 Research Article
 Open Access
Smooth Adaptation by Sigmoid Shrinkage
 Abdourrahmane M. Atto^{1}Email author,
 Dominique Pastor^{1} and
 Grégoire Mercier^{1}
https://doi.org/10.1155/2009/532312
© Abdourrahmane M. Atto et al. 2009
 Received: 27 March 2009
 Accepted: 6 August 2009
 Published: 4 October 2009
Abstract
This paper addresses the properties of a subclass of sigmoidbased shrinkage functions: the non zeroforcing smooth sigmoidbased shrinkage functions or SigShrink functions. It provides a SURE optimization for the parameters of the SigShrink functions. The optimization is performed on an unbiased estimation risk obtained by using the functions of this subclass. The SURE SigShrink performance measurements are compared to those of the SURELET (SURE linear expansion of thresholds) parameterization. It is shown that the SURE SigShrink performs well in comparison to the SURELET parameterization. The relevance of SigShrink is the physical meaning and the flexibility of its parameters. The SigShrink functions performweak attenuation of data with large amplitudes and stronger attenuation of data with small amplitudes, the shrinkage process introducing little variability among data with close amplitudes. In the wavelet domain, SigShrink is particularly suitable for reducing noise without impacting significantly the signal to recover. A remarkable property for this class of sigmoidbased functions is the invertibility of its elements. This propertymakes it possible to smoothly tune contrast (enhancement, reduction).
Keywords
 Wavelet Coefficient
 Wavelet Domain
 Speckle Noise
 Contrast Function
 Wavelet Shrinkage
1. Introduction
The Smooth SigmoidBased Shrinkage (SSBS) functions introduced in [1] constitute a wide class of WaveShrink functions. The WaveShrink (Wavelet Shrinkage) estimation of a signal involves projecting the observed noisy signal on a wavelet basis, estimating the signal coefficients with a thresholding or shrinkage function and reconstructing an estimate of the signal by means of the inverse wavelet transform of the shrunken wavelet coefficients. The SSBS functions derive from the sigmoid function and perform an adjustable wavelet shrinkage thanks to parameters that control the attenuation degree imposed to the wavelet coefficients. As a consequence, these functions allow for a very flexible shrinkage.
The present work addresses the properties of a subclass of the SSBS functions, the nonzeroforcing SSBS functions, hereafter called the SigShrink (Sigmoid Shrinkage) functions. First, we provide a discussion on the optimization of the SigShrink parameters in the context of WaveShrink estimation. The optimization exploits the new Stein Unbiased Risk of Estimation ((SURE), [2]) proposed in [3]. SigShrink performance measurements are compared to those obtained when using the parameterization of [3], which consists of a sum of Derivatives of Gaussian (DOG). We then address the main features of the SigShrink functions; artifactfree denoising and smooth contrast functions make SigShrink a worthy tool for various signal and image processing applications.
The presentation of this paper is as follows. Section 2 presents the SigShrink functions. Section 3 briefly describes the nonparametric estimation by wavelet shrinkage and addresses the optimization of the SigShrink parameters with respect to the new SURE approach described in [3]. Section 4 discusses the main properties of the SigShrink functions by providing experimental tests. These tests assess the quality of the SigShrink functions for image processing: adjustable and artifactfree denoising as well as contrast functions. Finally, Section 5 concludes this paper.
2. Smooth SigmoidBased Shrinkage
for , are shrinkage functions satisfying the following properties.
(P1) Smoothness. There is smoothness of the shrinkage function so as to induce small variability among data with close values;
(P2) Penalized Shrinkage. A strong (resp., a weak) attenuation is imposed for small (resp., large) data.
(P3) Vanishing Attenuation at Infinity. The attenuation decreases to zero when the amplitude of the coefficient tends to infinity.
Each is the product of the identity function with a sigmoidlike function. A function will hereafter be called a SigShrink (Sigmoid Shrinkage) function.
where is the indicator function of a given set : if if . It follows that acts as a threshold. Note that sets a coefficient with amplitude to half of its value and so minimizes the local variation (second derivative) around , since .
 (1)
fix threshold and angle of the SigShrink function, with and . Keep in mind that the larger , the stronger the attenuation,
 (2)
compute the corresponding value of from (4),
 (3)
shrink the data according to the SigShrink function defined by (1).
3. Sigmoid Shrinkage in the Wavelet Domain
3.1. Estimation via Shrinkage in the Wavelet Domain
Let us recall the main principles of the nonparametric estimation by wavelet shrinkage (the socalled WaveShrink estimation) in the sense of [4]. Let stand for the sequence of noisy data where is an unknown deterministic function, the random variables are independent and identically distributed (iid), Gaussian with null mean and variance , in short, for every .
