 Research Article
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Smooth Adaptation by Sigmoid Shrinkage
EURASIP Journal on Image and Video Processing volume 2009, Article number: 532312 (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).
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
The family of realvalued functions defined by [1]
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.
Note that tends to , which is a hardthresholding function defined by
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 .
In addition, it is easy to check that, in Cartesiancoordinates, the points , and belong to the curve of the function for every . Indeed, according to (1), we have and for any . It follows that parameterizes the curvature of the arc , that is, the arc of the SigShrink function in the interval ]−[. This curvature directly relates to the attenuation degree we want to apply to the wavelet coefficients. Consider the graph of Figure 1, where a SigShrink function is plotted in the positive half plan. Due to the antisymmetry of the SigShrink function, we only focus on the curvature of arc . Let be the intersection between the abscissa axis and the tangent at point to the curve of the SigShrink function. The equation of this tangent is . The coordinates of point are . We can easily control the arc curvature via the angle, denoted by , between vector , which is fixed, and vector , which is carried by the tangent to the curve of at point . The larger , the stronger the attenuation of the coefficients with amplitudes less than or equal to . For a fixed , the relation between angle and parameter is
It easily follows from (3) that ; when , then and is the hardthresholding function of (2). From (3), we derive that can be written as a function of and as follows:
In practice, when is fixed, the foregoing makes it possible to control the attenuation degree we want to impose to the data in by choosing , which is rather natural, and calculating according to (4). Since we can control the shrinkage by choosing henceforth denotes the SigShrink function where is given by (4). This interpretation of the SigShrink parameters makes it easier to find "nice" parameters for practical applications. Summarizing, the SigShrink computation is performed in three steps:

(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).
Hereafter, the terms "attenuation degree" and "threshold" designate and , respectively. In addition, the notation will be preferred for calculations and statements. The notation , introduced just above, will be used for practical and experimental purposes since the attenuation degree is far more natural in practice than parameter . Some SigShrink graphs are plotted in Figure 2 for different values of the attenuation degree (fixed threshold ).
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 .
In order to estimate , we assume that an orthonormal transform, represented by an orthonormal matrix , is applied to . The outcome of this transform is the sequence of coefficients
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
Consider the WaveShrink estimation described in Section 3.1. The risk function or cost used to measure the accuracy of a WaveShrink estimator of is the standard MSE. Since the transform is orthonormal, this cost is
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.
The quantity , where denotes norm and
is an unbiased estimator of the risk , where is a SigShrink function.
Proof.
From [3, Theorem 1], we have that
where can be any differentiable shrinkage function that does not explode at infinity (see [3] for details). A SigShrink function is such a shrinkage function. Taking into account that the derivate of the SigShrink function is
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).
The tests are carried out with the following values for the noise standard deviation: . For every value , 25 tests have been performed based on different noise realizations. Every test involves performing a DWT for the tested image corrupted by AWGN, computing the optimal SURE parameters (SigShrink and LDSigShrink), applying the SigShrink function with these parameters to denoise the wavelet coefficients, and building an estimate of the corresponding image by applying the inverse DWT to the shrunken coefficients. For every test, the PSNR is calculated for the original image and the denoised image. The PSNR (in deciBel unit, dB), often used to assess the quality of a compressed image, is given by
where stands for the dynamics of the signal, in the case of 8 bitcoded images.
Table 1 gives the following statistics for the 25 PSNRs obtained by the SURE SigShrink, SURE LDSigShrink, and "sum of DOGs" method: average value, variance, minimum, and maximum. Average values and variances for the SURE SigShrink and SURE LDSigShrink parameters are given in Tables 2, 3, 4, and 5.
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.
From Tables 2, 3, 4, and 5, we observe (concerning the optimal SURE SigShrink parameters) that

(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.
The SigStretch function, denoted , is defined as the inverse of the SigShrink function and is given by
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 .
Figures 4 and 5 give denoising examples for different values of and . The denoising concerns the "Lena" image corrupted by AWGN with standard deviation (Figure 3). The "Haar" wavelet and 4 decomposition levels are used for the wavelet representation (SWT). The classical minimax and universal thresholds [4] are used. In these figures, stands for the SigShrink function which parameters are and .
