EFFICIENT IMAGE COMPRESSION USING LAPLACIAN PYRAMIDAL FILTERS FOR EDGE IMAGES

International journal of Computer Networking and Communication (IJCNAC)Vol. 1, No. 2(November 2013) 13

www.arpublication.org

EFFICIENT IMAGE COMPRESSION USING

LAPLACIAN PYRAMIDAL FILTERS FOR

EDGE IMAGES

V.Karthikeyan1

and V.J.Vijayalakshmi2

1Department of ECE, SVS College of Engineering, Coimbatore, India

[email protected] 2Department of EEE, Sri Krishna College of Engg & Tech., Coimbatore, India

[email protected]

Abstract

This project presents a new image compression technique for the coding of retinal and

fingerprint images. Retinal images are used to detect diseases like diabetes or

hypertension. Fingerprint images are used for the security purpose. In this work, the

contourlet transform of the retinal and fingerprint image is taken first. The coefficients of

the contourlet transform are quantized using adaptive multistage vector quantization

scheme. The number of code vectors in the adaptive vector quantization scheme depends

on the dynamic range of the input image.

Keywords: Counterlet Transform, adaptive multistage vector quantization,

hypertension, Fingerprint image

1. INTRODUCTION

The retina is the only location where blood vessels can be visualized non-invasively in vivo.

The evaluation of retinal images is a diagnostic tool widely used to gather information about

patient retinopathy. Diabetic retinopathy is the leading cause of blindness in the adult population.

Mass screening efforts are necessary to detect diabetic retinopathy. Traditionally the retina has

been observed either directly through an ophthalmoscope or fundus camera. But high resolution

retinal images are necessary to in order to identify features such as exudates and micro

aneurysms. High resolution image in turn occupy large storage space. In order to store thousands

of records of patients, effective retinal image compression is necessary. The Contour let

transform has better performance in representing edges than wavelets for its anisotropy and

directionality; hence it is well suited for multi-scale image representation. The Contourlet

coefficients in the corresponding sub bands are quantized using adaptive multi-stage vector

quantization. Vector quantization (VQ) has been proven a powerful compression scheme for

coding of images and image sequences. The applicability of VQ is, however, limited by an

exponential growth of the computational complexity with the vector dimension. For this reason,

low dimensionality VQ are typically used in image compression, but such VQ limit the coding

efficiency and tend to yield highly visible block boundaries in low bit rate applications.

Multistage Vector Quantization (MSVQ) is a structured VQ scheme in which the search time and

codebook complexity reduction with respect to optimal VQ is obtainable. Compression is

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performed by a program that uses a formula or algorithm to determine how to compress or

decompress data. Graphic image file formats are usually designed to compress information as

much as possible (since these can tend to become very large files). Graphic image compression

can be either lossy (some information is permanently lost) or lossless (all information can be

restored).

Compressing an image is significantly different than compressing raw binary data. This also

means that lossy compression techniques can be used in this area. The present study is envisaged

to improve the image coding method by using the contourlet transform. The Contourlet transform

has better performance in representing edges than wavelets for its anisotropy and directionality;

hence it is well suited for multi-scale image representation. The Contourlet coefficient has less

computational complexity with the vector dimension. For this reason, low dimensionality VQ is

typically used in image compression, but such the corresponding sub-bands are quantized using

adaptive multi-stage vector quantization. Vector quantization (VQ) has been proven a powerful

compression scheme for coding of images and image sequences. Multistage Vector Quantization

(MSVQ) is a structured VQ scheme in which the search time and codebook complexity reduction

with respect to optimal VQ is obtainable.