where and . The random variables are iid and . The transform is assumed to achieve a sparse representation of the signal in the sense that, among the coefficients , only a few of them have large amplitudes and, as such, characterize the signal. In this respect, simple estimators such as "keep or kill" and "shrink or kill" rules are proved to be nearly optimal, in the Mean Square Error (MSE) sense, in comparison with oracles (see [4] for further details). The wavelet transform is sparse in the sense given above for smooth and piecewise regular signals [4]. Hereafter, the matrix represents an orthonormal wavelet transform. Let be the sequence resulting from the shrinkage of by using a function . We obtain an estimate of by setting where is the transpose, and thus, the inverse orthonormal wavelet transform.
In [4], the hard and softthresholding functions are proposed for wavelet coefficient estimation of a signal corrupted by Additive, White and Gaussian Noise (AWGN). Using these thresholding functions adjusted with suitable thresholds, [4] shows that, in AWGN, the waveletbased estimators thus obtained achieve within a factor of of the performance achieved with the aid of an oracle. Despite the asymptotic nearoptimality of these standard thresholding functions, we have the following limitations. The hardthresholding function is not everywhere continuous and its discontinuities generate a high variance of the estimate; on the other hand, the softthresholding function is continuous but creates an attenuation on large coefficients, which results in an over smoothing and an important bias for the estimate [5]. In practice, these thresholding functions (and their alternatives "nonnegative garrote" function [6], "smoothly clipped absolute deviation" function [7]) yield musical noise in speech denoising and visual artifacts or over smoothing of the estimate in image processing (see, e.g., the experimental results given in Section 4.1). Moreover, although thresholding rules are proved to be relevant strategies for estimating sparse signals [4], wavelet representations of many signals encountered in practical applications such as speech and image processing fail to be sparse enough (see illustrations given in [8, Figure 3]). For a signal whose wavelet representation fails to be sparse enough, it is more convenient to impose the penalized shrinkage condition (P2) instead of zero forcing since small coefficients may contain significant information about the signal. Condition (P1) guarantees the regularity of the shrinkage process, and the role of condition (P3) is to avoid over smoothing of the estimate (noise mainly affects small wavelet coefficients). SigShrink functions are thus suitable functions for such an estimation since they satisfy (P1), (P2), and (P3) conditions. The following addresses the optimization of the SigShrink parameters.
3.2. SUREBased Optimization of SigShrink Parameters
for a shrinkage function . The SURE approach [2] involves estimating unbiasedly the risk . The SURE optimization then consists in finding the set of parameters that minimizes this unbiased estimate. The following result is a consequence of [3, Theorem 1].
Proposition 3.1.
is an unbiased estimator of the risk , where is a SigShrink function.
Proof.
the result derives from (1), (8), and (9).
As a consequence of Proposition 3.1, we get that minimizing of (6) amounts to minimizing the unbiased (SURE) estimator given by (7). The next section presents experimental tests for illustrating the SURE SigShrink denoising of some natural images corrupted by AWGN. For every tested image and every noise standard deviation considered, the optimal SURE SigShrink parameters are those minimizing , the vector representing the wavelet coefficients of the noisy image.
3.3. Experimental Results
The SURE optimization approach for SigShrink is now given for some standard test images corrupted by AWGN. We consider the standard dimensional Discrete Wavelet Transform (DWT) by using the Symlet wavelet of order ("sym8" in the Matlab Wavelet toolbox).
The SigShrink estimation is compared with that of the SURELET "sum of DOGs" (Derivatives Of Gaussian). SURELET (free MatLab software is avalaible at http://bigwww.epfl.ch/demo/suredenoising/) is a SUREbased method that moreover includes an interscale predictor with a priori information about the position of significant wavelet coefficients. For the comparison with SigShrink, we only use the "sum of DOGs" parameterization, that is, the SURELET method without interscale predictor and Gaussian smoothing. By so proceeding, we thus compare two shrinkage functions: SigShrink and "sum of DOGs."
In the sequel, the SURE SigShrink parameters (attenuation degree and threshold) are those obtained by performing the SURE optimization on the whole set of the detail DWT coefficients. The attenuation degree and threshold thus computed are then applied at every decomposition level to the detail DWT coefficients. We also introduce the SURE LevelDependent SigShrink (SURE LDSigShrink) parameters. These parameters are obtained by applying an SURE optimization at every detail (horizontal, vertical, diagonal) subimage located at the different resolution levels concerned (4 resolution levels in our experiments).
where stands for the dynamics of the signal, in the case of 8 bitcoded images.
Means, variances, minima, and maxima of the PSNRs computed over 25 noise realizations, when denoising test images by the SURE SigShrink, SURE LDSigShrink, and "sum of DOGs" methods. The tested images are corrupted by AWGN with standard deviation . The DWT is computed by using the "sym8" wavelet. Some statistics are given in Tables 2, 3, 4, and 5 for the SigShrink and LDSigShrink optimal SURE parameters.
Image  "House"  "Peppers"  "Barbara"  "Lena"  "Flin"  "Finger"  "Boat"  "Barco"  

