simulated performance evaluation of an srr procedure based on … · 2019. 7. 17. · huber srr...

DARUN KESRARAT: SIMULATED PERFORMANCE EVALUATION OF SRR PROCEDURE PLACED ON ROBUST .

DOI 10.5013/IJSSST.a.20.03.14 14.1 ISSN: 1473-804x online, 1473-8031 print

Simulated Performance Evaluation of an SRR Procedure based on Robust Stochastic Huber Norm

Darun Kesrarat

Assumption University of Thailand Bangkok, Thailand.

e-mail: [email protected]

Abstract - The classical Super Resolution Reconstruction, SRR, procedures based on stochastic L1 or L2 norms are typically susceptible to stochastic modeling inaccuracies of image and noise. This led in 2006 to the proposal of an SRR procedure based on robust stochastic Huber norm for superior resolution enhancement and noise suppression performance in numerous noise environments. This paper aims to evaluate the performance of this proposed robust stochastic Huber SRR procedure and to analyze its shortcomings. This SRR procedure is stochastically based on Bayesian Maximum Likelihood (ML) used with regularization function to stochastically formulate the error function, which is based on the Huber norm to evaluate the computed SR image from poor-quality captured images. This error function is then minimized to evaluate the final solution (the computed SR image). In order to suppress artifacts from the final solution and to accelerate the convergence calculation, the classical Tikhonov and Huber-Tikhonov functions are used. Our simulation results clearly indicate the superior performance of the Huber norm compared with classical stochastic L1 or L2 norm for a range of noise environments: No-noise, Gaussian, Poisson and Impulsive Noise. Keywords - SRR (Super Resolution Reconstruction), Bayesian Maximum Likelihood (ML), Huber norm, Digital Image Processing

I. INTRODUCTION

The last two decades witnessed numerous comprehensive researches on SRR procedures [1-19] because the primary concept of SRR procedure is the stochastic accumulation of a group of distorted blurred LR portraits to synthesize a single portrait or numerous portraits with better quality [2, 9, 11, 18]. This literature part only surveys the SRR researches from the estimated error function perspective from the fact that the error function is a primary part of the SRR procedure and ultimately effects its performance. In 1996, Richard R. S. et. al. [12-13] initially presented the SRR procedure placed on both L2 error function and HMRF regularized function and, later, M. Elad et. al. [6] presented the SRR procedure placed on both L2 error function and non-ellipsoid regularized function in 1997. Next, M. Elad et. al. [8] presented the SRR procedure placed on both L2 error function (R-SD and R-LMS) in 1999. Subsequently, M. Elad et. al. [7] presented the accelerate SRR procedure placed on L2 error function for simple shifted warping, space invariant blurring and Gaussian noise in 2001. Thereafter, A. J. Patti et. al. [1] presented the SRR procedure placed on both L2 error function and POCS and Y. Altunbasak et. al. [19] presented SRR procedure placed on L2 error function for MPEG video in 2002. Closely, Deepu R. et. al. [2-3] presented the SRR procedure placed on both L2 error function and MRF regularized function for focusing objective in 2003. Following, S. Farsiu et. al. [15-16] presented the SRR procedure placed on both L1 error function and BTV regularized function in 2004 and they presented the accelerate SRR procedure placed on both L1 error function and BTV regularized function for color

portraits in 2006. All above surveyed researches of the SRR procedures are placed on either L1 error function or L2 error function. From the fact that the L1 error function has larger fluctuation range of final solution but the L2 error function is typically susceptible to stochastic modeling correctness of image and noise, a Huber error function [20], which is presented by the robust stochastic error function [10], is invented for stability and noise suppression. As a result, this stochastic article focuses to evaluate the performance of this robust stochastic Huber SRR procedure [20] and to analyze these shortcomings. Subsequently, the simulated outcomes obviously indicate the superior performance of the Huber norm compared with classical stochastic L1 or L2 norm for numerous noise environments: No-noise, Gaussian, Poisson, and Impulsive Noise.

II. STOCHASTIC IDEAL ROBUST HUBER SRR PROCEDURE

Let tY be stochastically expressed as a group of distorted blurred LR portraits, which consist of 1 2N N

elements, and tX is stochastically expressed as a computed SR portrait, which consists of 1 2qN qN elements ( q is a resolution expanding factor in both the horizontal and vertical dimensions). To simplify simulated computation, each portrait is divided into many overlay square areas. To simplify expression, each overlay square area is stochastically expressed as a vector, which is in a lexicographic format. The overlay square area LR portrait is



stochastically expressed as 2M

kY , which are constituted 2 1M elements and The overlay square area HR portrait is

stochastically expressed as 2 2q MX , which are constituted 2 2 21 or 1L q M elements. The stochastic relationship [20]

between the LR portrait and the HR portrait is stochastically expressed as the upcoming equation.

