Image Compression with Wavelet Transform
Amandeep Kaur Department of Mathematics, SGGS Khalsa College Mahilpur, INDIA.
email:amanmath437@gmail.com (Received on: August 28, 2016)
ABSTRACT
The objective of image compression technique is to reduce redundancy of the image data in order to be able to store or transmit data in an efficient form. Wavelet transform is very effective in image compression since wavelets allow both time and frequency analysis simultaneously. This paper presents the fundamental concept behind the wavelet transform and provides an overview of image compression with wavelet transform.
Keywords: Wavelet transforms, lossless and lossy compression, quantization, and threshold.
INTRODUCTION
Image compression is important for many applications that involve huge data storage, transmission and retrieval such as for multimedia, documents, videoconferencing, and medical imaging. Uncompressed images require considerable storage capacity and transmission bandwidth. Efficient image compression solutions are becoming more critical with the recent growth of data intensive, multimedia-based web applications. It has many applications in information theory, applied harmonic analysis and in many other fields. The objective of image compression is to minimize the size of an image by exploiting the redundancy within the data without degrading the quality of the image. The reduction in file size allows more images to be stored in a given amount of disk or memory space. The common redundancies are spatial redundancy, temporal redundancy, inter-pixel redundancy, psycho-visual redundancy and statistical redundancy. Image compression can be used to reduce duplicity, computational time, cost of image and quality of image. There are many image compression techniques such as JPEG, JPEG 2000, DCT, DWT, Wavelet, Huffman Coding, Quantization, Lossy Compression or Lossless Compression.
There are two types of image compression: lossless and lossy. In a lossless
compression algorithm, compressed data can be used to recreate an exact replica of the
original; no information is lost to the compression process. This type of compression is also
known as entropy coding. This name comes from the fact that a compressed signal is generally more random than the original; patterns are removed when a signal is compressed. While lossless compression is useful for exact reconstruction, it generally does not provide sufficiently high compression ratios to be truly useful in image compression. It is generally used for applications that cannot allow any difference between the original and reconstructed data. Lossless image compression is particularly useful in image archiving as in the storage of legal or medical records. Methods for lossless image compression includes: Entropy coding, Huffman coding, Bit-plane coding, Run-length coding and LZW (Lempel Ziv Welch) coding.
In lossy compression, the original signal cannot be exactly reconstructed from the compressed data. The reason is that, much of the detail in an image can be discarded without greatly changing the appearance of the image. As an example consider an image of a tree, which occupies several hundred megabytes. In lossy image compression, though very fine details of the images are lost, but image size is drastically reduced. Lossy image compressions are useful in applications such as broadcast television, videoconferencing, and facsimile transmission, in which a certain amount of error is an acceptable trade-off for increased compression performance. Methods for lossy compression include: Fractal compression, Transform coding, Fourier-related transform, DCT (Discrete Cosine Transform) and Wavelet transform.
With lossless compression, the original image is recovered exactly after decompression. Unfortunately, with images of natural scenes it is rarely possible to obtain error-free compression at a rate beyond 2:1. Much higher compression ratios can be obtained if some error, which is usually difficult to perceive, is allowed between the decompressed image and the original image. This is lossy compression. In many cases, it is not necessary or even desirable that there be error-free reproduction of the original image. For example, if some noise is present, then the error due to that noise will usually be significantly reduced via some denoising method. In such a case, the small amount of error introduced by lossy compression may be acceptable. Another application where lossy compression is acceptable is in fast transmission of still images over the Internet.
Over the past ten years, the wavelet transform has been widely used in signal processing research, particularly, in image compression. In many applications, wavelet-based schemes achieve better performance than other coding schemes like the one based on DCT.
Since there is no need to block the input image and its basis functions have variable length, wavelet based coding schemes can avoid blocking artifacts. Wavelet based coding also facilitates progressive transmission of images.