( Input PSNR = 34.1514).  
Mean(PSNR)  SigShrink  37.1570  36.4765  36.2587  37.3046  35.2207  35.3831  36.1187  36.6890 
LDSigShrink  37.4880  36.6827  36.3980  37.5518  35.3128  35.8805  36.3608  36.9928  
SURELET  37.3752  36.6708  36.3767  37.5023  35.3102  35.9472  36.3489  35.9698  
Var(PSNR)  SigShrink  0.4269  0.3635  0.0746  0.0696  0.0702  0.0630  0.0533  0.5338 
LDSigShrink  0.8786  0.3081  0.0879  0.0643  0.0262  0.0571  0.0937  0.5613  
SURELET  0.5154  0.4434  0.0994  0.1241  0.0413  0.0453  0.0479  0.3132  
Min(PSNR)  SigShrink  37.1067  36.4479  36.2409  37.2837  35.2021  35.3681  36.1060  36.6384 
LDSigShrink  37.4427  36.6502  36.3764  37.5377  35.3043  35.8695  36.3409  36.9220  
SURELET  37.3196  36.6280  36.3502  37.4799  35.2986  35.9355  36.3353  35.9190  
Max(PSNR)  SigShrink  37.2101  36.5211  36.2753  37.3202  35.2385  35.4043  36.1309  36.7345 
LDSigShrink  37.5405  36.7100  36.4175  37.5750  35.3244  35.8985  36.3790  37.0374  
SURELET  37.4218  36.7061  36.3967  37.5198  35.3255  35.9614  36.3636  35.9960  
( Input PSNR = 24.6090).  
Mean(PSNR)  SigShrink  31.0833  29.5395  28.9750  31.3434  27.9386  28.1546  29.6099  29.9200 
LDSigShrink  31.6472  30.0930  29.3972  32.0571  28.3815  29.4191  30.2895  30.4545  
SURELET  31.2834  29.9621  29.2817  31.9059  28.3502  29.4365  30.2706  27.4525  
Var(PSNR)  SigShrink  0.0016  0.0010  0.0003  0.0003  0.0001  0.0002  0.0003  0.0019 
LDSigShrink  0.0030  0.0009  0.0003  0.0008  0.0002  0.0002  0.0003  0.0015  
SURELET  0.0014  0.0008  0.0003  0.0004  0.0001  0.0002  0.0003  0.0005  
Min(PSNR)  SigShrink  31.0022  29.4883  28.9490  31.3068  27.9221  28.1188  29.5829  29.8443 
LDSigShrink  31.5005  30.0315  29.3741  31.9621  28.3647  29.3908  30.2563  30.3773  
SURELET  31.2056  29.9124  29.2378  31.8653  28.3339  29.3967  30.2468  27.4074  
Max(PSNR)  SigShrink  31.1630  29.6216  29.0129  31.3777  27.9555  28.1724  29.6416  30.0088 
LDSigShrink  31.7552  30.1848  29.4313  32.0952  28.4164  29.4604  30.3272  30.5144  
SURELET  31.3555  30.0225  29.3075  31.9350  28.3616  29.4571  30.3093  27.4843  
( Input PSNR = 20.1720).  
Mean(PSNR)  SigShrink  28.5549  26.5452  25.9539  28.7835  24.8761  25.1774  26.9844  27.2684 
LDSigShrink  29.2948  27.3111  26.5146  29.7435  25.6407  26.6262  27.8216  27.9599  
SURELET  28.8085  26.9941  26.4404  29.5937  25.5953  26.7659  27.8227  23.6221  
