2004 IEEE 11th Digital Signal Processing Workshop & IEEE Signal Processing Education Workshop

Real Transform Domain Operations for Image Compression and Segmentation

J. Layhue and R. Sundaram Gannon University - ECE Department

109 University Square Erie, PA 16541

Abstract - This paper describes a real transform that can be used for various image applications such as compression and edge detection. The transform is based on the symmetric extension of a two dimensional block of data and is similar to the DCT. This symmetric extension is shown to be effective for image compression and for detecting transitions in images. Future work on this subject is also discussed.

I. INTRODUCTION

Although Wavelets are becoming the standard in image processing, transform coding is efficient and effective for various imaging applications. Real transforms like the OCT have proven to be effective in the area of compression while maintaining simplicity in terms of implementation [1][2][4][5]. The OCT is simply a real extension of the OFT and has been one of the prominent choices for transform coding. In this paper another transform that leads to real transform coefficients is discussed, namely the Discrete Symmetric Cosine Transform (DSCT) [3]. This paper will show that the DSCT is easy to implement and can provide high quality results when used in certain image processing applications.

II. PROPOSED DSCT ALGORITHM

The proposed DSCT algorithm is created by symmetrically extending an image data matrix and is intended for block processing like that used by the JPEG standard [4). To symmetrically extend the data consider an M x M block of data from an N x N image where N » M. The original data in the M x M matrix like that shown in Fig. la is extended as shown in Fig. l b.

This idea of creating even symmetric matrices to provide teal Fourier coefficients is not uncommon. In general, however, zero extension is used to create the even symmetry [3). Zero extension makes the analysis of the extended data block relatively simple. For our analysis we used data extension. That is, we used copies of the data found in the last row and column of the original M x M matrix and therefore did not create the discontinuities that traditional zero insertion does.

0-7803-8434-2/04/$20.0002004 IEEE 139

2M

c c MxM

a c Block 1 Block 2

MxM 2M b

b d b d d

(a) Block 3

(b)

Fig . 1. (a) Original sub block of data and (b) expansion of original matrix into form used for DSCT.

To consider this process mathematically, the extended data matrix is analyzed in a piecewise fashion for Simplicity. The separate pieces include: the four M x M data blocks, the row of extended values and the column of extended values. The OFT of blocks 1 and 4 from figure Ib yields

N-J N-I X(kl, k2) = L Lx(nl'n2)W2N

",k,+n,k, + n,�O",�o (1) 2N-I 2N-I

L I x(2N - "I ,2N � "2 )W2N ",kl +n,k, n,=N+lrr2=N+l

where

W Ilk � -)�N [ .2Ir )""

2N � e (2)

using the standard form of the DFT [J]. From equation 1, one can see that block 4 is simply a copy of block 1 that has been mirrored both horizontally and vertically. Next, any values with a spatial index n equal to zero were then pulled out of the summation to get

(a) (b)

(c) (d)

Fig. 2. - (a) Lena, (b) medical image 1, (c) medical image 2, (d) astronomical image

}II-I X(kl,k�) = x(O,O)+ Lx(nl,O)W2JI/"" +

}II-I Lx(O,nZ)W2}11n,k, +

. (3)

A similar result corresponds to blocks 2 and 3 with merely a sign change in the exponent of the single summations. This fact leads to more cosine tenns and an entirely real calculation.

Similarly, for the horizontally extended row of data we get

N-I

X(kpk2) = Lx(N,n2)W2N

Nk!+n2kl + n'2=O 2N-J L x(N,2N - nZ)W2N Nk,t.,., +.

(4) /'I2�N+1

A similar expression is found for the vertically extended column of data. Combining all of these results we get an entirely real equation

Table 1 - Compression results using the DCT and the DSCT

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Image Name

M n h Compression Ratios (:1)

Low-nolse Noisy Images

Imaaes

OCT DSCT OCT DSCT

Lena e 5 30 7.43 9.06 3.11 4.40

Medical 8 5 30 14.55 16.65 Ima091 5.68 8.92

Medical 8 5 30 15.31 16.40

Image 2 5.57 10.76

Hubble 8 5 30 19.23 20.04

Imaae 1 4.87 8.53

X(IG,k2) = ± f a(�,n2)X(�,n2)CoS(!!..-�IG)cos(!!..-n2k2) .,.{I" =0 N N

where alpha is simply a mask of the form

1 2 ... 2 1 2 4 ." 4 2

2 4 4 2 1 2 ." 2 1

(5)

Equation 5 will be referred to as the DSCT using data insertion or DSCT for short. The DSCT is comparable to the DCT in terms of computational complexity given this entirely real form [4]. The DSCT can also be efficiently implemented with only slight increases in computational load from the DCT. This slight increase in computational complexity is a tradeoff for better results in given applications.· The following sections discuss the use of the DSCT algorithm for image compression and transition detection.

