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Year : 2020  |  Volume : 16  |  Issue : 5  |  Page : 1171-1176

Image smoothing using regularized entropy minimization and self-similarity for the quantitative analysis of drug diffusion

1 Physical Examination Office, Health Commission of Shandong Province, Jinan, Shandong, China
2 Department of Infrastructure Management, Qilu Hospital of Shandong University, Jinan, Shandong, China
3 Department of Intervention Medicine, The Second Hospital of Shandong University, Jinan, Shandong, China
4 School of Mathematics, Shandong University, Jinan, Shandong, China
5 School of Mathematics and Physics, Qingdao University of Science and Technology, Shandong, China
6 College of Basic Medicine, Jining Medical University, Shenghua, China
7 Department of Radiology, Peking University Third Hospital; Beijing Key Laboratory of Magnetic Resonance Imaging Equipment and Technique, Beijing, China

Date of Submission20-May-2020
Date of Decision15-Jun-2020
Date of Acceptance10-Aug-2020
Date of Web Publication29-Sep-2020

Correspondence Address:
Shujun Fu
School of Mathematics, Shandong University, Jinan
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/jcrt.JCRT_656_20

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 > Abstract 

Background: Targetable drug delivery is an important method for the treatment of liver tumors. For the quantitative analysis of drug diffusion, the establishment of a method for information collection and characterization of extracellular space is developed by imaging analysis of magnetic resonance imaging (MRI) sequences. In this paper, we smoothed out interferential part in scanned digital MRI images.
Materials and Methods: Making full use of priors of low rank, nonlocal self-similarity, and regularized sparsity-promoting entropy, a block-matching regularized entropy minimization algorithm is proposed. Sparsity-promoting entropy function produces much sparser representation of grouped nonlocal similar blocks of image by solving a nonconvex minimization problem. Moreover, an alternating direction method of multipliers algorithm is proposed to iteratively solve the problem above.
Results and Conclusions: Experiments on simulated and real images reveal that the proposed method obtains better image restorations compared with some state-of-the-art methods, where most information is recovered and few artifacts are produced.

Keywords: Entropy, extracellular space, image smoothing, low rank, nonconvex minimization, nonlocal self-similarity, targetable drug delivery

How to cite this article:
Wang L, Meng X, Liu B, Liao S, Xiang S, Zhou W, Fu S, Li Y, Li Y, Han H. Image smoothing using regularized entropy minimization and self-similarity for the quantitative analysis of drug diffusion. J Can Res Ther 2020;16:1171-6

How to cite this URL:
Wang L, Meng X, Liu B, Liao S, Xiang S, Zhou W, Fu S, Li Y, Li Y, Han H. Image smoothing using regularized entropy minimization and self-similarity for the quantitative analysis of drug diffusion. J Can Res Ther [serial online] 2020 [cited 2021 Sep 22];16:1171-6. Available from: https://www.cancerjournal.net/text.asp?2020/16/5/1171/296442

 > Introduction Top

Targetable drug delivery is an important method for the treatment of liver tumor.[1] For the quantitative analysis of drug diffusion, the establishment of a method for information collection and characterization of extracellular space is developed by imaging analysis of magnetic resonance imaging (MRI) sequences.[2],[3] With current image capturing technologies, digital images are inevitably contaminated by noise and other interferences in the process of image acquisition and transmission. Since these interferences degrade visual quality, image smoothing remains an indispensable issue in many image-processing tasks, image smoothing is not only an image-processing tool to provide high quality image but also an important pre-processing step for many high-level visual problems including biometrics, image segmentation, and interventional imaging.[4],[5],[6],[7],[8],[9],[10]

In the field of image denoising, our purpose is to restore the original signal from its corrupted version with the best quality we can achieve. To realize such image quality, many outstanding denoising schemes have been proposed. Generally, denoising algorithms can be roughly divided into three categories: Spatial domain, transform domain, and hybrid methods.[4],[5]

Spatial domain methods, also known as spatial filters, estimate each pixel by executing the weighted average of its local/nonlocal neighbors, where their similarities determines the weight. Therefore, local and nonlocal filters are the two main forms of spatial filters. Mean filtering is designed to replace each pixel value in an image with the mean value of its neighbors.[4] Bilateral filter maintains the edges while smoothing the image by a nonlinear combination of nearby pixel values, combining gray levels or colors based on their geometric and photometric similarities, it also prefers nearby values in both domain and range.[11] Nonlocal means filter (NL-means) is introduced and achieves excellent results,[12] it takes a mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. In comparison with a local filter, after filtering this method results in higher resolution and less detail loss. Recently, several variants of the NL-means have been proposed to improve the adaptability of nonlocal filters.[13],[14],[15]

The transform domain method assumes that the image can be sparsely represented by some representation bases, such as the wavelet basis. To overcome visual artifacts, many data-driven based methods are proposed. The clustering K-singular value decomposition (KSVD) is a dictionary learning algorithm for creating a dictionary for sparse representations [16] via a singular-value decomposition (SVD) approach.