Var(PSNR)  SigShrink  0.0015  0.0009  0.0004  0.0007  0.0002  0.0002  0.0002  0.0017 
LDSigShrink  0.0028  0.0022  0.0006  0.0013  0.0002  0.0003  0.0007  0.0024  
SURELET  0.0015  0.0024  0.0004  0.0004  0.0003  0.0003  0.0004  0.0006  
Min(PSNR)  SigShrink  28.4563  26.4906  25.9164  28.7256  24.8499  25.1474  26.9606  27.1534 
LDSigShrink  29.1894  27.2160  26.4642  29.6501  25.6143  26.5912  27.7927  27.8702  
SURELET  28.7439  26.8867  26.4128  29.5424  25.5599  26.7256  27.7803  23.5541  
Max(PSNR)  SigShrink  28.6309  26.5974  25.9921  28.8215  24.8962  25.1962  27.0133  27.3490 
LDSigShrink  29.4082  27.3887  26.5684  29.8135  25.6715  26.6726  27.8970  28.0518  
SURELET  28.8828  27.0884  26.4771  29.6331  25.6259  26.8062  27.8615  23.6703  
( Input PSNR = 17.2494).  
Mean(PSNR)  SigShrink  26.9799  24.6863  24.2771  27.1918  22.9274  23.3429  25.4271  25.7142 
LDSigShrink  27.7840  25.5818  24.8910  28.2782  23.9326  24.9625  26.3764  26.5068  
SURELET  27.2768  25.1307  24.8383  28.1462  23.8954  25.0756  26.3880  21.3570  
Var(PSNR)  SigShrink  0.0018  0.0014  0.0005  0.0011  0.0002  0.0002  0.0006  0.0020 
LDSigShrink  0.0071  0.0035  0.0006  0.0022  0.0007  0.0003  0.0011  0.0035  
SURELET  0.0021  0.0012  0.0004  0.0008  0.0003  0.0003  0.0006  0.0007  
Min(PSNR)  SigShrink  26.8957  24.6337  24.2299  27.1388  22.9031  23.3139  25.3856  25.6094 
LDSigShrink  27.6242  25.4966  24.8499  28.1395  23.8746  24.9369  26.3102  26.3964  
SURELET  27.1928  25.0577  24.7906  28.0753  23.8608  25.0446  26.3167  21.3180  
Max(PSNR)  SigShrink  27.0502  24.7740  24.3079  27.2623  22.9493  23.3813  25.4782  25.7942 
LDSigShrink  27.9473  25.7515  24.9507  28.3628  23.9717  24.9984  26.4346  26.5985  
SURELET  27.3627  25.2000  24.8701  28.1867  23.9375  25.1146  26.4311  21.4116 
Mean values (based on 25 noise realizations) for optimal DWT "sym8" SURE SigShrink parameters, when denoising the "Lena" image corrupted by AWGN. The SURE SigShrink parameters are the SigShrink parameters and obtained by performing the SURE optimization on the whole set of the detail DWT coefficients. It follows from these results that the threshold height as well as the attenuation degree tends to be increasing functions of the noise standard deviation .
Image  "House"  "Peppers"  "Barbara"  "Lena"  "Flinstones"  "Fingerprint"  "Boat"  "Barco" 