III. LOSSY COMPRESSION OF NOISY IMAGES USING THE DSCT

The block-based image compression process using the DSCT is very similar to that used by the JPEG standard where the DCT is replaced by the DSCT [5]. This scheme was used to compare the effectiveness of using the DSCT vs. the DCT for image compression. The commonly used Lena image was evaluated as well as medical and astronomical images. These images are seen in figures 2(a)-2(d). After evaluating the original images zero mean Gaussian noise with a standard deviation of 0.04 was added to the images and they were processed again. The standard deviation was chosen based on the visual affects of adding noise with varying standard deviations.

Fig. 3. Transition detection using DSCT

(a) (b) Fig. 4. (a) Lena and (b) I dimensional verification of

effectiveness of DSCT for transition detect

The results in table 1 show that the DSCT and DCT provide comparable compression results for low noise images of size M x M where n is number of quantization bits used and k is the number of Fourier coefficients that were kept. This idea of keeping only a given number of coefficients and setting the others to zero is commonly done when using the zigzag

scanning method of ordering transform coefficients [4]. Table 1 also shows that the DSCT leads to exceptional results in noisy environments.

The DSCT therefore provides an easy to implement algorithm for compressing images in noisy environments. The DSCT using data insertion therefore lends itself well to medical and astronomical images that are generally noisy because of the recording techniques used to obtain these types of images.

It should be noted that Wavelets should also be considered in terms of their effectiveness. However, the computational complexity of implementing wavelets is greater than the complexity needed to implement either the DCI or the DSCI [6]. At this time we are evaluating systems that provide high computational efficiency and therefore have not yet considered Wavelets. The comparison of our system to Wavelets will be completed as our work progresses.

IV. TRANSITION DETECTION USING DSCT

Transition detection can also be achieved with the DSCT. The block diagram in Fig. 3 shows the basic transition detection scheme. The transition detection is done in the frequency domain using a basic bandpass fIlter.

We are currently working on making the bandpass filtering process adaptive. Spatial statistics such as the variance and histogram of a given M x M block will be used to determine if there are any ''possible'' transitions in the current block. If the possibility of transitions does not exist the block will not be processed and no transition information will be stored thus decreasing the computations needed.

If there is a possibility that transitions occur in a given block it will be processed following the scheme in Fig. 3. After processing the block the system will recursively analyze the spatial domain transition map to determine if the optimum frequency domain filter was used. If the filter was determined

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to be suboptimal, the bandpass filter will be adjusted and the current block of data processed again.

Fig. 4 shows the result of processing a single row of image data without using this adaptive method. The commonly used Lena image in Fig. 4a shows a horizontal white line where the data was extracted. The plot in Fig. 4b shows the scaled image intensity values and the result of basic transition detection using the DCT and the DSCT plotted versus the discrete spatial index n. The zero crossing on the plot shows the detection ofa transition. The results of using both types are nearly identical showing that the DSCT can produce results comparable to the DCT.

Our goal is to create an adaptive scheme that is more reliable and robust than current DCI methods. Namely, our system will react better in noisy environments. Not much work has been done in this are, but the initial results are promising.

V. CONCLUSIONS AND FUTURE WORK

The results shown in this paper prove that the DSCT provides comparable or greater compression results in the area of image data compression when compared to the commonly used DCT. Namely, the DSCT provides exceptional results in the area of data compression in noisy environments. When compared the DCT, the DSCT provides a compression ratio up to twice that of the DCI.

For image data compression, our current scheme can be used to process 2D grayscale images. It is our desire to extend this system to process both color images and 3D data such as video. Producing a computationally efficient and effective compression algorithm In the area of transition detection, this paper shows that the DSCT can provide results comparable to those of the DCT with similar computational complexity.

The transition detection process is being made adaptive as discussed in section IV. The transition detection system will also be extended to work with color images and eventually 3D data.

The future work being done with the DSCT will focus on creating an extremely robust system that produces high quality results in both compression and transition detection and possibly for object recognition especially focused toward noisy envirorunents.

REFERENCES

[I ] R.C. Gonzalez, and R.E. Woods, Digital Image CompresSion, Second Edition, Prentice Hall, New Jersey, 2002.

[2] B.G. Haskell, and AN. Representation, Compression, Publishing, New York, 1997.

Netravali, Digital Pictures: and Standards, Perseus

[3] R. Sundaram, "Algorithms for adaptive transfonn edge detection," IEEE Trans. On Signal Processing, Vol. 47, No.8, pp. 1-10, Aug. 1999.

[4] P. Symes, Digital Video Compression, McGraw-Hill, New York, 2004.

[5] A. Skodras, C. Christopoulos, T. Ebrahimi, "The JPEG 2000 Still Image Compression Standard," IEEE Signal Processing Magazine, Vol. 18, Issue 5, pp. 36-58, Sept. 2001.

[6] B. Jerabek, P. Schneider, A. Ubi. "Comparison of lossy image compression methods applied to photorealistic and graphical images using public domain sources," Technical Report RIST ++ 15/98, Research Institute for Software technology, University of Salzburg, 1998.

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