1.1. Error Metrics

Two of the error metrics used to compare the various image compression techniques are the Mean

Square Error (MSE) and the Peak Signal to Noise Ratio (PSNR). The MSE is the cumulative

squared error between the compressed and the original image, whereas PSNR is a measure of the

peak error. The mathematical formulae for the two are

MSE = (1)

PSNR = 20 * log10 (255 / sqrt (MSE)) (2)

Where I(x, y) is the original image, I'(x, y) is the approximated version (which is actually the

decompressed image) and M, N are the dimensions of the images. A lower value for MSE means

lesser error, and as seen from the inverse relation between the MSE and PSNR, this translates to a

high value of PSNR. Logically, a higher value of PSNR is good because it means that the ratio of

Signal to Noise is higher. Here, the 'signal' is the original image, and the 'noise' is the error in

reconstruction. So, if you find a compression scheme having a lower MSE (and a high PSNR),

you can recognize that it is a better one.

2. PROPOSED COMPRESSION METHOD

In this project, we propose an image compression scheme using successive approximation

quantization of vectors of the contourlet transformed image for edge images like fundus image,

retinal image. The Contourlet Transform is a directional transform, which is capable of capturing

contours and fine details in images. The pyramidal directional filter bank (PDFB), was proposed

by MinhDo and Vetterli, which overcomes the block-based approach of curvelet transform by a



directional filter bank, applied on the whole scale also known as contour let transform (CT). In

essence, first a wavelet-like transform is used for edge (points) detection, and then a local

directional transform for contour segments detection. With this insight, one can construct a

double filter bank structure in which at first the Laplacian pyramid (LP) is used to capture the

point discontinuities, and followed by a directional filter bank (DFB) to link point discontinuities

into linear structures. The overall result is an image expansion with basis images as contour

segments, and thus it is named the contourlet transform. The combination of this double filter

bank is named pyramidal directional filter bank (PDFB).

The flow graph of the contourlet transform is depicted in below Fig. The LP decomposition at

each level generates a down sampled low pass version of the original and the difference between

the original and the prediction, resulting in a band pass image. Band pass images from the LP are

fed into a DFB so that directional information can be captured. The DFB is efficiently

implemented through an I-level binary tree decomposition that leads to 2/ sub bands with wedge-

shaped frequency partitioning as illustrated below. The cascading of LP and DFB results in

Pyramidal Directional Filter Bank (PDFB). In this contour let scheme, each generation doubles

the spatial resolution as well as the angular resolution. The PDFB provides a frame expansion for

images with frame elements like contour segments, and thus is also called the contourlet

transform.

2.1. Laplacian Pyramid

One way of achieving a multiscale decomposition is to use a Laplacian pyramid (LP),

introduced by Burt and Adelson. The LP decomposition at each level generates a down sampled

lowpass version of the original and the difference between the original and the prediction,

resulting in a bandpass image as shown in Fig. In this figure, ‘H’ and ‘G’ are called analysis and

synthesis filters and ‘M’ is the sampling matrix. The process can be iterated on the coarse version.

In Fig. the outputs are a coarse approximation ‘a’ and a difference ‘b’ between the original signal

and the prediction. The original image is convolved with a Gaussian kernel. The resulting image

is a low pass filtered version of the original image.

Fig.1. Laplacian pyramid scheme (a) analysis, and (b) reconstruction.

Thus the Laplacian pyramid is a set of band pass filters. By repeating these steps several times a

sequence of images, are obtained. If these images are stacked one above another, the result is a

tapering pyramid data structure, as shown in Fig. and hence the name.

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2.2. Directional Filter Bank

In 1992, Bamberger and Smith introduced a 2-D directional filter bank (DFB) that can be

maximally decimated while achieving perfect reconstruction. The directional filter bank is a

critically sampled filter bank that can decompose images into any power of two’s number of

directions. The DFB is efficiently implemented via an l-level tree structured decomposition that

leads to ‘2l’ sub bands with wedge-shaped frequency partition as shown in Fig. The original

construction of the DFB involves modulating the input signal and using diamond shaped filters.

Therefore, the LP permits further sub band decomposition to be applied on its band pass images.

Those band pass images can be fed into a DFB so that directional information can be captured

efficiently. The scheme can be iterated repeatedly on the coarse image. The end result is a double

iterated filter bank structure, named pyramidal directional filter bank (PDFB), which decomposes

images into directional sub bands at multiple scales. The scheme is flexible since it allows for a

different number of directions at each scale.

