Image Compression Using Wavelet Transform

Image Compression Using Wavelet Transform

Image Compression Using Wavelet Transform

Mridul Kumar Mathur

Department of Computer Science

Lachoo Memorial College of Science and Technology Jodhpur, India

mathur_mridul@yahoo.com

Gunjan Mathur

Department of Electronics and Communication Engineering Jodhpur Institute of Engineering and Technology

Jodhpur, India er_gunjan_mathur@yahoo.com

Abstract The swift development in digital technology has increased the use of images in practically all the applications. The extensive use of these images have raised the need of image compression, so as to save memory and transmission bandwidth

1. Mean Square Error (MSE)

   Mean square error is the cumulative squared error between the compressed image and the original image.

   MSE =

   [/(x, y) /r(x, y)]2

   MN

   y=1 x=1

   Keywords Image, Wavelet Transform, Compression, PSNR, MSE.

   1. INTRODUCTION

      A digital image is a rectangular array of pixels sometimes called a bitmap.It is represented by an array of N rows and M columns and usually N=M. Typical values of N and M are 128, 256, 512 and 1024 etc.A gray scale image that is 256 x 256 pixels has 65,536 elements. Image Compression is a procedureused to reduce the amount of data used to represent a digital image. The reduction in the data reduces the number of bits required to store or transmit the image over digital media.

      Image compression is also of two types: First, Lossless, in which the reconstructed image is exact replica of the original image.If the reconstructed image after the compression is exactly identical to the original image then the compression is known as lossless compression. Second, Lossy, where the reconstructed image is not an exact replica of the original image.If the reconstructed image after compression is not exactly same as the original image then the compression is known as lossy compression. In lossy compression, there is always some loss in the data.The extent of compression is more in lossy compression techniques compared to lossless compression techniques, but the superiority of reconstructed image is good in lossless compression.

   2. PERFORMANCE INDICATORS

      A. Compression Ratio

      Compression ratio is the ratio of numbers of bits required to represent original image to the number of bits required to represent compressed image.

      Uncompressed Size

      C. Peak Signal to Noise Ratio (PSNR)

      Peak Signal to Noise Ratio is the ratio if maximum power of the signal and the power of unnecessary distorting noise.

      255

      PSNR = 20 × log10 [ ]

      MSE

   3. WAVELET TRANSFORM

      Wavelets are functions defined over a finite interval and having an average value of zero. The main purpose of wavelet transform is to represent any arbitrary function as a superposition of a set of such wavelets or basis functions. The discrete wavelet transform of a finite length signal x(n) having N components is expressed by an NxN matrix.

      In many applications wavelet-based schemes (also referred as sub band coding) outperform other coding schemes like one based on DCT. Wavelet-based coding is more robust under transmission and decoding errors, and also facilitates progressive transmission of images.

      1. Threshold Values

         For the compression of image, firstly the DWT is applied in the image using threshold value. Threshold values neglects the certain wavelet coefficients, for doing this one has to decide the value of threshold. Value of threshold affects the quality of compressed image.

         Thresholding can be of two types:

         1. Hard Threshold:

            If x is the set of wavelet coefficients, then threshold value t is given by,

            T(t; x) = 0, if |x| < t

            x, otherwise

            i.e. all the values of x which are less than threshold t

            are equated to zero.

         2. Soft Threshold:

      Compression Ratio =

      Compressed Size

      In this case, all the coefficients x lesser than threshold t are mapped to zero. Then t subtracted

      from all x,t. This condition is depicted by following equation:

      Fig. 5. Original Image 3

      T(t; x) = 0 if x < t

      sign(x)(|x| t) otherwise

      Usually, soft threshold gives a better signal to noise ratio (PSNR) as compared to hard threshold.

   4. RESULTS

      Simulations of various images have been performed using MATLAB.

      A. Image 1

      Fig. 1. Original Image 1

      D. Image 4

      Fig. 6. Reconstructed Image 3

      Fig. 7. Original Image 4

      1. Image 2

      2. Image 3

      Fig. 2. Reconstructed Image 1

      Fig. 3. Original Image 2

      Fig. 4. Reconstructed Image 2

      Fig. 8. Reconstructed Image 4

      TABLE I. COMPRESSION RATIOS OF DIFFERENT IMAGES

      Image	Type of Image	Compression Ratio
      Image 1	JPEG	67.6743
      Image 2	PNG	84.2773
      Image 3	JPEG	10.7651
      Image 4	PNG	60.6735

   5. CONCLUSION

The wavelet transform can be used as a lossy image compression technique. This technique provides good compression to grayscale images. Wavelet transform is much suitable for low bit rate images. Wavelet transform can provide compression ratio of 60-80.

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