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Noiseless Codelength in Wavelet Denoising

Abstract

We propose an adaptive, data-driven thresholding method based on a recently developed idea of Minimum Noiseless Description Length (MNDL). MNDL Subspace Selection (MNDL-SS) is a novel method of selecting an optimal subspace among the competing subspaces of the transformed noisy data. Here we extend the application of MNDL-SS for thresholding purposes. The approach searches for the optimum threshold for the data coefficients in an orthonormal basis. It is shown that the optimum threshold can be extracted from the noisy coefficients themselves. While the additive noise in the available data is assumed to be independent, the main challenge in MNDL thresholding is caused by the dependence of the additive noise in the sorted coefficients. The approach provides new hard and soft thresholds. Simulation results are presented for orthonormal wavelet transforms. While the method is comparable with the existing thresholding methods and in some cases outperforms them, the main advantage of the new approach is that it provides not only the optimum threshold but also an estimate of the associated mean-square error (MSE) for that threshold simultaneously.

1. Introduction

We can recognize different phenomena by collecting data from them. However, defective instruments, problems with the data acquisition process, and the interference of natural factors can all degrade the data of interest. Furthermore, noise can be introduced by transmission errors or compression. Thus, denoising is often a necessary step in data processing and various approaches have been introduced for this purpose. Some of these methods, such as Wiener filters, are grouped as linear techniques. While these techniques are easy to implement, their results are not always satisfactory. Over past decades, researchers have improved the performance of denoising methods by developing nonlinear approaches such as [16]. Although these approaches have succeeded in providing better results, they are usually computationally exhaustive, hard to implement, or use particular assumptions either on the noisy data or on the class of the data estimator.

Thresholding methods are alternative approaches to the denoising problem. The thresholding problem is first formulated in [7] where VisuShrink is introduced. This threshold is a nonadaptive universal threshold and depends only on the number of data points and noise variance. VisuShrink is a wavelet thresholding method which is both simple and effective in comparison with other denoising techniques. When an orthogonal wavelet basis is used, the coefficients with small absolute values tend to be attributed to the additive noise. Taking advantage of this property, finding a proper threshold, and setting all absolute values of coefficients smaller than the threshold to zero can suppress the noise. The main issue in such approaches is to find a proper threshold. In using the thresholding method for image denoising, the visual quality of the image is of great concern. An improper threshold may introduce artifacts and cause blurring of the image. One of the first soft thresholding methods is SureShrink [8] which has a better effect on the image than Visushrink in many cases. This method uses a hybrid of the universal threshold and the SURE (Stein's Unbiased Risk Estimator) threshold. The SURE threshold is chosen by minimizing Stein's estimate.

In this research we focus on the mean-square error (MSE) associated with the denoising process. The importance of this error in any signal reconstruction and estimation is inevitable [9, 10]. After all, in any estimation process, in this case reconstructing the denoised signal from the noisy one, the major goal is to achieve the original signal as much as possible, and the MSE is one of the most used criteria for evaluation purposes. To find the optimum threshold we estimate the MSE of a set of completing thresholds and choose the one that minimizes this error. The fundamentals of MSE estimation are similar to the method proposed in Minimum Noiseless Description Length (Codelength) Subspace Selection (MNDL-SS) [10]. Each subspace in this approach keeps a subset of the coefficients and discards the rest. Thresholding also produces a subspace that includes a set of coefficients that are being kept. On the other hand, it was shown in [10] that an estimate of noiseless description length (NDL) can be provided for each subspace by using the noisy data itself. We also show that in the process of estimating the NDL, the estimates of MSE is also provided. Furthermore comparison of the NDL of competing subspaces is equivalent to comparison of their MSEs. Therefore, the method presented in this paper is denoted by MNDL thresholding. The competing thresholds that are the sorted coefficients generate competing subspaces. For these subspaces the estimate of MSEs are provided and compared. The optimum threshold is associated with the subspace with minimum MSE (equivalently minimum NDL). Because of the particular choice of competing subspaces in MNDL thresholding, the effect of the additive noise is different from that in MNDL-SS. While the independence of the additive noise in MNDL-SS is the main advantage in estimating the desired NDL, in the case of thresholding, the additive noise is highly dependent. The main challenge in this work is to develop a method for NDL estimation acknowledging the presence of this noise dependence. We provide a threshold that is a function of the noise variance , the data length , and the observed noisy data itself.