Fig.2. Laplacian pyramid structure.

3. MULTISTAGE VECTOR QUANTIZATION

In vector quantization, an input vector of signal samples is quantized by selecting the best

matching representation from a codebook of ‘2kr’ stored code vectors of dimension k. VQ is an

optimal coding technique in the sense that all other methods of coding a random vector in ‘k’

dimensions with a specific number b=kr of bits are equivalent to special cases of VQ with

generally suboptimal codebooks.



Fig.3. Multistage Vector Quantization Scheme

The resulting encoding and storage complexity, of the order of 2kr, may be prohibitive

for many applications. A structured VQ scheme which can achieve very low encoding

and storage complexity is multistage VQ (MSVQ). In MSVQ, the kr bits are divided

between L stages with bi bits for stage ‘i’. The storage complexity of MSVQ is

as vectors, which can be much less than the complexity of vectors for

unstructured VQ. MSVQ is a sequential quantization operation where each stage

quantizes the residual of the previous stage.

The structure of MSVQ encoder consists of a cascade of VQ stages as shown in Fig. 9.

For an Lstage MSVQ, an l th –stage quantizer l Q, l =0, 1, 2…L −1 is associated with a

stage codebook l C contains l K stage code vectors. The set of stage quantizes

are equivalent to a single quantizerQ, which is referred to as the

direct sum vector quantizer.

4. ADAPTIVE VECTOR QUANTIZATION

In this paper, we have proposed an adaptive vector quantizer which is based on the

dynamic range of the input image. A vector quantizer is defined as a mapping 'Q' of 'L'

dimensional Euclidean space into a finite subset Y which is given by Q: RL=Y

Where 'V' is the set of reproduction vectors, which is generally termed as VQ codebook

Y = (xi; i =1, 2, N). Here x represents the code vectors and 'N' is the number of vectors in

the code book 'V'. The first step in the proposed adaptive vector quantization scheme is to

split the image into a set of 'L' dimensions. Two important factors which decide the size

of the code book are rate (R) and dimension (L). The number of code vectors in the code

book is 2RL

• In our algorithm, the dynamic range of the input image is determined first. The dynamic

range is given by

Dynamic range = Maximum gray level value – Minimum gray level value.

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This dynamic range is used to fix the interval of the vector space to be partitioned. The

interval is calculated as

In our approach, the general expression of code vector for L=2 is given by

(3)

Where i, j are integers (i, j = 0, 1, 2.). Successive vectors are formed by adding the interval in

both dimensions to the initial code vector. After forming the code vectors, the next step is to

project the input vector into the code vector. Each input vector is matched with a closest

codeword in the codebook, and then the index of the codeword is transmitted instead of the code

vector itself the matching of the input vector to the nearest code vector is based on minimum

distortion. A distortion measured is used to assign the cost of reproducing any vector as

reproduction. A quantizer is optimal if it minimizes the average distortion. The squared error

distortion is

(4)

The multistage vector quantization scheme is shown in Fig. 4. In the figure, 'X' represents

the input vector, LUT stands for Look up table, i1, i2 represents the indices from different

stages.

Fig.4.Block Diagram of the Proposed System



5. PROPOSED ALGORITHM

The proposed retinal image coder scheme is summarized below

• The correlation present in the input retinal image is minimized by taking the

contourlet transform of the input retinal image. Different pyramidal and

directional filters are taken into consideration in this work.

• The transform coefficients obtained in step 1 are adaptively vector quantized in a

multistage manner where the residual coefficients due to quantization are

iteratively feedback and vector quantized. The number of code vectors depends on

the dynamic range of the input image; hence it is adaptive in nature.

• The output indices from step 2 are lossless coded using static Huffman code. In

decoding, the decoder basically performs the reverse process of the above steps.

