1. Effective of Some Mathematical Functions to Image Compression

2. Abstract
In This Study we Introduce the Effective of some Mathematical Transformations such as Fourier Transformation and its variants, as well as Wavelet Transformations, to Image Compression.
Digital images contain large amount of information that need evolving effective techniques for storing and transmitting the ever increasing volumes of data. Image compression addresses the problem by reducing the amount of data required to represent a digital image.

3. The discrete cosine transform (DCT) and the discrete wavelet transform (DWT) are techniques for converting a signal into elementary frequency components. They are widely used in image compression. These functions illustrate the power of Mathematics in the image compression. In this work, a comparison study between discrete wavelet transform and discrete cosine transform is introduced. Our results show that the discrete wavelet transform gives better performance than the discrete cosine transform in terms of peak signal to noise ratio as a quality measure.

4. Introduction
In the modern digital age, computer storage technology continues at a rapid pace, a means for reducing the storage requirements of an image is still needed in most situations. In images, The common characteristic of most of the images is that, the neighboring pixels are correlated, and image contains redundant information. Therefore the most important task in image compression is to find a less correlated representation of the image. The fundamental component of image compression is reduction of redundancy and irrelevancy. Redundancy reduction aims at removing duplication from image, and irrelevancy reduction omits parts of the signal that will not be noticed by Human Visual System (HVS).

5. There are various methods of compressing still Images, one of these methods is a transform coding is one of the most popular image compression techniques, and use a reversible, linear mathematical transform. For image compression, it is desirable that the selection of transform should reduce the size of resultant data set as compared to source data set. Some mathematical transformations have been invented for the sole purpose of image compression such as, Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), Hadamard-Haar Transform (HHT), Karhune-Loeve Transforms (KLT), Slant-Haar Transform (SHT), Walsh-Hadamard Transform (WHT), and Wavelet Transforms (WT). selection of proper transform is one of the important factors in data compression scheme.

6. DCT converts data (image pixels) into sets of frequencies
DCT converts data (image pixels) into sets of frequencies. DCT-based image compression relies on two techniques to reduce data required to represent the image. The first is quantization of the image’s DCT coefficients; the second is entropy coding of the quantized coefficients. However, Discrete wavelet transformation (DWT) transforms discrete signal from the time domain into time frequency domain. DWT have higher decorrelation and energy compression efficiency, so DWT can provide better image quality on higher compression ratios, and have some properties which makes it better choice for image compression than DCT.

7. Discrete cosine transform (DCT)
The discrete cosine transforms (DCT) is a technique for converting a signal into elementary frequency components. It represents an image as a sum of sinusoids of varying magnitudes and frequencies. The DCT has the property that, for a typical image, most of the visually significant information about the image is concentrated in just a few coefficients of the DCT. For this reason, the DCT is often used in image compression applications.
For compression, the input image is first divided into blocks, and the 2D – DCT is computed for each block. The DCT coefficients are then quantized, coded and transmitted.

8. The 1D – DCT is given by: Where is DCT of The inverse transform
of 1D – DCT is given by:
For

9. The 2D – DCT is a direct extension of the 1 – D case and is given by:
For u,v = 0,1,2,…..,N-1. and α (u) and α (v) are defined as:
To reconstruct the image, receiver decodes the quantized DCT coefficients, computes the inverse 2D – DCT of each block, and then puts the blocks back together into a single image. The inverse transform is defined as:
For x, y = 0,1,2,….,N-1.

10. Discrete Wavelet Transforms (DWT)
This transforms illustrate the power of Mathematics in image compression field. Image compression is one of the most important applications of wavelets. Wavelets are mathematical functions that satisfy certain properties and can be used to transform one function representation into another. Wavelet transform decomposes an image into a set of band limited components which can be reassembled to reconstruct the original image without error. Wavelet transform (WT) represents an image as a sum of wavelet functions (wavelets) with different locations and scales. any decomposition of an image into wavelets involves a pair of waveforms: one to represent the high frequencies corresponding to the detailed parts of an image (wavelet function ψ) and one for the low frequencies or smooth parts of an image (scaling function Ф) .

