Performance Comparison of Hartley Transform with Hartley Wavelet and Hybrid Hartley Wavelet Transforms for Image Data Compression

This paper proposes image compression using Hybrid Hartley wavelet transform. The paper compares the results of Hybrid Hartley wavelet transform with that of orthogonal Hartley transform and Hartley Wavelet Transform. Hartley wavelet is generated from Hartley transform and Hybrid Hartley wavelet is generated from Hartley transform combined with other orthogonal transform which contributes to local features of an image. RMSE values are calculated by varying local component transform in hybrid Hartley wavelet transform and changing the size of it. Sizes of local component transform is varied as N=8, 16, 32, 64. Experiments are performed on twenty sample color images of size 256x256x3. Performance of Hartley Transform, Hartley Wavelet transform and Hybrid Hartley wavelet Transform is compared in terms of compression ratio and bit rate. Performance of Hartley wavelet is 35 to 37% better than that of Hartley transform whereas performance of hybrid Hartley wavelet is still improved than Hartley wavelet transform by 15 to 20%. Hartley-DCT pair gives best results among all Hybrid Hartley Transforms. Using hybrid wavelet maximum compression ratio up to 32 is obtained with acceptable quality of reconstructed image


INTRODUCTION
Transmission of uncompressed multimedia data like audio, video and still images requires large amount of memory an d more time. Though there are advancements in processor speed, communication technologies and storage density in terms of hardware, there is a dire need of increasing data storage capacity and transmission bandwidth as todays communication is very much data intense. Compression of data provides an efficient way to increase storage capacity and reduce transmission ti me. The idea behind the compression is to reduce the number of bits required to represent the image. Compression is classified in two ways: lossless compression and lossy compression. In lossless compression reconstructed image is identical to original image. Hence it is used in medical image compression, text data compression etc. Methods for lossless image compression include Huffman coding, Run leng th coding, Bit plane coding etc. In lossy compression reconstructed image is not exactly same as original image. Hence it is useful in applications like video conferencing, and facsimile transmission, in which certain amount of error is acceptable to get increased compression ratio [1].
Mainly transform based coding is used for lossy image compression. It is based on the fact that pixels in an image exhibit a certain level of correlation with their neighboring pixels. Transformation is a process of mapping these correlated (spatial domain) coefficients to uncorrelated (frequency domain) coefficients [2].JPEG image compression is most widely used technique for lossy image compres sion. This technique is DCT oriented. DCT separates the image into set of different frequencies. From these frequencies high frequency elements are eliminated as they represent less important contents of the image [3].Transforms other than DCT has also been us ed for image compression. In [4] DST is used to compress the image. It gives acceptable values of RMSE but they are slightly higher than one obtained by DCT. Also non sinusoidal transforms like Haar, Walsh, Slant and Kekre transform have been experimented in [4]. Walsh and Haar give advantage of less computation but error in reconstructed image is quite higher than DCT. Generally to apply DCT image is divided into blocks. But it eliminates correlation across the boundaries and hence results in blocking artifacts. This drawback can be avoided by using wavelet transforms. Larger the block, more efficient is the coding. But it requires more computational power. When smaller block size is considered, lesser image distortion is observed at the cost of coding efficiency. Hence normally 8x8 or 16x16 size blocks are used.
Wavelets provide solution to this problem. Hence in recent years researchers have focused on wavelet transforms. Basic concept of wavelet is to select an appropriate wavelet function called "Mother Wavelet" and then perform an analysis using shifted and d ilated version of mother wave. Wavelet transform gives time-frequency analysis of signal. [5].It has high energy compaction capability [6]. Wavelet transform is applied on entire image rather than a block of an image. It allows a uniform distribution of compression error across entire image. Wavelet transform provides better image quality as compared to DCT at higher compression ratio [7].

RELATED WORK
Initially Haar wavelets were emphasized. But in recent years wavelet transforms of Walsh [8], Slant [9], and Kekre transforms [9] have also been studied and implemented. Wavelet transform of these respective transforms is generated using the procedure mentioned in [10]. Image compression using biorthogonal wavelet transform is proposed by Liu in [11]. A lifting scheme wavelet based transform with a modified entropy coding algorithm is proposed in [12]. It discusses effect of block sub-band coding on compression factor and quality of an image. Multi wavelet transform based on zero tree coefficient shuffling has been proposed in [13]. A simpler method of generating wavelet transform is presented in [10]. Wavelets of Haar, Cosine, Walsh, Sine, Slant are generated using the method proposed in [10,14] and Column, Row and Full Wavelet transform is used in image compression [15]. If image quality of reconstructed image is negotiable then Column wavelet Transform is used instead of Full Wavelet transforms to save number of computations. Also image compression using Real Fourier transform and its wavelet transform is proposed in [16]. Results showed that wavelets performed better than respective orthogonal transform.

PROPOSED TECHNIQUE
This paper proposes a hybrid Hartley Wavelet transform method for image compression. Results obtained by applying Hybrid Hartley Wavelet transform are compared with the results of Orthogonal transform applied on image. Wavelet transform of respective orthogonal transform is generated using equation

Im
Bn(n) Febru a r y 1 8 , 2 0 1 4 Where "A" is of size MxM and "B" is of size NxN and A and B are same orthogonal component matrices. Wavelet transform TAB generated will be of size MNxMN. Size of "A" and "B" is selected and varied as power of two and it generates TAB of size MNxMN. Hybrid wavelet transform is generated using the same kronecker product mentioned above but as it is a "hybrid" wavelet, component transforms "A" and "B" are different.

EXPERIMENTS AND RESULTS
Twenty different color images are selected for experimental purpose. Each image is of size 256x256x3. Experiments are performed using Matlab 7.0 on AMD dual core processor with 4 GB RAM. Images chosen are shown in Figure 1.   Avg.