The different pyramidal filters taken into consideration are 'Haar', 'biorthogonal

9/7'. The different directional filters taken into consideration are 'pkva' and

'biorthogonal 9/7'. In the case of wavelets, the different wavelet filters chosen are

'Haar', 'biorthogonal 9/7' and 'la8'. Table 1 shows the performance of the proposed

algorithm for the first, second and third level of decomposition by varying the bits per

dimension (bpd) from 0.125 to 1.0. From the table it is obvious that in the case of contour

lets, as the level of decomposition increases, the Peak Signal to Noise Ratio (PSNR)

value slightly increases

6. SIMULATION OUTPUTS

In this project work we proposed a new retinal and fingerprint image compression

scheme using successive approximation quantization of vectors of the contourlet

transformed image. In this a new image compression scheme using successive

approximation quantization of the image using Laplacian filter with noise variance of

about γ = 0.01 and we have to prove that we preserve the meta information like edges.

So we also take edge images as a data base. Here we may achieve even upto 1:80

compression ratio with PSNR around 30 db.

(a) (b)

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(c)

Fig.5. (a) Input image (b).Contour Coefficient Level 1 (c) the compressed image using the

contourlet transforms

6. CONCLUSION

From the experimental results, it can be observed that for the same bits per dimension, the PSNR

obtained using contourlet transform is better than that of the wavelet transform. Hence, a better

image reconstruction is possible with less number of bits using contourlet transform. In retinal

images, the optic nerves are having higher amount of contoured patterns hence contourlet

performs superior than wavelet. Higher PSNR results can be obtained by including more number

of stages in multi-stage vector quantization, which will result in increased computational

complexity.

REFERENCES

[1] M.N. Do and M. Vetterli, "The contourlet transform: an efficient directional multiresolution image

representation," IEEE Trans. of Image Processing, vol. 14, no. 12, pp. 2091-2106, December

2004.

[2] A. Gersho and R. Gray, Vector Quantization and Signal Compression, Kluwer Academic

Publishers, M.A., 1995.

[3] R.M. Gray, "Vector Quantization," IEEE ASSP magazine, vol. 1, no. 2, pp. 4 - 29, Apr. 1984.

[4] M.N. Do, Directional multiresolution image representations, Ph.D. thesis, EPFL, Lausanne,

Switzerland, Dec. 2001.

[5] P. J. Burt and E.H. Adelson, "The Laplacian pyramid as a compact image code," IEEE Trans.

Commun, vol. 31, pp. 532-540, Apr. 1983.

[6] R. H. Bamberger and M. 1. T Smith, "A filter bank for the directional decomposition of images:

Theory and design," IEEE Trans. Signal Proc. vol. 40, pp. 882-893, Apr. 1992.

[7] R.C.Gonzalez, R.E.Woods, Digital Image Processing, New Delhi, Pearson Education Pvt. Ltd.,

2nd Edition.



[8] Khalid Sayood, Introduction to Data Compression, New Delhi, Harcourt India Private Limited,

2nd edition, 2000.

[9] Vivien Chappelier, Christine de Beaulieu, “Image Coding with Iterated Contourlet and Wavelet

Transforms”, IEEE International Conference on Image Processing (ICIP 2004),

Vol. 5, pp. 3157-3160, Oct. 2004.

[10] Ramin Eslami, Hayder Radha, “Wavelet-Based Contourlet Transform and its Application to

Image Coding”, IEEE International Conference on Image Processing (ICIP 2004), Vol. 5, pp.

3189-3192, Oct. 2004.

Authors

Prof.V.Karthikeyan has received his Bachelor’s Degree in Electronics and

Communication Engineering from PGP college of Engineering and Technology in

2003, Namakkal, India, He received Masters Degree in Applied Electronics from KSR

college of Technology, Erode in 2006 He is currently working as Assistant Professor in

SVS College of Engineering and Technology, Coimbatore. He has about 8 years of

Teaching Experience

Prof.V.J.Vijayalakshmi has completed her Bachelor’s Degree in Electrical &

Electronics Engineering from Sri Ramakrishna Engineering College, Coimbatore,

India. She finished her Masters Degree in Power Systems Engineering from Anna

University of Technology, Coimbatore, She is currently working as Assistant Professor

in Sri Krishna College of Engineering and Technology, Coimbatore She has about 5

years of teaching Experience.