The paper is arranged as follows. Section 2 describes the considered thresholding problem. Section 3 briefly describes the fundamentals of the existing MNDL subspace selection. Section 4 introduces the MNDL thresholding approach. Hard and soft MNDL thresholdings are presented in Sections 5 and 6. Section 7 provides the simulation results and Section 8 is our conclusion.

2. Problem Statement

Noiseless data of length has been corrupted by an additive noise:

(1)

where is an independent and identically distributed (i.i.d) Gaussian random process with zero mean and variance . (The method presented here is for real data. However, it can also be used for complex data. )In the considered denoising process, we project the noisy data into an orthogonal basis. The goal is to provide the optimum threshold for the resulting coefficients that minimizes the mean square error.

Assume that the noiseless data vector is generated by space . The space can be expanded by orthogonal basis vectors:

(2)

where is the inner product of vectors and .

The data in (1) is represented in this basis as follows:

(3)
(4)

where is the th coefficient of the noiseless data, is the -th coefficient of the noisy data, and is the -th coefficient of the additive noise. Note that since the basis vectors of are orthogonal, is also a sample of Gaussian distribution with zero mean and variance, .

The thresholding approach uses the available noisy coefficients, , to provide the best estimate of the noiseless coefficients denoted by . There are two general thresholding methods: hard and soft thresholdings. Hard thresholding eliminates or keeps the coefficients by comparing them with the threshold

(5)

where is the hard threshold. Soft thresholding eliminates the coefficients below and reduces the absolute value of the rest of the coefficients:

(6)

There are different approaches for calculation of the proper and . In this paper, we expand the existing theory of the MNDL subspace selection method in [10] to provide new hard and soft thresholding methods.

Important Notation

In this paper a random variable is denoted by a capital letter, such as and , while a sample of that random variable is represented by the same letter in lower case such as and .

3. MNDL Subspace Selection (MNDL-SS)

The MNDL Subspace Selection (MNDL-SS) approach has been introduced in [10]. This approach addresses the problem of basis selection in the presence of a noisy data. In MNDL-SS, competing subspaces represent a projection of the noisy data on a complete orthogonal basis such as an orthogonal wavelet basis. each subspace contains a subset of the basis. The subspace keeps the coefficients of the noisy data in that subset and sets the rest of the coefficients to zero. Among competing subspaces, MNDL-SS chooses the subspace that minimizes the description length (codelength) of "noiseless" data. In this setting subspaces of the space are chosen as follows: Each is a subspace of that is spanned by the first elements of the bases. The estimate of the noiseless coefficients in subspace is

(7)

and the estimate of the noiseless data in , , is

(8)

In each subspace the description length (codelength) of the noiseless data is defined as [10] (this criterion is different from MDL criterion and the explanation is provided in [10])

(9)

where is the reconstruction error (the equality of the error in the time domain and the error in coefficients of data is the result of the Parseval's Theorem):

(10)

which is a sample of random variable .

The optimum subspace can be chosen by minimizing the average description length of noiseless data among the competing subspaces. Minimizing the average of noiseless data length in (9) is equivalent to minimizing the mean square error (MSE) in the form of as the term is a constant and not a function of :

(11)

MNDL-SS estimates the MSE for each subspace by using the available data error in that subspace. The data error is defined in the following form:

(12)

which is a sample of random variable . MNDL-SS studies the structure of the two random variables and and uses the connection between these two random variables to provide an estimate of the desired criterion for different .