Spatial and transform domain methods have achieved great success in image denoising. However, their overall performance does not exceed the hybrid methods. Acting on the spatial and transform domains, hybrid methods have obtained remarkable achievements. Block-matching and 3D collaborative filtering [17] is a denoising method based on the fact that an image has a locally sparse representation in the transform domain. The sparsity is enhanced by grouping similar 2D image patches into 3D groups. In the principal component analysis (PCA) with local pixel grouping-PCA algorithm,[18] the input dataset for PCA is obtained from a patch-matching technique, the training samples are determined by selecting the pixels with similar properties within the local window. In addition, many excellent denoising algorithms have been proposed.[19],[20]

Recently, recovering the latent low-rank matrix structure from its noisy observation, methods based on the low-rank matrix approximation have attracted increasing attention in image processing due to its popularity and effectiveness.[21] For example, the spatially adaptive iterative singular-value thresholding algorithm [22] sparsely represents image patches using SVD and removes noise in image by the iterative singular-value shrinkage using the BayesShrink technique. Guo et al.[23] and Nejati et al.[24] adopted similar strategies, except that they employed SVD based on soft threshold processing. The method based on weighted nuclear norm minimization [25] has achieved excellent performance, which assigns different weights to singular values such that the values of soft thresholding becomes more reasonable. Presently, the above algorithms are considered the most advanced methods in the field of image denoising. A weighted p-norm minimization-denoising model has been proposed for denoising 3D MRI images.[26]

Medical images are usually rich in texture and complex in structure, and they are merely of an approximately low rank. In this paper, the nonlocal similarity on an image block level is used to process image blocks, and an optimization problem is solved based on the low-rank property of the matrix of grouped similar blocks rather than an entire image. Making full use of low-rank priors, nonlocal self-similarity (NSS), and regularized sparsity-promoting entropy, our proposed method obtained better image restoration results, whereby most lost information is recovered and few artifacts are produced. The main contributions of our research are summarized as follows.

Sparsity-promoting function produces a much sparser representation of images by solving a nonconvex minimization problem.[27],[28] A low-rank image restoration algorithm based on the entropy function of singular values has been proposed.[28] However, this entropy minimization method (ENM) does not satisfy the condition of the optimal solution.[28] A regularized ENM technique is proposed to ensure an optimal convergence solution in numerical computing.

After analysis of the ill-posedness of the ENM method, we therefore, proposed the block-matching regularized entropy minimization (BMREM) algorithm, which is based on the regularized entropy and NSS in the framework of low-rank minimization. Moreover, the alternating direction method of multipliers (ADMM) algorithm was proposed to iteratively solve the problem above.

The remainder of the paper is organized as follows. Related works of low-rank minimization is introduced in Section 2. The proposed BMREM algorithm is presented in Section 3. In Section 4, we revealed experimental results and stated the merits of the proposed approaches compared to other classic methods. Finally, we conclude this paper in Section 5.

 > Low-Rank Minimization Based On Entropy Function Top

Sparse representation and compressed sensing have achieved large success in image processing.[29],[30] SVD is often an effective approach to solve the low-rank model using some special thresholding operations on the singular values of the observation matrix.[21],[31],[32],[33]

Here, our objective is to recover an unknown clear matrix X from its observed degraded data , where H is a degeneracy operator, Y is observed degraded data, and ɲ is noise and interferences. For the low-rank matrix X, the nuclear norm approximation is a natural relaxation of rank minimization, which is difficult to solve as a NP-hard problem. However, in many practical applications the solution to the nuclear norm approximation is only suboptimal one of low-rank problem. Some nonconvex sparsity-promoting functions of singular-value vector σ(x) are proposed as a surrogate of the above nuclear norm-minimizing problem:[27],[28]