 
Mean  0.3183  0.2615  0.2655  0.3054  0.1309  0.1309  0.1913  0.3122 
Mean  2.3420  1.9289  1.9156  2.3861  1.1145  1.1375  1.6885  2.1334 
 
Mean  0.5113  0.4407  0.4256  0.5158  0.3429  0.3491  0.4264  0.4584 
Mean  3.0439  2.6016  2.6259  3.1045  2.3897  2.4181  2.8454  2.8954 
 
Mean  0.5640  0.4931  0.4638  0.5764  0.4305  0.4310  0.4997  0.5185 
Mean  3.2612  2.7893  2.9397  3.3283  2.7167  2.7670  3.1414  3.2043 
 
Mean  0.5925  0.5151  0.4900  0.6066  0.4761  0.4802  0.5389  0.5505 
Mean  3.3885  2.9240  3.2249  3.4733  2.8835  2.9493  3.3459  3.4142 
Variances (based on 25 noise realizations) for the optimal SURE SigShrink parameters whose means are given in Table 2.
Image  "House"  "Peppers"  "Barbara"  "Lena"  "Flinstones"  "Fingerprint"  "Boat"  "Barco" 

 
Var :  0.1550  0.2625  0.0877  0.0592  0.0002  0.0004  0.0642  0.2138 
Var :  0.0932  0.2204  0.0591  0.0209  0.0015  0.0017  0.1454  0.1500 
 
Var :  0.4569  0.2777  0.0468  0.1946  0.0722  0.0297  0.0478  0.5645 
Var :  0.0002  0.0001  0.0003  0.0011  0.0003  0.0003  0.0018  0.0001 
 
Var :  0.4858  0.3753  0.0968  0.1594  0.0433  0.0586  0.1100  0.6510 
Var :  0.6270  0.1439  0.0504  0.1215  0.0184  0.0227  0.0452  0.3095 
 
Var :  0.7011  0.3639  0.1123  0.2463  0.0662  0.1041  0.0982  0.8360 
Var :  0.9610  0.4325  0.1219  0.1720  0.2287  0.0445  0.1570  0.7928 
Mean values of the optimal SURE LDSigShrink parameters, for the denoising of the "Lena" image corrupted by AWGN. The DWT with the "sym8" wavelet is used. The SURE LDSigShrink parameters are obtained by applying a SURE optimization at every detail (Hori. for Horizontal, Vert. for Vertical, Diag. for Diagonal) subimage located at the different resolution levels concerned. We remark first that the threshold height, as well as the attenuation degree, tends to be increasing functions of the noise standard deviation . In addition, for every considered, the attenuation degree as well as the threshold tends to decrease when the resolution level increases.
 


 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  
 0.2864  0.2738  0.3172  3.1072  2.3829  4.2136 
 0.2298  0.1722  0.3057  1.8747  1.4181  2.1687 
 0.0863  0.0657  0.1868  0.7361  0.4852  1.3251 
 0.1154  0.1558  0.4071  0.4957  0.4867  1.4383 
 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  
 0.5397  0.4517  0.9361  4.9893  4.0930  4.6560 
 0.4209  0.3767  0.4641  2.9436  2.4534  3.1053 
 0.2622  0.1794  0.3481  1.9541  1.3087  2.2195 
 0.2128  0.3161  0.4528  1.0539  1.0125  1.8657 
 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  
 0.8934  0.5412  0.9712  4.5129  5.0167  4.4367 
 0.4633  0.4217  0.5209  3.5723  2.8134  3.8653 
 0.3294  0.2642  0.4135  2.4032  1.7920  2.5764 
 0.2644  0.3264  0.4655  1.5004  1.3231  2.0720 
 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  
 0.8772  0.8785  0.9575  4.6843  4.5268  4.6499 
 0.4963  0.4389  0.5746  4.2031  3.2062  4.5700 
 0.3643  0.2745  0.4424  2.6642  1.9881  2.8343 
 0.2700  0.3119  0.4743  1.6543  1.3744  2.2185 
Variances (based on 25 noise realizations) for optimal SURE SigShrink parameters whose means are given in Table 4.
 


 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  




























 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  




























 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  




























 

 
Hori.  Vert.  Diag.  Hori.  Vert.  Diag.  




