4. MNDL Thresholding

To use the ideas of subspace selection in thresholding, we first have to explain how a particular choice of subspaces serves the problem of thresholding. In MNDL-SS, the competing subspaces are chosen a priori and are not functions of the observed data. However, if the method is going to be used for thresholding, forming the competing subspaces is based on the observed data. In this case, we first sort the bases based on the absolute value of the observed noisy coefficients . This will dictate a particular indexing on the bases such that

(13)

Therefore, the first subset represents the basis associated with the largest absolute value of the coefficients. The subset with two coefficients includes the two bases with the largest absolute value of the sorted coefficients and so on. The subspace includes of the basis and represents the first largest absolute values of the coefficients and as a result this subspace represents thresholding with a threshold value of .

Back to the MNDL-SS, the subspace that minimizes the average codelength of the noiseless data (equivalent to the subspace MSE in (11)) is the optimum subspace. Due to the indexing in the form of (13), the choice of this subspace results in the optimum threshold .

To estimate the MSE of the subspaces, we follow the fundamentals of the MNDL-SS method. Due to the random choice of subspaces in MNDL-SS, the random variables s that represent the additive noise of the coefficients in (4) are independent Gaussian random variables. In MNDL thresholding, the additive noise of coefficients is still Gaussian. However, due to the particular choice of the index for the coefficients, s are no longer independent. This will cause a major challenge in estimating the MSE of subspaces and is the main focus of this paper.

5. MNDL Hard Thresholding

In [10] it is shown that the expected value of the reconstruction error and that of the data error in subspace can be written in the form of

(14)
(15)

where is the -norm of the discarded coefficient vector in subspace :

(16)

and the noise parts are

(17)
(18)

where is the Gaussian random variable with samples defined in (3).

If the noise parts and are available, then by estimating the expected value of the data error with the available sample we have

(19)

and then the estimate of the desired MSE in (14) is

(20)

The main challenge in MNDL thresholding is in calculating and in (18) and (17). In MNDL-SS, due to the independence of s in (4), random variables and are Chi-square random variables and calculation of the expected values of these terms is straight forward. However, in MDNL thresholding, since the s are not independent, calculation of these expected values is not easy. In the following section we focus on estimating these desired expected values for the case of thresholding.

5.1. Estimate of MSE in MNDL Hard Thresholding

In order to calculate the expected value of the additive noise in (17) and (18), we need to study the noise effects that are associated with the sorted noisy coefficients. Each in (4) has a Gaussian distribution with mean and variance . Figure 1 shows the distribution of , , and . The expected value of the noise part of under the condition is as follows:

Figure 1
figure1

Distributions of θ(m + 1), θ(m), and θ(m – 1).

(21)

where

(22)
(23)

and is

(24)

where and and are defined as

(25)
(26)

Details of the calculation are provided in Appendix .

Therefore, for the desired noise parts we have

(27)
(28)

Similarly we have

(29)
(30)

The noise part of the MSE in (27) and the noise part of the expected value of data error in (29) are dependent on , that are not available. We suggest estimating them by using the available data as follows [11].

  1. (i)

    Generate Gaussian vectors of data length with variance of the additive noise.

  2. (ii)

    Sort the absolute value of the associated noise coefficients, .

  3. (iii)

    Find the estimate of the expected value of this vector by averaging over 50 samples of these vectors:

    (31)
  4. (iv)

    Estimate as follows:

    (32)
  1. (i)

    Estimate and by replacing with in (22) and (23):

    (33)
(34)

The estimates of the noise parts in (27) and (29) are

(35)
(36)

5.2. Calculating the Threshold

By using the provided noise part estimates in (35) and (36) we can estimate the desired MSE for subsets of different order with the following steps.

  1. (i)

    Estimate the noise parts and using (36) and (35).

  2. (ii)

    Estimate the MSE in (20) as follows:

    (37)

In MNDL thresholding the goal is to find by minimizing the MSE in (11). Here we provide an estimate of using the MSE estimate:

(38)

6. MNDL Soft Thresholding

In some applications, such as image denoising, soft thresholding generally performs better and provides a smaller MSE than hard thresholding [12]. In soft thresholding, not only are the values smaller than the threshold set to zero, but also the value of coefficients larger than the threshold is also reduced by the amount of the threshold. Thus, we need to take into account this changing level of coefficients in MSE estimation. For MNDL soft thresholding we follow the same procedure as in MNDL hard thresholding. Here, the MSE in subspace is

(39)

where is the smallest coefficient in subspace (which is ).