The concept of entropy originated from an equilibrium system in the field of thermodynamics, and then it gradually extended to the fields of image processing.[34],[35],[36],[37],[38] In statistics, information entropy is a measure of the degree of chaos in the distribution of random variables, and it is defined as the expected value of an amount of information in each event of a random distribution. The information theory reveals that the entropy of a random variable is maximized when its distribution is uniform. In other words, driving the distribution of a variable skewing toward a few of its values greatly, or equally making a signal more sparse, can decrease its entropy.[28]

An entropy function is introduced to promote sparsity in low-rank minimization:[28]

Actually, in the case of discrete setting the above entropy function only replaces the 0-norm of the Shannon entropy with a 1-norm. As shown in the following experiments, in the low-rank minimization problem (1) the entropy function will lead to sparser singular values of recovered image, so that the main components of the obtained image are more concentrated for ease of reduced information loss.

The low-rank minimization (1) based on the entropy function (2) is a nonconvex optimization problem. To solve this puzzle, the concave property of the entropy function and the iterative reweighted minimization technique were utilized, therefore leading to the proposal of the following ENM problem:[28]


Here l is iteration time.

 > Proposed Method Top

After analysis of the ill-posedness of the ENM method (3), we proposed, based on the regularized entropy and NSS in the framework of low-rank minimization a BMREM algorithm.

Regularized entropy minimization

When examining the weights in (4), we noticed that if . Then, in Lemma 3[28] is violated. The optimal solution in Lemma 4[28] will not be guaranteed, and one cannot expect a convergence solution in numerical computing. This incorrect setting leads to poor image recovery as shown in the following experiments.

To solve this problem, we proposed the following regularized weights:

A simple computation, therefore, gives

Where c>1 is a big constant such that . Further means the information of the singular-value vector σl as its entropy, means the self-information of . When is satisfied, we had , Thus, an optimal solution in the minimization problem (3) is guaranteed.[25],[27] This technique is the regularized ENM method.

Block-matching entropy minimization

The NSS is widely used in image processing and computer vision.[12],[20],[25] There are many repeated local patterns across a medical image. For a given block with information loss its nonlocal similar blocks can afford supplementary information for its upgraded reconstruction. A simpler and effective grouping of mutually similar blocks can be realized by image block matching to find blocks that exhibit a high correlation to a given one. The correlation between matrix rows and columns is naturally associated with the rank of the matrix, thus the formed group matrix by block matching is more likely to be of low rank than an entire image.

Considering the correlation between local image blocks, we proposed an image-smoothing algorithm called BMREM, where the SVD driven by the entropy function provides highly sparser representation of image data. Because bigger singular values mainly describe image structures, while smaller singular values are mainly related to interferences and noise, through singular-value shrinking we can recover the useful information of degraded image. This method is an iterative collaborative filtering from image block estimation to pixel estimation based on a sparse low-rank prior. The proposed method includes image block matching and grouping, singular-value shrinkage filtering, and aggregation.[25]

The proposed BMREM method first estimates latent image blocks, then it estimates each image pixel included in multiple image blocks to recover the entire image by aggregating estimated image blocks. Specifically, after the overlapping image blocks yi with size W×W are extracted from a degraded image y, for each image blocks yi block matching is performed to assemble a group of similar blocks based on a certain similarity criterion in a square L×L search window centered at yi. The reference block yi and its (N-1)-most similar blocks denoted by yj(j = 1, 2, ..,N–1) are chosen to construct a group matrix Yi= [yi, yi,...,yN–1] using each similar block as its column. In the grouped matrix Yi the corresponding columns from similar image blocks lead to a lower rank of the similar matrix,[25],[39] and consequently a highly sparse representation of image blocks is obtained from the entropy-driven SVD and shrinkage. Here, the singular energy of the similar matrix concentrates on a few foregoing bigger singular values, which benefits the recovery of image information from the observed image. Finally, the estimated image blocks by the proposed singular-value shrinkage are aggregated to obtain a restored image.

More specifically, to restore a degraded image with noisy pixels, for each of the grouped similar matrix Yi we solved the following matrix minimization model:

By introducing a variable E, we can employ the ADMM algorithm [40] to solve the above problem:

Where Li is a Lagrange multiplier and a positive constant. We can update {Ei, Xi, Li} sequentially by solving the following series of subproblems:

1. Update of Ei: Xi and Li are fixed, we solve

2. Update of Xi: Ei and Li are fixed, we first compute the SVD of can be obtained as in (8). Then we solve

3. Update of Li: Ei and Xi are fixed, Li is easily obtained by

The above optimization procedure is described in [Algorithm 1].

