Image Compression Based on 2D Dual Tree Complex Wavelet Transform (2D DT-CWT)

By removing the redundant data, the image can be represented in a smaller number of bits and hence can be compressed. There are many different methods of image compression. This paper investigates a proposed form of compression based on 2D Dual Tree complex wavelet transform (2D DT-CWT). Many wavelet coefficients are closed to zero. Thresholding can modify the coefficients to produce more zeros that are allowed a higher compression rate. The wavelet analysis does not actually compress a signal. Therefore Huffman coding is used with a signal processed by the wavelet analysis in order to compress data. Wide range of threshold values is used in the proposed form. From the results the proposed form give higher rate of compression and lower RMS error compared with that forms based on Discrete Wavelet Transform (DWT), Dual Tree Real Wavelet Transform (DT-RWT) and the well known method based Discrete Cosine Transform (DCT).


Introduction
A typical still image contains a large amount of spatial redundancy in plain areas where adjacent picture elements (pixels, pels) have almost the same values.It means that the pixel values are highly correlated.The redundancy can be removed to achieve compression of the image data.The basic measure for the performance of a compression algorithm is compression ratio (CR), defined as a ratio between original data size and compressed data size.In a lossy compression scheme, the image compression algorithm should achieve a tradeoff between compression ratio and image quality.Higher compression ratios will produce lower image quality and vice versa.Quality and compression can also vary according to input image characteristics and content [1].
In recent years, many studies have been made on wavelets.Image compression is one of the most visible applications of wavelets.The rapid increase in the range and use of electronic imaging justifies attention for systematic design of an image compression system and for providing the image quality needed in different applications [1].
Although the discrete wavelet transform (DWT) has dominated the field of image compression for well over a decade, DWTs in their traditional critically sampled form are known to be somewhat deficient in several characteristics, lacking such properties as shift invariance and significant directional selectivity [2].

Complex Wavelet
The critically sampled discrete wavelet transform (DWT) has been successfully applied to a wide range of signal processing tasks.However, its performance is limited because of the following problems [2].
• Oscillations of the coefficients at a singularity (zero crossings).
• Shift variance when small changes in the input cause large changes in the output.
(2) And for their Fourier transform pairs The same concept of analytic or quadrature formulation is applied to the filter bank structure of standard DWT to produce complex solutions and in turn the CWT.The real-valued filter coefficients are replaced by complex-valued coefficients by proper design methodology that satisfies the required conditions for convergence.Then the complex filter can again be decomposed into two real-valued filters.Thus, two real-valued filters that give their respective impulse responses in quadrature will form the Hilbert transform pair.The combined pair of two such filters is termed as an analytic filter.The formulation and interpretation of the analytic filter is shown in figure (1) [4].

1D Dual-Tree CWT
Similarly to the DWT, the DT-CWT is a multiresolution transform with decimated subbands providing perfect reconstruction of the input.In contrast, it uses analytic filters instead of real ones and thus overcomes problems of the DWT at the expense of moderate redundancy.Running in two DWT trees a and b (real and imaginary) of real filters, this transform produces real and imaginary parts of the coefficients.
The wavelet functions ) (t PDF created with pdfFactory Pro trial version www.pdffactory.comwhere t denotes continuous time and n the discrete time index The half-sample delay condition is derived from a strategy of designing filters so that the wavelets generated by them form the Hilbert transform pair.This happens when the scaling filters 0 h and 0 g are offset from one another by a half sample.
As a result, the lowpass filters of one tree interpolate midway between the lowpass filters of the other tree and the DT-CWT is shift invariant.In the frequency domain, the magnitude condition is given by.
And the phase condition runs The filterbank structures for both DT-CWT are identical.Figure (2) shows 1-D analysis and synthesis filterbanks spanned over three levels.It is evident from the filterbank structure of DT-CWT that it resembles the filterbank structure of standard DWT with twice the complexity.It can be seen as two standard DWT trees operating in parallel.One tree is called as a real tree and other is called as an imaginary tree.Sometimes in future discussions the real tree will be referred to as tree-a and the imaginary tree as tree-b.
The form of conjugate filters used in 1-D DT-CWT is given as ) (  The 2-D DT-CWT structure has an extension of conjugate filtering in 2-D case.The filterbank structure of 2-D dual-tree is shown in figure (4).2-D structure needs four trees for analysis as well as for synthesis.The pairs of conjugate filters are applied to two dimensions (x and y) directions, which can be expressed as: The filterbank structure of treea, similar to standard 2-D DWT spanned over 3-level, is shown in figure (5) pairs respectively in the synthesis filterbank similar to their corresponding analysis pairs [4].