6.1. Noise Effects in the MSE

The noise part of the MSE in (39) is

(40)

where s are the associated noise parts of coefficients s in subspace . The expected value of the noise part of under the condition is

(41)

where and are defined similar to those in (22) and (23), and is defined as

(42)

where and are defined as

(43)
(44)

Details of this calculation are provided in Appendix .

Using the estimates of and from (33) and (34), the estimate of MSE's noise part in (40) is

(45)

6.2. Estimate of the Noiseless Part of MSE

To complete the estimation of MSE in (39), we need also to estimate the noiseless part using the data error. The expected value of the data error in the case of soft thresholding is

(46)

(Details are provided in Appendix ). The last component is the same as noise part in MNDL hard thresholding in (29) and can be estimated by using (36). Therefore, by estimating with its available sample , from (46) we have

(47)

where is defined in (24) and and are defined in (33) and (34).

6.3. Calculating the Threshold

The two components of MSE in (39) were estimated in previous sections. Therefore, the MSE can be estimated as follows.

  1. (i)

    The noise part is estimated by using (45).

  2. (ii)

    The noiseless part is estimated by using (47).

  3. (iii)

    The estimate of the MSE in (39) is the sum of (45) and??(47):

(48)

Similar to MNDL hard thresholding, the optimum subspace is the one for which the estimate of the MSE is minimized:

(49)

6.4. Subband-Dependent MNDL Soft Thresholding

In image denoising, soft thresholding methods outperform hard thresholding methods in terms of MSE value and visual quality. In addition, it has been shown that subband-dependent thresholding performs better than universal thresholding methods [8]. In the subband-dependent method, a different threshold is provided for every subband of the wavelet transform. Here, we present the subband-dependent MNDL soft thresholding method. In every subband the MSE is estimated as a function of its subspaces. The subspace and its equivalent threshold that minimizes the MSE are chosen. The process of subband dependent MNDL thresholding with wavelet thresholds is as follows.

  1. (i)

    The discrete wavelet transform of the image is taken.

  2. (ii)

    In every subband the MSE, in (39), is estimated as a function of the : the noise part is estimated using (45), and the noiseless part is estimated using (47). MSE estimate is the sum of the noiseless part and the noise part estimates.

  3. (iii)

    In each subband, the MSE is minimized over values of , and in (38) is chosen, where , and is the number of coefficients in the subband.

  4. (iv)

    In each subband, the th largest absolute value of the coefficients is the optimum threshold.

  5. (v)

    The image is denoised using the subband thresholds.

  6. (vi)

    The inverse discrete wavelet transform is taken.

The unknown noise variance is estimated by the median estimator, , where MAD is the median of absolute value of the wavelet coefficients at the finest decomposition level (the diagonal direction of decomposition level one).

In the following section we provide simulation results of the method. The low complexity of this algorithm is an additional strength of the method. Our future plan is to utilize the approach for potential applications in areas such as biomedical engineering [13].

7. Simulation Results

We first demonstrate the performance of MNDL hard and soft thresholding by using two well-known examples in wavelet denoising. The first signal is the Blocks signal of length 1024 with few nonzero coefficients. The signal along with its wavelet coefficients is shown in Figure 2. Wavelet transform employs Daubechies's wavelet with eight vanishing moments with four scales of orthogonal decomposition [14]. The other signal is the Mishmash signal of length 1024 with no nonzero coefficients in Figure 3.

Figure 2
figure2

(a) Blocks signal with length 1024, (b) wavelet coefficients, (c) noisy blocks signal with additive white noise, σ w = 1, and (d) noisy coefficients.

Figure 3
figure3

(a) Mishmash signal with length 1024, (b) wavelet coefficients, (c) noisy Mishmash signal with additive white noise, σ w = 5, and (d) noisy coefficients.

The MSE and its estimates using the existing MNDL-SS method [10] and the developed MNDL hard thresholding method are shown in Figure 4. The that minimizes the unavailable MSE and its estimate, , with these approaches are provided in Table 1 for different noise variances. The results in this table and the rest of the results in this section are averages of five runs. As the figure and the table show, as was expected MNDL thresholding outperforms the MNDL-SS approach.