We use the Matlab routine fmincon to compute the optimal SURE SigShrink parameters. This function computes the minimum of a constrained multivariable function by using nonlinear programming methods (see Matlab help for the details). Note the following. First, one can use a test set and average the optimal parameter values on this set for application to images other than those used in the test set. By so proceeding, we avoid the systematic use of optimization algorithms such as fmincon on images that do not pertain to the test class. The low variability that holds among the optimal parameters given in Tables 2, 3, 4, and 5 ensures the robustness of the average values. Second, instead of using optimal parameters, one can use heuristic ones (calculated by taking into account the physical meaning of these parameters and the noise statistical properties) such as the standard minimax or universal thresholds, which are shown to perform well with SigShrink (see Section 4).
From Table 1, it follows that the 3 methods yield PSNRs of the same order. The level dependent strategy for SigShrink (LDSigShrink) tends to achieve better results than the SigShrink and the "sum of DOGs." For every method, the difference (over the 25 noise realizations) between the minimum and maximum PSNR is less than 0.2 dB.
 (i)
the threshold height as well as the attenuation degree tends to be increasing functions of the noise standard deviation ,
 (ii)
for every tested , the SURE leveldependent attenuation degree and threshold tend to decrease when the resolution level increase (see Table 4),
 (iii)
for every fixed , the variance of the optimal SURE parameters over the 25 noise realizations is small; optimal parameters are not very disturbed for different noise realizations,
 (iv)
as far as the level dependent strategy is concerned, the attenuation degree as well as the threshold tends to decrease when the resolution level increases for a fixed .
4. Smooth Adaptation
In this section, we highlight specific features of SigShrink functions with respect to several issues in image processing.
Besides its simplicity (function with explicit close form, in contrast to parametric methods such as Bayesian shrinkages [9–14]), the main features of the SigShrink functions in image processing are the following.
Adjustable Denoising
The flexibility of the SigShrink parameters allows to choose the denoising level. From hard denoising (degenerated SigShrink) to smooth denoising, there exists a wide class of regularities that can be attained for the denoised signal by adjusting the attenuation degree and threshold.
ArtifactFree Denoising
The smoothness of the nondegenerated SigShrink functions allows for reducing noise without impacting significantly the signal; a better preservation of the signal characteristics (visual perception) and its statistical properties is guaranteed due to the fact that the shrinkage is performed with less variability among coefficients with close values.
Contrast Function
The SigShrink function and its inverse, the SigStretch function, can be seen as contrast functions. The SigShrink function enhances contrast, whereas the SigStretch function reduces contrast.
In what follows, we detail these characteristics. The following proposition characterizes the SigStretch function.
Proposition 4.1.
for any real value , with being the Lambert function defined as the inverse of the function: .
Proof.
[See appendix].
In the rest of the paper, the wavelet transform used is the Stationary (also call shiftinvariant or redundant) Wavelet Transform (SWT) [15]. This transform has appreciable properties in denoising. Its redundancy makes it possible to reduce residual noise due to the translation sensitivity of the orthonormal wavelet transform.
4.1. Adjustable and ArtifactFree Denoising
The shrinkage performed by the SigShrink method is adjustable via the attenuation degree and the threshold .
For a fixed attenuation degree, we observe that the smoother denoising is obtained with the larger threshold (universal threshold). Small value for the threshold (minimax threshold) leads to better preservation of the textural information contained in the image (compare in Figure 4, image (a) versus image (d); image (b) versus image (e); image (c) versus image (f); or equivalently, compare the zooms of these images shown in Figure 5).
Now, for a fixed threshold , the SigShrink shape is controllable via (see Figure 2). The attenuation degree , reflects the regularity of the shrinkage and the attenuation imposed to data with small amplitudes (mainly noise coefficients). The larger , the more the noise reduction. However, SigShrink functions are more regular for small values of , and thus, small values for lead to less artifacts (in Figure 5, compare images 5(d), 5(e), and 5(f)).
It follows that SigShrink denoising is flexible thanks to parameters and , preserves the image features, and leads to artifactfree denoising. It is thus possible to reduce noise without impacting the signal characteristics significantly. Artifact free denoising is relevant in many applications, in particular for medical imagery where visual artifacts must be avoided. In this respect, we henceforth consider small values for the attenuation degree.
Remark 3.2.
At this stage, it is worth mentioning the following. Some parametric shrinkages using a priori distributions for modeling the signal wavelet coefficients can sometimes be described by nonparametric functions with explicit formulas (e.g., a Laplacian assumption leads to a softthresholding shrinkage). In this respect, one can wonder about possible links between SigShrink and the Bayesian Sigmoid Shrinkage (BSS) of [14]. BSS is a oneparameter family of shrinkage functions; whereas SigShrink functions depend on two parameters. Fixing one of these two parameters yields a subclass of SigShrink functions. It is then reasonable to think that depending on the distribution of the signal and noise wavelet coefficients, these functions should somehow relate to BSS. Actually, such a possible link has not yet been established.
with being the SigStretch function (inverse of the SigShrink function , see (11)). Thus, SigShrink has several interpretations depending on the model used.
4.2. Speckle Denoising