Image Compression
Images require much storage space, large transmission bandwidth and long transmission time.The only way currently to improve on these resource requirements is to compress images, such that they can be transmitted quicker and then decompressed by the receiver.
There are many different methods of data compression.This investigation will concentrate on transform coding and then more specifically on Wavelet Transforms.Image data can be represented by coefficients of discrete image transforms.Coefficients that make only small contributions to the information contents can be omitted.Usually the image is split into blocks (subimages) of 8x8 or 16x16 pixels, then each block in discrete cosine transform is transformed separately.However this does not take into account any correlation between blocks, and creates "blocking artifacts" , which are not good if a smooth image is required.
However wavelets transform is applied to entire images, rather than subimages, so it produces no blocking artifacts.This is a major advantage of PDF created with pdfFactory Pro trial version www.pdffactory.comwavelet compression over other transform compression methods.
For some signals, many of the wavelet coefficients are close to or equal to zero.Thresholding can modify the coefficients to produce more zeros.In Hard thresholding any coefficient below a threshold λ, is set to zero.This should then produce many consecutive zero's which can be stored in much less space, and transmitted more quickly by using entropy coding compression.
An important point to note about Wavelet compression is "The use of wavelets and thresholding serves to process the original signal, but, to this point, no actual compression of data has occurred".This explains that the wavelet analysis does not actually compress a signal, it simply provides information about the signal which allows the data to be compressed by standard entropy coding techniques, such as Huffman coding.Huffman coding is good to use with a signal processed by wavelet analysis, because it relies on the fact that the data values are small and in particular zero, to compress data.It works by giving large numbers more bits and small numbers fewer bits.Long strings of zeros can be encoded very efficiently using this scheme.Therefore an actual percentage compression value can only be stated in conjunction with an entropy coding technique.To compare different wavelets, the number of zeros is used.More zeros will allow a higher compression rate, if there are many consecutive zeros, this will give an excellent compression rate.
which keeps the input if it is larger than the threshold T ; otherwise, it is set to zero.The wavelet thresholding procedure removes noise by thresholding only the wavelet coefficients of the detail subbands, while keeping the low resolution coefficients unaltered [9].

Compression Algorithm
The different method for compression, investigate differ only in the selection of the method.The basic procedure remains the same: (1) Digitise the source image into a signal.
(2) Decompose the signal s into wavelet coefficients using DT-CWT.
(5) Reconstruct using the original approximation coefficients using inverse DT-CWT.Figure (7) illustrates that the compression algorithm using DT-CWT.
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Experimental Results and Discussions
Many standard files of images (lena, monarch, lmess and stone) are taken in this paper to know the strong and weak sides of the proposed method.Another methods such as image compression based DWT, 2D DT-RWT and the well known method (image compression based DCT) are implemented for the standard files of images that mentioned above.
Table (1) represents the result of compression ratio and RMS error for may standards images, each image is compressed by using the compression methods based on DWT,2D-DTRWT, DCT and the proposed form that based 2D DT-CWT.The proposed method give higher rate of compression and lower RMS error compared with other methods for many standard methods.
The various methods of compression are affected by threshold value.High values of threshold give higher rate of compression.From Table (1), it is clear that the different method of compression gives the same compression ratio for different values of thresholds.For example when lena image is compressed the compression ratio is (25:1), the thresholds values equals to 65, 50, 35 and 23 for the methods based DCT, DWT, 2D DT-RWT and 2D DT-CWT respectively.Hence the lena compressed image as shown in figure (8) is more smoothing and have lower RMS error when the method based 2D DT-CWT is used.The results of various methods are explained in figure (9) with the same threshold value (T=40) for lena image.
Figure (10) illustrates that the compression ratio is improved with increasing the threshold value.On the other hand the RMS error value is increased when the compression rate (CR) is increased as shown in figure (11).

Conclusions
Various functions are related to the corresponding filters through the dilation and wavelet equations[5].

∑
) form orthogonal or biorthogonal pairs with their respective counterpart filters of analysis tree as shown in figure (2 a).2.2 2D DT-CWTApart from that, the 2D DT-CWT also more selectively discriminates features of various orientations.While the critically decimated 2D DWT outputs three orientational selective sub-bands per level conveying image features oriented at the angles of o 2D DT-CWT produces six directional subbands per level to reveal the details of an image in o implementation of 2-D DT-CWT consists of two steps.Firstly, an input image is decomposed up to a desired level by two separable 2-D DWT branches, branch a and branch b, whose filters are specifically designed to meet the Hilbert pair requirement.Then six high-pass subbands are PDF created with pdfFactory Pro trial version www.pdffactory.comlevel.Secondly, every two corresponding subbands which have the same pass-bands are linearly combined by either averaging or differencing.As a result, subbands of 2-D DT-CWT at each level are defined by oriented, as illustrated in figure (3).Because the sum/difference operation is orthonormal; this constitutes a perfect reconstruction wavelet transform.The imaginary part of 2D DT-CWT has similar basis function as the real part [8].
All other trees-(b,c,d) have similar structures with the appropriate combinations of filters for row-and column-filtering.The overall 2-D dual-tree structure is 4-times redundant (expensive) than the standard 2-D DWT.The tree-a and tree-b form the real pair, while the tree-c and tree-d form the imaginary pair of the analysis filterbank.Trees-(a ~, b ~) and trees-( c ~, d ~) are the real and imaginary There are two thresholding methods frequently used.The softthreshold function (also called the shrinkage function) is shown in figure (6 a).
takes the argument and shrinks it toward zero by the threshold T .The other popular alternative is the hardthreshold function is shown in figure (6 b).

Figure ( 3 )Figure ( 4 ) 4 ]Figure ( 5 ) 4
Figure (3) Typical wavelets associated with the 2-D dual-tree oriented wavelet transform.The top row illustrates the wavelets in the spatial domain, the second row illustrates the (idealized) support of the spectrum of each wavelet in the 2-D frequency plane [8]