2 The second option - send a significance symbol for ni , as well as the quantization bins corresponding to Ŵj rate and distortion costs of nodes in Ci

Ri (1) = Σ -log2[P(Ŵj)] + Σ Rj

Di (1) = Σ (Wj - Ŵj)] + Σ Dj

Lagrangian cost Ji = Di + λRi

PHASE II

The tree-pruning is adjusted to account for the costs of transmitting map symbolsMap symbols are predicted based on the variance of local, causal quantized wavelet coefficients. High variances indicate a likelihood of a significant symbol.Low variances indicate a likelihood of a zerotree symbol.rate-distortion performance of a node may be improved by switching its symbol if the gain in map symbol rate exceeds the loss in “Phase I” ratedistortion efficiency.In Phase II, the map symbol at node is switched from y to x if λ Δ R i, map > Δ J i, data

TRANSMITTING WEDGELETSwe use a rate-distortion coding framework.Wedgelet parameters (d, θ, h) is quantized separately; The quantization stepsizes are chosen to ensure the correct operating point on the R/D curveInfluence of each parameter’s distortion on the squared-error image distortion must be estimatedThe height parameter h is coded first.For large values of h , errors in transmitting d_ and θ_ will create significant distortion in the coded wedgelet blockAdaptive arithmetic coding is used to transmit the indices of the smoothness parameters.Once coded for a node ni , a wedgelet may be used to predict the wavelet coefficients at all descendants. One way to obtain a prediction for these coefficients is to– create an image containing the coded wedgelet at the appropriate

location,– take the wavelet transform, and – extract the appropriate coefficients.

CONCLUSIONImage quality degrades because of the artifacts resulting from the block-based DCT schemewavelet-based coders facilitate progressive transmission of images.Edges are not efficiently described by wavelets. Geometric modeling captures the inherent simplicity of these edges.Wedgelets is such a scheme which exploits the simple geometric structure of pixels near edges, and which allows wavelets to efficiently represent those regions away from edgeswedgelets can be used to improve visual quality and increase PSNR.

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