

ORIGINAL ARTICLE 

Year : 2020  Volume
: 16
 Issue : 5  Page : 11711176 

Image smoothing using regularized entropy minimization and selfsimilarity for the quantitative analysis of drug diffusion
Lu Wang^{1}, Xiangbin Meng^{2}, Bin Liu^{3}, Shenghai Liao^{4}, Shibing Xiang^{4}, Weifeng Zhou^{5}, Shujun Fu^{4}, Yixiao Li^{6}, Yuliang Li^{3}, Hongbin Han^{7}
^{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 Submission  20May2020 
Date of Decision  15Jun2020 
Date of Acceptance  10Aug2020 
Date of Web Publication  29Sep2020 
Correspondence Address: Shujun Fu School of Mathematics, Shandong University, Jinan China
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/jcrt.JCRT_656_20
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 selfsimilarity, and regularized sparsitypromoting entropy, a blockmatching regularized entropy minimization algorithm is proposed. Sparsitypromoting 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 stateoftheart methods, where most information is recovered and few artifacts are produced.
Keywords: Entropy, extracellular space, image smoothing, low rank, nonconvex minimization, nonlocal selfsimilarity, 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 selfsimilarity for the quantitative analysis of drug diffusion. J Can Res Ther 2020;16:11716 
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 selfsimilarity for the quantitative analysis of drug diffusion. J Can Res Ther [serial online] 2020 [cited 2021 Apr 13];16:11716. Available from: https://www.cancerjournal.net/text.asp?2020/16/5/1171/296442 
> Introduction   
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 imageprocessing tasks, image smoothing is not only an imageprocessing tool to provide high quality image but also an important preprocessing step for many highlevel 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 (NLmeans) 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 NLmeans 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 datadriven based methods are proposed. The clustering Ksingular value decomposition (KSVD) is a dictionary learning algorithm for creating a dictionary for sparse representations ^{[16]} via a singularvalue 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. Blockmatching 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 groupingPCA algorithm,^{[18]} the input dataset for PCA is obtained from a patchmatching 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 lowrank matrix structure from its noisy observation, methods based on the lowrank matrix approximation have attracted increasing attention in image processing due to its popularity and effectiveness.^{[21]} For example, the spatially adaptive iterative singularvalue thresholding algorithm ^{[22]} sparsely represents image patches using SVD and removes noise in image by the iterative singularvalue 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 pnorm minimizationdenoising 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 lowrank property of the matrix of grouped similar blocks rather than an entire image. Making full use of lowrank priors, nonlocal selfsimilarity (NSS), and regularized sparsitypromoting 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.
Sparsitypromoting function produces a much sparser representation of images by solving a nonconvex minimization problem.^{[27],[28]} A lowrank 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 illposedness of the ENM method, we therefore, proposed the blockmatching regularized entropy minimization (BMREM) algorithm, which is based on the regularized entropy and NSS in the framework of lowrank 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 lowrank 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.
> LowRank Minimization Based On Entropy Function   
Sparse representation and compressed sensing have achieved large success in image processing.^{[29],[30]} SVD is often an effective approach to solve the lowrank 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 lowrank matrix X, the nuclear norm approximation is a natural relaxation of rank minimization, which is difficult to solve as a NPhard problem. However, in many practical applications the solution to the nuclear norm approximation is only suboptimal one of lowrank problem. Some nonconvex sparsitypromoting functions of singularvalue vector σ(x) are proposed as a surrogate of the above nuclear normminimizing 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 lowrank minimization:^{[28]}
Actually, in the case of discrete setting the above entropy function only replaces the 0norm of the Shannon entropy with a 1norm. As shown in the following experiments, in the lowrank 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 lowrank 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]}
Where,
Here l is iteration time.
> Proposed Method   
After analysis of the illposedness of the ENM method (3), we proposed, based on the regularized entropy and NSS in the framework of lowrank 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 singularvalue vector σ^{l} as its entropy, means the selfinformation 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.
Blockmatching 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 imagesmoothing 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 singularvalue 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 lowrank prior. The proposed method includes image block matching and grouping, singularvalue 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 y_{i} with size W×W are extracted from a degraded image y, for each image blocks y_{i} 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 y_{i}. The reference block y_{i} and its (N1)most similar blocks denoted by y_{j}(j = 1, 2, ..,N–1) are chosen to construct a group matrix Y_{i}= [y_{i}, y_{i},...,y_{N–1}] using each similar block as its column. In the grouped matrix Y_{i} 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 entropydriven 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 singularvalue 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 Y_{i} 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 L_{i} is a Lagrange multiplier and a positive constant. We can update {E_{i}, X_{i}, L_{i}} sequentially by solving the following series of subproblems:
1. Update of E_{i}: X_{i} and L_{i} are fixed, we solve
2. Update of X_{i}: E_{i} and L_{i} are fixed, we first compute the SVD of can be obtained as in (8). Then we solve
3. Update of L_{i}: E_{i} and X_{i} are fixed, L_{i} is easily obtained by
The above optimization procedure is described in [Algorithm 1].
> Results and Analysis   
As a fundamental inverse problem in image processing and lowlevel 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 imagesmoothing methods, including average (Average) and median (Median) filters, wavelet thresholding method (Wavelet),^{[4]} nonlocal means method (NLmeans),^{[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 imagesmoothing 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 signaltonoise 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 x_{j} 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 (NLmeans, 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 lowrank minimization. The lower rank of the group matrix is key to restoring information from the observed images in the lowrank minimization with the sparsitypromoting 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 signaltonoise 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 gadoliniumDTPA (diethylenetriamine pentaacetic acid) (GdDPTA). 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 GdDPTA 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 blockmatching 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   
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 sparsitypromoting 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.
Acknowledgment
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
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1]
[Table 1]