Table 1 Comparing and its estimates using MNDL hard thresholding and MNDL-SS methods for the Blocks signal.
Figure 4
figure4

Desired unavailable MSE (solid red line), and its estimate using MNDL hard thresholding (- -) and MNDL-SS (. -) as a function of m. (a) Blocks signal with σ w = 1, (b) Blocks signal with σ w = 5, (c) Mishmash signal with σ w = 1, and (d) Mishmash signal with .

Table 2 compares the MSE of the proposed hard thresholding method with that of two hard thresholding methods, VisuShrink and MDL. The comparison includes the optimum hard MSE, which represents the minimum MSE when the noisy coefficients are used as hard thresholds, along with the resulting MSE of different approaches. As the table shows, in most cases, MNDL hard thresholding provides the minimum MSE among the approaches.

Table 2 MSE comparison of different thresholding methods with the MNDL hard thresholding approach.

The MSE and its estimate with MNDL soft thresholding are shown in Figure 5. The results in this figure are for Blocks and Mishmash signals and for two different levels of the additive noise. As the figure shows, MSE estimates are very close to the MSE itself.

Figure 5
figure5

The MSE and its estimate in MNDL soft thresholding as a function of m. (a) Blocks signal, σ w = 1, (b) Blocks signal with σ w = 5, (c) Mishmash signal with σ w = 1, and (d) Mishmash signal with σ w = 5.

The MSE results for soft threshoding are compared in Table 3. The table provides optimum MSE with both optimum thresholding and optimum subband thresholding along with the results for Sureshrink and MNDL soft thresholding. As the table shows for Blocks the MNDL subband-dependent has the best results while for Mishmash MNDL soft thresholding outperforms the other approaches in almost all cases. While here we have shown the simulation results for two of six test signals in [8], the results for the other two signals (Heavy sine, Doppler) are similar to those of the provided signals.

Table 3 MSE for Optimum soft thresholding; Optimum Subband-dependent soft thresholding; MNDL soft thresholding, MNDL Subband-dependent soft thresholding, Sureshrink.

7.1. Image Denoising

There are many image denoising approaches, such as recent work in [15, 16]. These approaches have succeeded in providing good results. They usually use a particular assumption either on the noisy image and/or on the class of the data estimator. A well-known image denoising thresholding approach is BayesShrink. BayesShrink [12] is a thresholding method that is also widely used for image denoising. This method attempts to minimize the Bayes' Risk Estimator function assuming a prior Generalized Gaussian Distribution (GGD) and thus yields a data adaptive threshold [17]. Note that our method does not make any particular assumption on the data and is not especially proposed for image denoising. Here we use the approach for images as an example of a class of two-dimensional data.

To explore the application of MNDL soft thresholding in image denoising we use four images: Cameraman (a sample of a soft image), Barbara (a sample of a highly detailed image), Lena, and Peppers. These images are shown in Figure 6 and with size 512512. The wavelet transform employs Daubechies's wavelet with eight vanishing moments and with four scales of orthogonal decomposition.

Figure 6
figure6

Test images. From top left, clockwise: Peppers, Lena, Cameraman, and Barbara.

In Table 4, we compare the MSE of the MNDL with two well-known thresholding methods: BayesShrink [12] and SureShrink [8]. All these thresholds are soft subband-dependent. As the table shows, MNDL thresholding performs better than SureShrink in most cases and is comparable with the BayesShrink. The MNDL soft thresholding is compared visually with BayesShrink and SureShrink in Figure 7. As the figure shows, the ringing effect at the edges of the image with the MNDL soft thresholding is less than that with the BayesShrink approach. The importance of the new approach is that it can provide an estimate of MSE simultaneously. Note that it can also provide estimate of MSE for other thresholding methods as follows. Find the closest absolute value of the coefficients to the given threshold and use the index of that coefficient and check the estimate of MSE for the associated . Table 5 shows the optimum subband threshold for Cameraman and Table 6 shows the thresholds for BayesShrink and MNDL. As the tables show, the thresholds of MNDL are slightly larger than the optimum ones. On the other hand the Bayes thresholds are smaller than the optimum ones, especially for the coarsest level. While the MSE at this noise level is almost the same for these methods, the thresholds indicate that MNDL keeps fewer coefficients compared to BayesShrink and its threshold is much closer to the optimum one.

