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 Total Downloads : 1156
 Authors : Srinivasareddy Putluri
 Paper ID : IJERTV1IS7507
 Volume & Issue : Volume 01, Issue 07 (September 2012)
 Published (First Online): 25092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Reconstruction of 3D Sparse images using Nonlinear Filtering Compressed Sensing 3D Techniques
1Srinivasareddy Putluri
Department of Electronics and Communication Engg., RRS College of Engineering and Technology, Muthangi, Post Graduate student of Electronics and Communication Engineering,
Jawaharlal Nehru Technological University, Hyderabad, India.
AbstractThe objective of this project is to reconstruct 3D sparse images using Nonlinear Filtering Compressed Sensing (NFCS) 3D Anisotropic and 3D Isotropic techniques. Compressed Sensing is a new paradigm for signal recovery and sampling. Using proposed NFCS techniques, a random mask is generated for the selected input 3D image using a sampling matrix. Now, a sparse image is generated for the selected 3D image. Applying Normalization, the original image will be reconstructed from the sparse 3D
is an unknown signal which is Ksparse in the transform domain. This is described by orthogonal matrix and = is the coefficients vector of reconstruction in that domain.
The perfect recovery obtained from the number of given measurements depends upon length N and sparsity level K of original signal and on the acquisition matrix [1], [3].
If the unknown signal has sparse gradient, it has been shown in [2], [3] that it can be recovered by casting problem (1) as
1, subject to = . (2)
The above formulation is particularly suited to the image recovery problem, since many images can be modelled as piecewise smooth functions containing a number of jump discontinuities.
Exact measurements are often not possible in real problems, so if the measurements are corrupted with random noise, namely we have
2
a1, subject to  + 2 2
(3)
or
 2 2
image in iterations. The obtained reconstructed 3D image will be filtered using an appropriate adaptive nonlinear filter in proposed NFCS 3D
(4)
1, subject to 
+ 2
techniques. The output parameters like Normalized Factor, PSNR, and CPU elapsed time will be calculated and compared along with simulation results. The results obtained using proposed techniques will confirm that these possess properties of efficiency, stability, low computational cost, fast recovery of 3D images in few seconds and its performance is competitive with those of state of art algorithms.
Keywords – NFCS, Anisotropic, Isotropic, Compressed Sensing, Normalized Factor
I.INTROCUCTION
Compressed Sensing is a new process of acquiring and reconstructing a signal that is supposed to be sparse or compressible. It states that a relatively small number of linear measurements of a sparse signal can contain most of its salient information. In transform domain, the signals that that have a sparse representation can be exactly recovered from these measurements by solving an optimization problem of the form
minimize 1 subject to = (1) where is an mxn measurement matrix, (m<<n) which possess restricted isometry property [1], [3],
with the original signal can be reconstructed with an error comparable to the noise level by solving the above minimum problem.
The idealized sparse signals are rarely encounter in applications, but real signals are quite often compressible with respect to an orthogonal basis. This means that, if expressed in that basis, their coefficients exhibit exponential decay when sorted by magnitude. The compressible signals are well approximated by Ksparse signals and the compressed sensing paradigm guarantees that from m linear measurements, we can obtain a reconstruction with an error comparable to that of the best possible Kterms approximation within the sparsing basis [2].
In data processing, the traditional practice is to measure data in full length and then compress the resulting measurements before storage or transmission. In such a scheme, recovery of data is generally straightforward. This traditional data acquisition process can be described as full sensing plus compressing.
Compressive Sensing (CS) represents a paradigm shift in which the number of measurements is reduced during acquisition so that no additional compression is necessary. The price to pay is that more sophisticated recovery
procedures become necessary. In this paper, NFCS 3D Anisotropic and Isotropic techniques for image recovery in a fast and efficient way are described. These NFCS Anisotropic and NFCS Isotropic Adaptive nonlinear filtering strategies are discussed in an iterative framework in comparison with 2D image techniques for image recovery.
In the NFCS 3D methods, a sparse image is generated by applying FFT for the input image. From the sparse image, original image will be reconstructed using NFCS strategies by normalization process. NFCS 3D Anisotropic and NFCS Isotropic methods use nonlinear filters in an iterative approach to filter the reconstructed image. These methods possess the required properties of efficiency, stability and low computational cost and its performance is competitive than those of existing algorithms.