11. .
Discrete wavelet transforms for two – dimensional can be derived from one – dimensional DWT. The Easiest way for obtaining scaling and wavelet function for two-dimensions is by multiplying two one-dimensional functions. The scaling function for 2D – DWT can be obtained by multiplying two 1 – D scaling function
Wavelet functions for 2D – DWT can be obtained by multiplying two wavelet functions or wavelet and scaling functions for one – dimensional analysis. From that follows that for 3D case there exist three wavelet functions that analysis details in
horizontail
vertical
and diagonal

12. Wavelet compression technique uses the wavelet filters for image decomposition, image is divided into approximation and detail sub image. The filter is applied along the row and then along the columns, the filters divide the input image into four non – overlapping multi-resolution coefficient sets, a lower resolution approximation image ( LL1) as well as horizontal (HL1) , vertical (LH1) and diagonal (HH1) detail components. The sub – band LL1 represents the coarse – scale DWT coefficients while the coefficient sets LH1, HL1 and HH1 represent the fine – scale of DWT coefficients.

13. comparative study of DCT & DWT
JPEG 2000 uses the wavelet transform (WT) to reduce the amount of information contained in a picture, while JPEG systems use the discrete cosine transform (DCT). It is true that the WT requires more processing power than the DCT. The DCT, or any type of Fourier transform, expresses the signal in terms of frequency and amplitude—but only at a single instant in time. The WT transforms a signal into frequency and amplitude over time, and is therefore more efficient. Undesirable blocking artifacts affect the reconstructed images (high compression ratios or very low bit rates. DCT function is fixed can not be adapted to input data. DWT No need to divide the input image into non-overlapping 2-D blocks, it has higher compression ratios avoid blocking artifacts. Disadvantages of DWT the cost of computing DWT as compared to DCT may be higher. The use of larger DWT basis functions or wavelet filters produces blurring and ringing noise near edge regions in images. Longer compression time and Lower quality than JPEG at low compression rates

14. Simulation Results
Simulations were carried out to test the effect of some mathematical transformations such as DCT as a variant of Fourier transformation, as well as wavelets transformation, to image compression. Matlab code was written for the generation of the studied techniques. The test set used is four 512×512 monochromatic images of 8-bit intensity (256 grey levels), Lena512, Baboon512, Barbara512 and Peppers512 (as shown in Figure 1).
As the image content being viewed influences the perception of quality irrespective of technical parameters of the system, test images that have different spatial and frequency characteristics have been selected: Lena512, Baboon512, Barbara512 and Peppers512 (shown in Figure 1).

15. Figure 1: Test images Lena (512×512) Baboon (512×512)
Barbara (512×512)
Peppers (512×512)

16. The test image Baboon512 has a lot of details, it contains components in high frequency area and low predictability, so it presents low redundant image and consequently difficult for compression. On the other hand, the test images Lena512 and Barbara512 are images with less detail than Baboon512. The test image Lena512 has higher predictability than the image Baboon512 since the latter has components in high frequency area more than the image Lena512. The performance of these schemes is usually characterised using the mean square of the error (MSE) and the Peak Signal to Noise Ratio (PSNR) .Table 1 shows the performance of the studied methods while Figure 2 shows the reconstructed compressed images of the methods, respectively.

17. Table 1: Simulation results for the various images using JPEG and JPEG2000 standards.
Bit-rate (bpp)
PSNR (dB)
Compression ratio
Lena
512
0.2098
0.1939
Baboon 512
0.2222
0.1964
Barbara 512
0.2308
0.2593
Peppers 512
0.2239
0.2271

18. Figure 2: Some of the reconstructed compressed images using JPEG and JPEG2000 standards
Lena 512
Baboon 512

19. Conclusion
This paper proposed an efficient implementation of some mathematical functions to Image Compression. DCT is used for transformation in JPEG standard. DCT performs efficiently at medium bit rates. at higher compression ratios, image quality degrades because of the artifacts resulting from the block-based DCT scheme. DWT is used as basis for transformation in JPEG 2000 standard. DWT provides high quality compression at low bit rates because of overlapping basis functions and better energy compaction property of wavelet transforms. DWT performs better than DCT in the context that it avoids blocking artifacts which degrade reconstructed images. However DWT provides lower quality than JPEG at low compression rates. In the future we hope to improve the performance of the DCT by emerging other compression techniques to reduce the staircase problem that results from using the DCT.