; 1, 2, ,k k k k kY D H F X V k N (1-a)

The warping procedure of X and kY is stochastically expressed as kF and the procedure noise is stochastically expressed as kV .

In general, SRR procedure is classified as an ill-posed problem [4–7] therefore there are numerous solutions, which satisfy the SRR degradation procedure in Eq. (1), for under-determined cases. However, the solution will be unstable even though there is a small noise for over-determined cases. In order to suppress artifacts from the final solution and to accurate the convergence computation, the regularized function is incorporated in the SRR procedure.

The SRR procedure placed on the error function is stochastically expressed as the following minimization equation.

1

ArgMinN

kk k kX k

X f D H F X Y

(1-b)

The error function is stochastically expressed as f where the classic error function is L1 stochastic function and L2 stochastic function. Alternatively, the Huber error function [10,20], which is presented by the robust stochastic error function [10], is presented for robustness and suppressing noise because the Huber error function is more forgiving about noise due to the fact that the influence error function should increase less expeditious then L2 stochastic function. The Huber error function is stochastically expressed as the following minimization equation.

2

2

;2 ;HUBER

x x Tf x

T T x T x T (2)

where T is Huber threshold. By cooperating the Laplacian regularized function, the

SRR procedure placed on error function is stochastically expressed as the following minimization equation.

21

ArgMinN

kHUBER k k kX k

X f D H F X Y X

(3)

The regularized threshold of this minimization problem is stochastically expressed as .

Figure 1. The relationship framework of the LR portrait and the HR portrait.



By applying the gradient descent technique, the solution of problem (3) is stochastically expressed as the following equation.

11

ˆˆ ˆ

ˆ

NT T T

k nk k k HUBER k k kkn n

Tn

F H D Y D H F XX X

X

(4)

2 ;

2 sign ;HUBER HUBERx x T

x f xT x x T

(5)

The step-size threshold of the gradient descent technique

is stochastically expressed as By using jointly the Huber-Laplacian regularized

function, the SRR procedure placed on the error function is stochastically expressed as the following minimization equation.

1

ArgMinN

kHUBER k k k HUBERX k

X f D H F X Y g X

(6)

2

2

;

2 ;g

HUBERg g g g

x x Tg x

T T x T x T

(7)

By applying the gradient descent technique, the solution of problem (6) is stochastically expressed as the following equation.

11

ˆˆ ˆ

ˆ

NT T T

k nk k k HUBER k k kk

n nT

nHUBER

F H D Y D H F XX X

X

(8)

2 ;

2 sign ;g

HUBER HUBERg

x x Tx g x

T x x T

(9)

Figure 2. The SRR procedure placed on stochastic error function.



TABLE I. THE RESULT OF EVALUATED SIMULATION OF SRR PROCEDURE (SUSIE PORTRAIT (40TH FRAME))

PSNR (dB) Noisy Model

No-noise AWGN (25dB) AWGN (22.5dB)

AWGN (20dB)

AWGN (17.5dB)

AWGN (15dB) Poisson

S&P D=0.005

S&P D=0.010

S&P D=0.015

Speckle V=0.01

Speckle V=0.02

Corrupted LR Image 32.1687 30.1214 29.0233 27.5316 25.7332 23.7086 27.9071 29.0649 26.4446 25.276 27.6166 25.3563

L1 SRR Image with Lap Reg 32.1687 30.3719 29.6481 28.7003 27.5771 26.2641 28.9197 29.5041 27.7593 26.9247 28.8289 27.5527

L2 SRR Image with Lap Reg 34.2000 32.3688 31.6384 30.6898 29.3375 27.6671 30.7634 31.5021 29.8395 28.7614 30.6139 28.9409Huber SRR Image with Lap Reg 35.1436 32.3936 31.6806 30.7518 30.0448 29.1977 30.8496 34.4428 34.4171 34.362 30.4619 29.3749Huber SRR Image with H-Lap Reg 35.1436 32.3936 31.6806 30.7989 30.1448 29.2958 30.9016 34.503 34.4591 34.5041 30.4693 29.7165