LITERATURE REVIEW
A crucial point in the practical application of compressed sensing is the numerical solution of
step on f2, followed by a backward (implicit) step on f1.
This algorithm makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. But, signal must be recovered from a collection of measurements of lower dimensional signals; in phase retrieval, holography, or band limited extrapolation. Another disadvantage is that it is not efficient for multidimensional signals like 3D signals. [7]
C.BOUNDCONSTRAINED RECONSTRUCTION
When dealing with image reconstruction problems it is well known that image intensity values have to be not negative and R, with R>0. From this, we could insert more information in the compressed sensing reconstruction problem by adding a bound constraint to (5) and (6), and considering: find that solves
2
problems (1)(4). A large amount of research has been aimed at finding fast algorithms for solving such problems. Previously there are many reconstruction algorithms which are used for recovery of 2 dimensional images. Following are
or
F (u), subject to = (5)
few previous algorithms that will operate on 2 D images for image recovery.
F (u), subject to  2 < 2 (6)

ITERATIVE REWEIGHTED ALGORITHM
The theory of compressive sensing has shown that sparse signals can be reconstructed exactly from many fewer measurements than traditionally
Where C is the closed convex set
C= { 12 : 0 , R, i=1,1, j=1. 2} and linear operator orresponds to a subsampled orthogonal transform.
believed necessary. In [4], it was shown
empirically that using `p minimization with p < 1
NFCS 2D ALGORITHMS
can do so with fewer measurements than with p =

In this algorithm, the local minima of the non convex problem can be computed. In particular, a regularization strategy is found to greatly improve the ability of a reweighted leastsquares algorithm to recover sparse signals, with exact recovery being observed for signals that are much less sparse than required by an unregularized version (such as FOCUSS, [5]). However, Improvements in terms of speed and multidimensional applicability ae also observed for the reweighted`1 approach of [6]

B.FORWARDBACKWARD SPLITTING
Various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. The principle of this algorithm is to use at every iteration the functions f1 and f2 separately; more specifically the core of iteration consists of a forward (explicit) gradient
Here, an extension of nonlinear filtering approach to the 2D case is considered. As in [9], we focus on a recovery problem where the optimal solution, in addition to satisfying the acquisition constraints, has minimal bounded variation norm, namely, it minimizes. The optimal reconstruction is evaluated by solving a sequence of total variation regularized unconstrained subproblems, where both isotropic and anisotropic TV estimates have been considered. For each value of the penalization parameter the unconstrained subproblems are approached making use of a twostep iterative procedure in fixed number of iterations based upon the forwardbackward splitting method [8].
Interestingly, in compressed sensing where the acquisition matrix is obtained as randomly chosen rows of an orthogonal transform, the two steps of the iterative procedure become an enforcing of the current iterate to be consistent with the given measurements, and a total variation filtering step. The image recovery speed can be increased better
and these algorithms can only work efficiently for 2 D signals shown in Fig. 1.
. 2
Original image Reconstructed image II. 2
Original image Reconstructed image Fig.1.Test Images – 2 and 2
Methods
(a) Horse image (b) Reconstructed image
In NFCS 2D Anisotropic and Isotropic methods, a random mask and sparse image are generated for the input Horse image. A nonlinear filter is used to recover original image from the sparse image. The output parameters like Error, PSNR, No. of iterations and CPU elapsed time are tabulated in Table I.
The image quality has been evaluated using the PSNR value, defined as
Method
Error
PSNR
Iterations
CPU Time
2
1.5648e004
119.0459
14
11.886620 s
2
1.8419e005
49.7175
14
42.789134 s


PROPOSED METHODOLOGY
In this paper, the image reconstruction and nonlinear filtering are extended to 3 dimensional images. Compressed Sensing image recovery problem can be overcomed using proposed NFCS 3D Anisotropic and Isotropic methods.
Using NFCS 3D methods, a sparse image is generated by applying FFT for the input image. From the sparse image, original image will be reconstructed using NFCS strategies by applying normalization process. NFCS 3D Anisotropic and NFCS 3D Isotropic methods use nonlinear filters in an iterative approach to filter the reconstructed image. These methods possess the required properties of efficiency, stability and low computational cost and its performance is competitive than those of existing algorithms.