In SAR, oceanography and medical ultrasonic imagery, sensors record many gigabits of data per day. These images are mainly corrupted by speckle noise. If postprocessing such as segmentation or change detection have to be performed on these databases, it is essential to be able to reduce speckle noise without impacting the signal characteristics significantly. The following illustrates that SigShrink makes it possible to achieve this because of its flexibility (see the shapes of SigShrink functions given in Figure 2) and the artifactfree denoising they perform (see Figures 4 and 5). In addition, since SigShrink is invertible, it is not essential to store a copy of the original database (thousands and thousands of gigabits recorded every year); one can retrieve an original image by simply applying the inverse SigShrink denoising procedure (SigStrech functions). More precisely, the following illustrates that SigShrink performs well for denoising speckle noise in the wavelet domain.
Speckle noise is a multiplicative type noise inherent to signal acquisition systems using coherent radiation. This multiplicative noise is usually modeled as a correlated stationary random process independent of the signal reflectance.
Two different additive representations are often used for speckle noise. The first model is a "signaldependent" stationary noise model; noise, assumed to be stationary, depends on the signal reflectance. This model is simply obtained by noting that , with being the signal reflectance and being a stationary random process independent of . The second model is a "signalindependent" model obtained by applying a logarithmic transform to the noisy image.
In addition, we consider the speckle signalindependent model. We use the estimation procedure described above for denoising the logarithmic transformed noisy image. The results are given in Figures 7(d) and 7(e).
By comparing the results of Figure 7, we observe that the PSNRs achieved are of the same order whatever the model. However, the denoising obtained with the additive independent noise model (logarithmic transform) has a better visual quality than that obtained with the additive signaldependent speckle model. In fact, one can note, from this figure, the ability of SigShrink functions to reduce speckle noise without impacting structural features and textural information of the image. Note also the gain in PSNR is larger than 10 dBs, performance of the same order as that of the best uptodate speckle denoising techniques ([17–22] among others).
4.3. Contrast Function
To conclude this section, we now present the SigShrink and SigStretch functions as contrast functions. Contrast functions are very useful in medical image processing. As a matter of fact, medical monitoring for arthroplasty (replacement of certain bone surfaces by implants due to lesions of the articular surfaces) requires 2D3D registration of the implant, and thus, requires knowing exactly the position of the implant contour. Precise edge detection is no easy task [23] because edge detection methods are sensitive to contrast (global contrast for the image and local contrast around a contour). The following briefly describes how to use SigShrinkSigStretch functions as contrast functions.
5. Conclusion
This work proposes the use of SigShrinkSigStretch functions for practical engineering problems such as image denoising, image restoration, and image enhancement. These functions perform adjustable adaptation of data in the sense that they can enhance or reduce the variability among data, the adaptation process being regular and invertible. Because of the smoothness of the function used (infinitely differentiable in ), the data adaptation is performed with little variability so that the signal characteristics are better preserved. The SigShrink and SigStretch methods are simple and flexible in the sense that the parameters of these classes of functions allow for a fine tuning of the data adaptation. This adaptation is nonparametric because no prior information about the signal is taken into account. A SUREbased optimization of the parameters is possible.
The denoising achieved by a SigShrink function is almost artifactfree due to the little variability introduced among data with close amplitudes. This artifactfree denoising is relevant for many applications, in particular for medical imagery where visual artifacts must be avoided. In addition, a fine calibration of SigShrink parameters allows noise reduction without impacting the signal characteristics. This is important when some postprocessing (such as a segmentation) must be performed on the signal estimate.
As far as perspectives are concerned, we can reasonably expect to improve SigShrink denoising performance by introducing interscale or/and intrascale predictor, which could provide information about the position of significant wavelet coefficients. It could also be relevant to undertake a complete theoretical and experimental comparison between SigShrink and Bayesian sigmoid shrinkage [14].
In addition, application of SigShrink to speech processing could also be considered. Since SigShrink yields denoised images that are almost artifactfree, would it be possible that such an approach denoises speech signals corrupted by AWGN without returning musical noise, in contrast to classical shrinkages using thresholding rules?
Another perspective is the SigShrinkSigStretch calibration of contrast in order to improve edge detection in medical imagery. Exact edge detection is necessary for 2D3D registration of images. Subpixel measurement of edge is possible by using, for example, the momentbased method of [24]. However, the method is very sensible to contrast. Low contrast varying images result in multiple contours; whereas high varying contrast in image leads to good precision for certain contour points but induces lack of detection for points in lower contrast zones. The idea is the use of the SigShrinkSigStretch functions for improving image contrast so as to alleviate edge detection in medical imagery. For instance, we can expect that combining SigShrinkSigStretch with edge detection methods such as [24] can lead to good subpixel measurement of the contour in an image.
Appendix
Proof of Proposition 4.1
for . The result then follows from (A.1), (A.6), and the fact that since .
Authors’ Affiliations
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