TABLE II. THE RESULT OF EVALUATED SIMULATION OF SRR PROCEDURE (LENA PORTRAIT)

PSNR (dB) Noisy Model

No-noise AWGN (25dB) AWGN (22.5dB)

AWGN (20dB)

AWGN (17.5dB)

AWGN (15dB) Poisson

S&P D=0.005

S&P D=0.010

S&P D=0.015

Speckle V=0.03

Speckle V=0.05

Corrupted LR Image 28.8634 27.8884 27.2417 26.2188 24.9598 23.3549 26.5116 26.8577 25.2677 24.219 23.5294 21.7994



Huber SRR Image with Lap Reg 32.0186 29.7224 29.1935 28.6305 27.8725 27.1945 28.7282 30.9462 30.9329 30.9124 26.6723 26.0595

Huber SRR Image with H-Lap Reg 32.0186 29.8030 29.1956 28.6313 27.8812 27.2119 28.7330 31.3351 31.1113 31.0234 26.8838 26.2596

III. EVALUATION BY SIMULATION

In this section, two portraits were used in QCIF format,

short for Quarter Common Intermediate Format, a videoconferencing format that specifies data rates of 30 frames per second (fps), with each frame containing 144 lines and 176 pixels per line, which is one fourth the resolution of Full CIF. The portraits were of Susie (40th frame), which consists of 176x144 elements, and of Lena, which consists of 256x256 elements. They were employed in these simulations, which were executed by MATLAB software, see figure 3-A, B, C and D.

First, a group of distorted blurred LR portraits is stochastically constructed from two authentic portraits (Susie portrait and Lena portrait) by Eq. (1). At first step in this construction, the authentic HR portrait is warped by a single element in the vertical part. At second step in this construction, this computed portrait is blurred by executing 3x3 Gaussian low-pass filter with unity SD. At third step in this construction, this computed portrait is resized in resolution by half in both parts. At final step in this construction, this computed portrait is accumulated by SRR procedure noise. The identical four step procedures are executed with alternative warping directions in vertical and horizontal parts.

(The principle for parameter adjustment in this simulation was to adjust parameters which make the SRR procedure the highest quality in PSNR aspect. As a result, to establish integrity, each simulation was rerun numerous times with various parameter and the computed SR portraits with the highest quality were recorded. [15-17])

The simulated outcomes of Susie portrait (40th frame) and Lena portrait can be exhibited in Table I and Table II,

respectively. From these simulated results in these tables, the computed SR portraits, which are reconstructed by the SRR procedure placed on Huber error function, have higher quality in PSNR aspect than the computed SR portraits, which are reconstructed by the SRR procedure placed on L1 or L2 error function.

For the simulated outcomes of Susie Portrait, SR portraits of the Huber error function has higher than SR portraits of the L2 error function about 1.4740 1.9564 dB and, moreover, higher than SR portraits of the L1 error function about 3.3159 1.9985 dB.

For the simulated results of Lena Portrait, SR portraits of the Huber error function have higher values than SR portraits of the L2 error function, about 1.0351 1.3192 dB and, moreover, higher than SR portraits of the L1 error function about 2.7162 1.4563 dB.

Due to limitations on the number of pages, this paper provides only the simulated portrait results of Lena portrait, which are depicted in Figure 3-A, B, C and D.

IV. CONCLUSION

This paper aimed to evaluate the performance of SRR procedure for reconstructing the HR portrait with higher resolution and better quality under numerous noise environments: No-noise, Gaussian, Poisson, and Impulsive Noise. In the evaluated simulation, two standard images of Susie and Lena portraits at a range of noise density conditions were employed to reconstruct the SR portrait in Peak Signal-to-Noise Ratio, PSNR aspect. Our simulation results show that SR portraits using the Huber error function are clearly of higher quality than SR portraits using the L1 and L2 error functions, especially for impulsive noise.



ACKNOWLEDGMENT

The authors are grateful to Assoc. Prof. Dr. Vorapoj Patanavijit for simulation source code and test data.

REFERENCES

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Figure 3-A. The Portrait Result of Evaluated Simulation of SRR Procedure



Figure 3-B. The Portrait Result of Evaluated Simulation of SRR Procedure



Figure 3-C. The Portrait Result of Evaluated Simulation of SRR Procedure



Figure 3-D. The Portrait Result of Evaluated Simulation of SRR Procedure

simulated performance evaluation of an srr procedure based on … · 2019. 7. 17. · huber srr...

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