A non-negative matrix factorization approach to dimensionality reduction is proposed to aid classification of images. The original images can be stored as  lower-dimensional columns of a matrix that hold degrees of belonging to feature components, so they can be used in the training phase of the classification at lower runtime and without loss in accuracy. The extracted features can be visually examined and images reconstructed with limited error. The proof of concept is performed on a benchmark of handwritten digits, followed by the application to histopathological colorectal cancer slides. Results are encouraging, though dealing with real-world medical data raises a number of issues.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Atencia-Ruiz, Miguel Alejandro

Stoean, Ruxandra

English

Repositorio Institucional Universidad de Málaga

Non-negative Matrix Factorization for MedicalImagingMiguel Atencia1 and Ruxandra Stoean21- Universidad de Ma´laga - Departamento de Matema´tica AplicadaEscuela de Ingenier´ıas Industriales, 29071-Ma´laga - Spain2- University of Craiova - Department of Computer ScienceFaculty of Sciences, 200585-Craiova - RomaniaAbstract. A non-negative matrix factorization approach to dimension-ality reduction is proposed to aid classification of images. The originalimages can be stored as lower-dimensional columns of a matrix that holddegrees of belonging to feature components, so they can be used in thetraining phase of the classification at lower runtime and without loss in ac-curacy. The extracted features can be visually examined and images recon-structed with limited error. The proof of concept is performed on a bench-mark of handwritten digits, followed by the application to histopathologi-cal colorectal cancer slides. Results are encouraging, though dealing withreal-world medical data raises a number of issues.1 IntroductionA non-negative matrix factorization (NMF) method for learning reduced featuresets for image classification is introduced in the current study. The approach isimportant for real-world learning tasks, where images have a massive dimensionthan may hinder the classification process. On the other hand, distinguishingbetween the redundant knowledge and the discriminative indicators is a criticalaspect of medical images, whose supervised labelling is often noisy. Nevertheless,the task is worthwhile, given the decisive information that classification canprovide to diagnosis support.In this context, the aim of the proposed approach is twofold. Firstly, it willbe able to discover the inside feature components that can be further visuallyinspected. Secondly, the reduced matrix obtained from the factorization canbe subsequently used in the training part of the classification instead of thelarger initial data matrix, with the given labels. Additional test examples canbe encoded and classified, allowing for efficient prediction.A first test case demonstrates the effectiveness of the methodology in bench-mark conditions. The second application of the NMF is performed on the chal-lenging task of histopathological slide interpretation in search of cancerous pat-terns for the colorectal disease.The structure of the paper is set as follows. The NMF technique for imagereinterpretation is put forward in section 2. The proof of concept on handwrittendigit data is outlined in subsection 3.1, whereas the application to the medicaltask is described in subsection 3.2. Conclusions are drawn in the enclosingsection of the paper.2 Non-negative Matrix Factorization for Image EncodingThe current study proposes a NMF methodology to model dimensionality re-duction and subsequent classification for image data sets. The approach followsseveral steps. Non-negative factorization of an n×m-sized matrix Y ≥ 0, wherethe inequality is considered component-wise, yields two factors W,H, both alsonon-negative, so that the approximation error ‖Y −W H‖ is minimized. Severalchoices for the matrix norm are possible, but in this work we restrict ourselves tothe most usual Frobenius norm. If the sizes of W and H are n× r and r×m re-spectively, the algorithm is meaningful when r  min(n,m) and, as such, it hasbeen related to Principal Components Analysis and Low Rank Approximation.For notation, background and examples, see [1] and references therein.NMF has been applied to several tasks related to feature selection and di-mensionality reduction, including text mining and source separation. In theparticular context of image processing, p×q resolution images are first reshapedto