WAVELET TRANSFORM
Wavelets are mathematical functions in which data should be divided into different
frequency components and then matched the resolution into its scale. Wavelet transforms
change a signal into a series of wavelets. In Wavelet Image Processing a single image can store
different parts of resolutions, which should be divided into many parts. Wavelet is applicable
for compressing the image using less storage space and also containing the full details of the
image. An image can be decomposed into approximate, horizontal, vertical and diagonal
details. Joseph Fourier discovered sines and cosines that are used to symbolize the approximation functions of an image and these functions are non-local functions. Scale used to represent data, which play important role in wavelet analysis and wavelet algorithms method representing at diverse scales or different resolutions. If window should be large it provides large features and if window should be small it provides minor features. Wavelets are right for estimating data with sharp breaks. The wavelet analysis determines wavelet prototype function that is known as analyzing wavelet.
The Fourier transform may be defined as a mathematical process with many uses in physics and engineering that states a mathematical function of time as the function of frequency, called as its frequency spectrum. Fourier transforms used to transform time domain signals into a frequency domain. Time domain representation is the function of time that the frequency spectrum and the frequency domain representation. It works by transforming a function from a time domain to the task in the frequency domain. The signal can be examined for its frequency content because the Fourier coefficient of the changed function shows the influence of every sine and cosine function at every frequency. An inverse Fourier transform do just the same as what you expect, alter data from the frequency domain to the time domain.
The inverses Fourier transforms represent the frequency domain function, which contain in the time domain function.
Wavelets are signals which are local in time and scale and generally have an irregular shape. A wavelet is a waveform of effectively limited duration that has an average value of zero. The term ‘wavelet’ comes from the fact that they integrate to zero; they wave up and down across the axis. Many wavelets also display a property ideal for compact signal representation: orthogonality. This property ensures that data is not over represented. A signal can be decomposed into many shifted and scaled representations of the original mother wavelet. A wavelet transform can be used to decompose a signal into component wavelets.
Once this is done the coefficients of the wavelets can be decimated to remove some of the details. Wavelets have the great advantage of being able to separate the fine details in a signal.
Very small wavelets can be used to isolate very fine details in a signal, while very large wavelets can identify coarse details. In addition, there are many different wavelets to choose from. Various types of wavelets are: Morlet, Daubechies, etc. One particular wavelet may generate a more sparse representation of a signal than another, so different kinds of wavelets must be examined to see which is most suited to image compression.
A wavelet function ψ(t) has two main properties,
∫
−∞∞𝜓(𝑡)𝑑𝑡 = 0
That is, the function is oscillatory or has wavy appearance.
∫ |𝜓(𝑡)|
−∞∞ 2𝑑𝑡 < ∞
That is, the most of the energy in ψ(t) is confined to a finite duration.
There are three types of wavelets: continuous wavelet transforms, discrete wavelet transforms
and fast wavelet transforms which are briefly explained below:
Continuous Wavelet Transform (CWT): A continuous wavelet transform is an addition of scaled and mother wavelet. Continuous wavelet transform is used for breakdown time function into wavelets. This continuous wavelet transform has the capacity to make time frequency, which is used to represent signal. The continuous wavelet transforms containing C coefficients with containing functions of scale and translations.
Discrete Wavelet Transform (DWT): The Discrete wavelet transform is used for transform image. In functional analysis and discrete analysis, a Discrete Wavelet Transform (DWT) is a wavelet transform that has the wavelets as discretely sampled. Instead of continuously varying parameters, the signal can be analyzed with a small number of scales with number of translations at each scale. In CWT there are wavelet coefficients for every combination of scale and translation parameters whereas in DWT, there are wavelet coefficients only at very few points. The DWT produces only the minimal number of coefficients necessary to reconstruct the original signal or image.
Fast Wavelet Transforms: The fast Wavelet transform is the mathematical algorithm made to turn a signal or a waveform in time domain to a series of coefficients based on a orthogonal basis of small finite waves, or wavelets. The transform can be simply stretched to a multidimensional signs, like image, where the time domain is exchanged by space domain.
Designing a model wavelet-transform image coder
A typical black-and-white image is an 𝑀 × 𝑀 array of integers chosen from some specified range, say, 0 through 𝐿 − 1. Each element of this array is referred to as a picture element or pixel, and the value of each pixel is referred to as a grayscale value and represents the shade of gray of the given pixel. Usually a pixel value of 0 is colored black, and 𝐿 − 1 is colored white. If M = 256 (hence 65536 pixels) and L = 256 (hence 8 bits per pixel), then the storage requirements for an image would be 256 x 256 x 8 = 524288 bits. The goal of image compression is to take advantage of hidden structure in the image to reduce these storage requirements.