Table 4 MSE for various images with Optimum soft MSE, MNDL soft thresholding, BayesShrink, and SureShrink. Averaged over five runs.
Table 5 Optimum threshold of subbands for Cameraman and .
Table 6 (a): MNDL threshold, (b): BayesShrink threshold of subbands for Cameraman and .
Figure 7
figure7

(a) Noiseless image, (b) Noisy image with σ w = 15, (c) Optimum soft thresholding, (d) Bayeshrink, (e) Sureshrink, (f) MNDL soft thresholding.

8. Conclusion

We proposed thresholding method based on the MNDL-SS approach. This approach uses the available data error to provide an estimate of the desired noiseless codelength for comparison of competing subspaces. In this approach, the statistics of the data error plays an important role. Unlike MNDL-SS, in MNDL thresholding the involved additive noises of the error are highly dependent. The main challenge of this work was to estimate the desired criterion in the presence of such dependence. We developed a method to estimate the desired criterion for the purpose of thresholding.

Experimental results show that the proposed MNDL hard thresholding method outperforms VisuShrink most of the time and the proposed MNDL soft thresholding method outperforms SureShrink most of the time. Application of MNDL soft thresholding for image denoising is also explored and the method proves comparable with the BayesShrink approach. Unlike the image denoising approaches, with MNDL thresholding no assumption on the structure of the signal is necessary. The main advantage of the method is in estimating both the desired noiseless description length and the mean square error (MSE). The calculated MSE estimate provides a quantitative quality measure for the proposed threshold simultaneously. An additional strength of the approach is its ability to estimate the MSE for any given threshold and it can be used for quality evaluation of any thresholding method.

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Appendices

Appendices

A. in Hard and Soft Thresholding

The expected value of the data error in hard thresholding, in (12), is

(A1)
(A2)

Since the noiseless coefficients s are independent of the noise part s, the third term becomes zero and we conclude with (15).

The expected value of the data error in soft thresholding is

(A3)
(A4)

The second part of the expected value of the data error in (A.4) is the same as the expected value of the data error in the hard thresholding in (A.1). Therefore, (A.4) can be written in the form of (46).

B. Calculating in MNDL Hard Thresholding

The distribution of noise coefficients is a Gaussian one and we have . On the other hand, for a sorted version of coefficients we have while . Therefore, the following extra condition holds on the noise coefficients:

(B1)

Under the above condition, the desired conditional expected value is

(B2)

where the numerator is

(B3)

and the denominator is

(B4)

where and are

(B5)
(B6)

The conditional expectation of in (B.2) can be simplified to

(B7)

where , , and are defined in (22), (23), and (24).

Any in the paper is the conditional expectation in (B.2). For notation simplicity, the condition of the expectation is eliminated throughout the paper.

C. Calculating in MNDL Soft Thresholding

Similar to calculating the conditional expected value of for MNDL hard thresholding, under the same condition , we have

(C1)

where the denominator is provided in (B.4) and the numerator is

(C2)

There are three integrals in . The first integral is

(C3)

The second integral is

(C4)

and the third integral is

(C5)

The numerator of (C.1) is calculated by adding up (C.3), (C.4) and (C.5). Therefore, the simplified version of in (C.1) is

(C6)

where , , and are defined in (22), (23), and (42).

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Beheshti, S., Fakhrzadeh, A. & Krishnan, S. Noiseless Codelength in Wavelet Denoising. EURASIP J. Adv. Signal Process. 2010, 641842 (2010). https://doi.org/10.1155/2010/641842

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Keywords

  • Mean Square Error
  • Additive Noise
  • Minimum Mean Square Error
  • Image Denoising
  • Thresholding Method