Image Denoising Using Hybrid Transforms

In this paper a new family of transformation for image denoising is presented, Multiridgelet and Walidlet transforms, which have been proposed as alternatives to Discrete Wavelet and Multiwavelet transforms. Walidlet transform is an intelligent tool for solving image processing problems such as image denoising with straight edges, the general algorithm of image denoising using discrete multiwavelet transform is introduced, then followed by the general algorithm of image denoising using Walidlet transform that is proposed here in order to achieve better results than that for Walidlet thresholding, finally, a comparative study is presented to show the differences between a mentioned algorithms


Introduction
The transformation is the process that processes the spatial domain of the signal and translates it to another domain [1], this processing isolates the approximation information (which represents the intelligent information) from the details (which contain the noise), such as discrete Fourier transform, discrete Wavelet transform, discrete Multiwavelet transform, discrete Ridgelet transform and so on.
Denoising of images is an important task in image processing and analysis, and it plays a significant role in modern applications in different fields, including medical imaging and preprocessing for computer vision.Denoising goal is to remove that noise, resulting in minimal damage to the image.Since in most cases the result consumer is human, the criterion for denoising fidelity would be the human visual perception of the result, rather than any of known mathematical criteria [2].
With the fastest growing areas of applying these transforms in multiple domains, here it is necessary to introduce a new transform with high performance and intelligent properties to improve the performance of the previous transforms.Therefore, we proposed a new hybrid transform for image denoising.
The main structure of the proposed transform contains two fundamental transforms, these are:  In fact, the Multiridgelet transform leads to a large family of orthonormal and directional bases for digital images, including adaptive schemas, However, the Multiridgelet transform overcomes the weakness point of the wavelet and Ridgelet transforms in higher dimensions, since, the wavelet transform in two dimensions is obtained by a tensor-product of one dimensional wavelets and they are thus good at isolating the discontinuity across an edge, but will not see the smoothness along edge.

Multiridgelet Transform
The geometrical structure of the Multiridgelet transform consists of two fundamental parts, these are : a-The Radon Transform .b-The One Dimensional Mltiwavelet Transform .Wavelet is a useful tool for signal processing applications such as image processing.Until recently, only scalar wavelets were known: wavelets generated by one scaling function.However, one can imagine a situation when there is more than one scaling function.This leads to the notion of muliwavelets, which have several advantages in comparison to scalar wavelets.
Such features such as short support, orthogonality, symmetry, and vanishing moments are known to be important in signal processing.A scalar wavelet cannot possess all these properties at the same time.
On the other hand, a multiwavelet system can have all of them simultaneously and provide good reconstruction while preserving length (orthogonality) good performance at the boundaries (via linear-phase symmetry), and a high order of approximation (vanishing moments).
Thus, multiwavelets offer the possibility of superior performance for image processing applications, compared with scalar wavelets [5].
Therefore the 1-D DMWT was applied on each slice of the Radon transform, in the other word the 1-D DMWT was applied on each row on the Radon transform coefficients, to get the Multiridgelet coefficients.The intelligent idea behind Walidlet transform is grouping the good properties of the two local transforms in one hybrid transform to obtain a new transform with strong and intelligent properties.

Walidlet transform
The performance of this transform is very well in many domains, and we expect it becomes very useful tool in different cases.
The algorithms of the new hybrid transforms (Walidlet, local Multiridgelet transforms and best sequence of directions) are described in the next paragraphs.

A General Algorithm For Computing Multiridgelet Transform
To compute Multiridgelet transform, the next steps should be followed :

1.
Resizing : Here it is necessary to check for image dimensions, image matrix should be a square matrix, N*N matrix, where N must be the power of two.If the image is not a square matrix, a zero padding operation should be performed to the image (adding rows or columns of zeros to get a square matrix and they must be a prime number).

Computing The Best Sequence of Directions
To compute the best sequence of directions, the following steps should be followed: 1.
Resizing, image matrix should be a square matrix, N*N matrix, and image dimensions should be a prime numbers, as mentioned before.

3.
Add horizontal and vertical slices.

4.
Centralize these slices mod (the size of the transform). 5.
Polarization, converting from cartesian to polar coordinate .6. Find

A General Algorithm For Computing Walidlet Transform
This transform is proposed here for image enhancement, it has intelligent mathematical properties.The next steps exhibit the sequence of this algorithm : 1. Resizing, as mentioned before.

2.
Applying 2-D DMWT : Here it is required to apply the two dimensions discrete multiwavelet transform ( 2_D DMWT ), repeated row method of compution to each image.3. Subband decomposition, as mentioned before.

The Multiridgelet transform :
Each subband is analyzed via the discrete multiridgelet transform, described previously.

5.
Integration : Grouping the multiridgelet coefficients of each subband in the result matrix according to their positions (LL, LH; HL, HH).The result matrix is called the Walidlet coefficients.

A General Algorithm For Computing Inverse Multiridgelet Transform
To compute the Inverse Multiridgelet transform, the next steps should be followed: 1. Apply 1_D IDMWT : Here it is required to apply the one dimension inverse discrete multiwaelet transform (1_D IDMWT ) for each row.

2.
Inverse resizing : Check coefficients matrix length, image length should be a prime number.If the coefficients matrix length is not a prime number, a zero padding operations should be performed to the coefficients matrix size, such as removing rows and columns from the coefficients matrix (which are added in the forward steps).

Apply 1-D DFT :
Here it is required to apply the one dimensional discrete Fourier transform .

Compute the best sequence of directions:
The same geometrical algorithm of computing the best sequence of directions was applying in the forward steps and backward steps, described previously.

5.
Compute the Fourier slices: Find the Fourier slices that must be taken.

Applying 2_D IDMWT :
Here it is required to apply the inverse two dimensions discrete multiwavelet transform (2-D IDMWT), using inverse repeated row method of computation to the result coefficients from the previous step, to have an N * N original reconstructed two dimensions signal matrix.

Denoising Algorithm Using Walidlet Transform
The algorithm of the proposed method is as follows : 1. Obtain the Walidlet Transform coefficients of the observed noisy image: where : DWLT: Discrete Walidlet Transform .
x :original image .g i j :Walid Transform coefficients.i , j : image dimensions .

Select the threshold type. If
the selection is of soft thershold type, then filter the Walidlet transform coefficients using these equations: Or in other form [7]: where: Thv is the threshold value, where: X ˆ: estimated image .
(DWLT ) in other words, hybrid the properties of the (2-D DMWT with the 2-D DMRT) in one transform named Walidlet transform. 671

Figure ( 2
Figure (2): Parallel-Beam Projection at Angle Theta For example, the line integral of f(x,y) in the vertical direction is the projection of f(x,y) onto the x-axis; the line integral in the horizontal direction is the projection of f(x,y) onto the y-axis [4], as shown in figure (3).

Figure
Figure (3): Horizontal and Vertical Projections of Simple Function Figure(5) exhibits the flow diagram of the hybrid transform (Discrete Walidlet Transform). 680