Any transform coding scheme consists of three steps: (1) the Transform Step, (2) the Quantization Step, and (3) the Coding Step. These are explained below:
(1) The Transform Step. In this step, the image data are acted on by some invertible transform T whose purpose is to decorrelate the data as much as possible. This means to remove redundancy or hidden structure in the image. Such a transform usually amounts to computing the coefficients of the image in some orthonormal or nonorthogonal basis. Because any such transform is exactly invertible, the transform step is referred to as lossless.
Since wavelet bases are very good at efficiently representing functions that are smooth
except for a small set of discontinuities. Any image that has large regions of constant grayscale
can therefore be well represented in a wavelet basis. Hence a wavelet basis with sufficient
vanishing moments can be used effectively in the transform step. It is also possible to find the
best wavelet packet basis for an image and use the expansion in that basis as the transform.
The advantage of this approach is that the resulting coefficients will be optimized relative to some appropriate measure of efficiency.
To choose wavelet filters in transform step there are several things to consider which are as follows:
(a) Symmetric filters are most valuable for minimizing so-called edge effects in the wavelet representation of a function. Since orthogonal filters (except the Haar filter) cannot be symmetric, biorthogonal filters are almost always chosen for image compression applications
(b) For efficient representation, we require filters with a large number of vanishing moments.
This way, the smooth parts of an image will produce very small wavelet coefficients.
Vanishing moments on the analysis filter are desirable as they will result in small coefficients in the transform, whereas vanishing moments on the reconstruction filter are desirable as they will result in fewer blocking artifacts in the compressed image. Hence sufficient vanishing moments on both filters are desirable.
(c) Long analysis filters mean greater computation time for the wavelet or wavelet packet transform. Long reconstruction filters can produce unpleasant artifacts in the compressed image. Therefore, we seek both analysis and reconstruction filters that are as short as possible. The more vanishing moments a filter has, the longer that filter must be. Therefore there is a tradeoff between having lots of vanishing moments and short filters.
(2) The Quantization Step. The coefficients calculated in the transform step will in general be real numbers, or at least high-precision floating point numbers, even if the original data consisted of only integer values. As such, the number of bits required to store each coefficient can be quite high. Quantization is the process of replacing these real numbers with approximations that require fewer bits to store. This "rounding off" process is necessarily lossy, meaning that the exact values of the coefficients cannot be recovered from their quantized versions. In a typical transform coding algorithm, all error occurs at this stage. It is often desirable to specify an independent parameter or threshold 𝜆 > 0 such that all coefficients less than 𝛌 in absolute value are quantized to zero.
(3) The Coding Step. Typically, most of the coefficients computed in the transform step will be close to zero, and in the quantization step will actually be set to zero. Hence the output of Steps (1) and (2) will be a sequence of bits containing long stretches of zeros. It is known that bit sequences with that kind of structure can be very efficiently compressed. This is what takes place at this step.
Fig.1 shows these steps in an image compression system. The image compression
system is composed of two distinct structural blocks: an encoder and a decoder. Image (data)
𝑓(𝑥, 𝑦) is fed into the encoder, which creates a set of symbols from the input data and uses
them to represent the image. Image 𝑓̂(𝑥, 𝑦) denotes an approximation of the input image that
results from compressing and subsequently decompressing the input image.
Fig 1: Basic steps in an image compression system.
Several quality measurement variables like, PSNR (peak signal-to-noise ratio), MSE (mean square error) etc. can be measured to find out how well an image is reproduced with respect to the reference image. Numerical values of these variables for any image tell us about the quality of that image. The measure of peak signal-to-noise ratio (PSNR) is defined as the following formula:
𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔
10( 255
2𝑀𝑆𝐸 ) 𝑑𝐵
𝑀𝑆𝐸 = 1
𝑀𝑁 ∑ ∑[𝑓(𝑥, 𝑦) − 𝑓̂(𝑥, 𝑦)]
2𝑁
𝑦=1 𝑀
𝑥=1