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.
Note that the SURELET "sum of DOGs" parameterization does not allow for such a heuristically adjustable denoising because the physical interpretation of its parameters is not explicit, whereas the SigShrink and the standard hard, soft, NNG, and SCAD thresholding functions mentioned in Section 3.1 depend on parameters with more intuitive physical meaning (threshold height and an additional attenuation degree parameter for SigSghink). Denoising examples achieved by using the hard, soft, NNG, and SCAD thresholding functions are given in Figure 6, for a comparison with the SigShrink denoising. The minimax threshold is used for the denoising (the results are even worse with the universal threshold). As can be seen in this figure, artifacts are visible in the image denoised by using hard thresholding, whereas images denoised by using soft, NNG, and SCAD thresholding functions tend to be over smoothed. Numerical comparison of the denoising PSNRs performed by SigShrink and these standard thresholding functions can be found in [1].
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.
To conclude this section, note that shrinkages and regularization procedures are linked in the sense that a shrinkage function solves to a regularization problem constrained by a specific penalty function [16]. Since SigShrink functions satisfy assumptions of [16, Proposition 3.2], the shrinkage obtained by using a function can be seen as a regularization approximation [7] by seeking the vector that minimizes the penalized least squares
where is the penalty function associated with is defined for every by
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.
We begin with the speckle signaldependent model. The denoising procedure then involves applying an SWT to the noisy image, estimating the noise standard deviation in each SWT subband by the robust Median of the Absolute Deviation ((MAD), normalized by the constant 0.6745) estimator [4], shrinking the wavelet coefficients by using a SigShrink function adjusted with the minimax threshold [4], and reconstructing an estimate of the signal by means of the inverse SWT. The results obtained for the "Lena" image corrupted by speckle noise (Figure 7(a)) are shown in Figures 7(b) and 7(c).
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.
The SigShrink function applies a penalized shrinkage to data with small amplitudes. The smaller the data amplitude, the higher the attenuation imposed by the SigShrink function. Thus, a SigShrink function is a contrast enhancing function; this function increases the gap between large and small values for the pixels of an image. As a consequence, a SigStretch function reduces the contrast by lowering the variation between large and small pixel values in the image. Figure 8 gives the original "Lena" image as well as the SigShrink and SigStretch shrunken images. This figure highlights that the contrast of the image can be smoothly adjusted (enhancement, reduction) by applying SigShrink and SigStretch functions without introducing artifacts. Note that, as for denoising, SigShrink allows for choosing the attenuation degree imposed to the data, when the threshold height is fixed. Figure 9 illustrates the variability that can be attained by varying the SigShrink attenuation degree for enhancing the contrast of a fluoroscopic image.
To conclude this section, we now illustrate the combination of SigShrink denoising and contrast enhancement for an ultrasonic image of breast cancer. The combination involves denoising the image by using the SigShrink method in the wavelet domain. A SigShrink function is then applied to the denoised image to enhance its contrast. The results are presented in Figure 10. It is shown that SigShrink denoises the image and preserves feature information without introducing artifacts. The parameter is chosen so as to avoid visual artifacts. Different thresholds are experimented to highlight how we can progressively reduce noise without affecting the image textural information. The threshold is the detection threshold of [8]. This threshold is smaller than the minimax threshold. It is close to when the sample size is large.
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
Because is antisymmetric, has the form
for every real value and where is such that
Therefore, for any real value . We thus have
which is also equivalent to
It follows that
which leads to
for . The result then follows from (A.1), (A.6), and the fact that since .
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Atto, A.M., Pastor, D. & Mercier, G. Smooth Adaptation by Sigmoid Shrinkage. J Image Video Proc 2009, 532312 (2009). https://doi.org/10.1155/2009/532312
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Keywords
 Wavelet Coefficient
 Wavelet Domain
 Speckle Noise
 Contrast Function
 Wavelet Shrinkage