Block Diagram
Figure 1 shows the block diagram for NFCS 3D Anisotropic and NFCS 3D Isotropic methods. Firstly, an orthogonal transform is applied for the original input image and a random mask is generated using a sampling matrix for the transformed input image. Now, sparse image is generated after applying FFT for the input image. From the sparse image, original image will be reconstructed using NFCS strategies using normalization.
10
= 20
(7)
MSE=
12
(8)
Figure 2. Block Diagram for NFCS 3D Methods
where R>0 is the maximum value of the image gray level range and
Error= 2= 1 . 2 (, , )2 (9)
. 3 and 3 Methods
Using these proposed 3 and
3 methods, nonlinear filters are used in
=1 =1
TABLE I: Output Parameters with Horse Image
an iterative approach to filter the reconstructed image. The output parameters like Normalized factor, PSNR ratio and CPU elapsed time will be calculated and compared with the 2D results.
3 method uses a Median filter to filter the reconstructed image whereas 3 Method uses a digital TV filter for filtering.
C.Reconstruction Approach
Let S belongs to R(N1XN2) be a randomly generated binary mask, such that the pointtopoint product with any v belongs to R(N1XN2), denoted by S x v , represents a random selection of the elements of v, namely, we have
In the present NFCS 3D methods, there is no limitation, since approach for these problems is done implicitly, thus, avoiding the need to deal with illconditioned linear systems.
The corresponding bound constrained twostep iterative algorithm is the following:
= + } (14)
= .
, = 1
=
(10)
=
+ 1  2
0, = 0
+1
2
2
denotes the sampling matrix, which is considered as the random generated mask for the selected input image.
Let T be an orthogonal transform acting on an image X denoted by
= . () (11)
The randomly subsampled orthogonal transform considered here is the Fast Fourier Transform.
Then the input data can be represented as
= . =
To find u that belongs to R (N1xN2) can solve
12
, . =
In the case of input data perturbed by additive white Gaussian noise with standard deviation .
= . + = + (12) The problem can be stated as –
,  2 <2
(15)
The proposed penalized splitting approach corresponds to an algorithm whose structure is characterized by twolevel iteration. There is an outer loop, which progressively diminishes the penalization parameter in order to obtain the convergence to the global minimum, and an inner loop, which iteratively, using the twostep approach, minimizes the penalization function for the given value of LAMBDA.
The generalized approach of the proposed NFCS 3D methods is applied for 3D images considering three dimensions in every step is as following:
Step A0: Initialization
Given F (.), y, , >0, > 0,0 < < 1,
,
And 0 0 < 0
Set k=0, 0,0 =0 and 0,0=0
Step A0: Start with the outer Iterations=10 While (,0> and  ,0 y2 > Toll)
Step B0: Start with the inner Iterations=4
12
2
i=0;
2= 2( + 2 2) (13)
To overcome this problem, penalization approach that considers a sequence of unconstrained
Step B1:
Updating Step: , = , + ,
Constrained Nonlinear filtering step:
,+1 {
2
minimization sub problems of the form
= 1 2
2
, ,
+
12
+
1
2
 2
Convergence Test:
( )
If (,+1) ( ,)
,+1
i=i+1 and , =,1
go to step B1.
,
Otherwise go to Step A2.
Step A2: Outer Iteration Updating K=k+1
,0= .,
,0 = ,+1
end while
Terminate with ,0 as an approximation of
The automatic stopping criterion of the outer loop depends upon which problem that is considering.


RESULTS AND ANALYSIS
The output parameters like Error, PSNR, No. of iterations and CPU elapsed time from (7), (8) and

are tabulated in Table I.
. 3
Original image Reconstructed image II. 3
Original image Reconstructed image Fig.3.Test Images – 3 and 3
Methods

Handsome boy image (b) Reconstructed image

TABLE II: Output Parameters with Handsome boy Image
Method
Error
PSNR
Iterations
CPU Time
3
5.5434e005
128.9371
14
16.996123
s
3
4.6344e005
99.6398
14
67.128490
s
Comparing the results of NFCS 2D and NFCS 3D methods, 3 method provides faster recovery and more PSNR value than all other methods as tabulated in Table III.
TABLE III. Comparison of Results – NFCS 2D and NFCS 3D Methods
Method
Error
PSNR
Iterations
CPU Time
2
1.5648e
004
119.0459
14
11.88662 s
2
1.8419e
005
49.7175
14
42.789134
s
3
5.5434e
005
128.9371
14
16.996123
s
3
4.6344e
005
99.6398
14
67.128490
s
Analyzing the above results from both NFCS 2D and NFCS 3D methods, the NFCS 3D methods provide a faster and efficient way of recovery of 3D images. In both NFCS 2D and NFCS 3D methods, Anisotropic methods provide more PSNR values, thereby less noise in comparison with the Isotropic methods.


CONCLUSIONS
For the solution of the compressed sensing reconstruction problem we have proposed an efficient iterative algorithm for 3D images, based upon an adaptive nonlinear filtering strategy, and its convergence property has been established. The capabilities, in terms of accuracy, stability, and speed of NFCS3D, are illustrated by the results considering 3D images and compared the results from NFCS 2D methods for the selected images. Of these proposed NFCS 3D methods, NFCS 3D Anisotropic methods offers faster image recovery and offers more PSNR value in comparison other NFCS 2D and existing methods.
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AUTHOR DETAILS:
SRINIVASAREDDY P,
Post Graduate Student, Department of Electronics and Communication Engg, RRS College of Engineering and Technology, Muthangi, Patancheru, Hyderabad, Andhra Pradesh, India.