 > Results and Analysis Top

As a fundamental inverse problem in image processing and low-level vision, image smoothing by removing interferences reconstructs a clear image from the damaged data. Here, the proposed method (BMREM) is verified on test images (Cameraman, House, Barbara) by comparing it with some classic image-smoothing methods, including average (Average) and median (Median) filters, wavelet thresholding method (Wavelet),[4] nonlocal means method (NL-means),[12] and KSVD method.[16] All methods are implemented using MATLAB programming with a gray scale ranging from 0 to 255. In comparison to different image-smoothing methods, parameters in each method are selected with a tradeoff such that better quantitative and visual results are obtained.

To evaluate the quality of image recovery, we used the peak signal-to-noise ratio (PSNR) as a quantitative measure of indices. Although PSNR is usually inconsistent with human visual perception, it is widely used as a standard quantitative evaluation index of image quality, and is closely related to visual perception. Larger PSNR values mean higher quality of image recovery. The mathematical formulation of PSNR is defined as follows:

Where xj and represents the clear image and its estimated one at pixel j, respectively.

First, these smoothing methods are compared on three test images (Cameraman, House, and Barbara) with different image noise deviations (Sigma = 5, 10, and 20), respectively. To sum up, simple methods (Average, Median, and Wavelet) can obtain common denoising results, while the advanced methods (NL-means, KSVD, and BMREM) can better remove image noise and interferences. Using NSS, the proposed BMREM method can best recover images with different noise by first restoring the group matrix from the matched image blocks by low-rank minimization. The lower rank of the group matrix is key to restoring information from the observed images in the low-rank minimization with the sparsity-promoting entropy function. [Table 1] shows the result of the quantitative evaluation of PSNR (dB) on test images, which verifies the above demonstration.
Table 1: Quantitative evaluation (peak signal-to-noise ratio [dB]) of entropy based smoothing methods

Click here to view

Second, in [Figure 1] we displayed the proposed BMREM method in smoothing two frames in a Ktran scanned MRI series of a rabbit inserted in gadolinium-DTPA (diethylenetriamine penta-acetic acid) (Gd-DPTA). Here, healthy New Zealand white rabbits were used. After the inguinal area of a rabbit is cut open, its femoral artery is punctured under direct supervision, and a 4F vertebral artery catheter and microcatheter are inserted into the left hepatic artery. Then, it is transferred to our imaging center for magnetic resonance scanning using the Ktran sequence. After that, 1 ml of 10% gadolinium agent was injected via the hepatic artery and portal vein, and 1 ml embolizing microspheres of diameter 40–100 μm were injected, and once again the Ktran sequence scanning was performed. After 30 min of metabolism, 0.8 ml Gd-DPTA was injected through an indwelling needle around the auricular vein (calculated according to 0.2 ml/Kg, 0.5 ml, with an addition of 0.3 ml addition to avoid some loss), followed by the Ktran sequence scanning.
Figure 1: Comparison of smoothed magnetic resonance imaging images with noise and interferences. (a and c) two observed magnetic resonance imaging images; (b and d) corresponding smoothed results by the block-matching regularized entropy minimization method, respectively

Click here to view

As shown in these restored images, the proposed BMREM filter obtains relatively better results specially in the preservation of important features of MRI images, which also verifies the higher performance of the proposed method.

 > Conclusion Top

Image smoothing is an important step in information collection and characterization of extracellular space for the quantitative analysis of drug diffusion, while preserving the image features in the imaging analysis of MRI sequences. In this paper, we propose a BMREM algorithm to restore the damaged part in scanned digital images using priors of low rank, NSS, and regularized sparsity-promoting entropy. An ADMM algorithm was used to iteratively solve the above nonconvex minimization problem. Experiments on simulated and real images verified the effectiveness of the proposed algorithm. In the future, we will develop an effective automatic algorithm to estimate the dynamic flow field of drug diffusion.


The research has been supported in part by the National Natural Science Foundation of China (61671276, 11971269), the Natural Science Foundation of Shandong Province of China (ZR2019MF045), and the National Science Fund for Distinguished Young Scholars (61625102).

Financial support and sponsorship


Conflicts of interest

There are no conflicts of interest.

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