n = p q-dimensional vectors, so that each image forms a column of an n×mmatrix Y , where m is the number of images available. The matrix is subse-quently decomposed as Y ≈ W H, where the key parameter r is set manually.In this work we implement the Hierarchical Alternating Least Squares (HALS)algorithm, which solves the unconstrained least squares problem alternativelyfor one column of W and one row of H at a time, followed by projection toensure non-negativity. Initialization, which is known to be a critical aspect ofthe algorithm, is based upon Singular Value Decomposition [2]. Also note thatthe algorithm allows for easy scaling or normalizing the columns of W : for eachcolumn W.j , compute αj = max(W.j) and replace W.j ← W.jαj , as well as therows of H by Hj. = αj Hj., so the product W H remains constant, and eachcolumn of W is distributed in the interval (0, 1).Since W,H are non-negative, if the columns of W are scaled to the samemagnitude as those of Y , they can be interpreted as features or cluster centreswithin Principal Components Analysis and Self-Organizing Maps [3]. In turn,each column H.i constitutes an encoded or compressed version of the correspond-ing original image Y.i, which could be later recovered by Yˆ.i = W H.i, and theencoding error can be assessed by ‖Y.i − Yˆ.i‖.In this work we propose the inclusion of the NMF algorithm as a previousstep to the application of a classification method, which is expected to have anenhanced performance and reduced computational cost due to the much lowerdimensionality of the trained vectors Hi, in contrast to the original images Yi.Thus NMF will play a similar role to that of autoencoders in Deep Learningarchitectures [4], with the added benefit of easier interpretation of results asfeature vectors. The W matrix is constructed as a result of the NMF algorithm,which can be considered a sort of training. Then, a subsequent test image yshould be encoded into a compressed vector h, in such a way that y ≈ W h.The compression can be achieved by the Moore-Penrose pseudoinverse W+ =(W>W)−1W> and the low-dimensional code is obtained by h = W+ y, assuggested by [5] in the context of source separation.1 2 3 4 56 7 8 9 1011 12 13 14 1516 17 18 19 2021 22 23 24 2526 27 28 29 30Fig. 1: Features or basis images in the resulting matrix W provided by NMFwith r = 30 on the handwritten digit data.3 Experimental ResultsThe application of NMF to image data is expected to provide dimensionalityreduction and subsequent faster learning of the classifier.3.1 Experiment 1: Proof of Concept on Handwritten DigitsThe first experiment assesses the NMF algorithm by considering the benchmarksynthetic handwritten digit data set provided by Matlab, which consists of train-ing and test sets containing 5000 28x28 resolution sample images each. The taskis then to categorize the 784-dimensional vectors into the 10 classes.3.1.1 EncodingThe key hyperparameter r is manually set as r = 30, which triggered the bestperformance within the experimental trials. The corresponding features appear-ing in the columns of W are shown in Figure 1. For the training set, the encodedimages are directly recovered by yˆ = W h from the columns of H computed in thefactorization. For the test set, the encoded version h = W+ y is first computed,and decoded again by yˆ = W h. The encoding error for 5 random columns of thetest set are depicted in Figure 2, where it can be seen that the reduced versionslook similar to the original ones with a visually assessable small error.error: 2.4586error: 2.4776error: 3.2339error: 4.0137error: 2.3595error: 2.4586error: 2.4776error: 3.2339error: 4.0137error: 2.3595Fig. 2: Five handwritten digit test samples in original (top row) and en-coded/decoded version (bottom row). Mean error: 2.46, 2.48, 3.23, 4.01, 2.36.Table 1: Results of shallow classifiers on the original and reduced digit data.Method Configuration Acc. (%) Time (sec.)Initial dataSVM linear, 1a1 90.96 137MLP 1 hidden layer (30), 100 maxit, SCG 86.88 321RF 5 variables at split, 30 trees 97.12 48Reduced H dataSVM radial, 1a1 97.5 17MLP 1 hidden layer (100), 300 maxit, SCG 95.98 191RF 5 variables at each split, 30 trees 97.56 23.1.2 LearningThe next step is to use the labels and try to learn the image-label correspondenceby means of a classifier. Three state-of-the-art shallow approaches are imple-mented in R, namely Support Vector Machines (SVMs), Multilayer Perceptron(MLP) and Random Forests (RF). These choices allow to assess the encodingability before a variety of classification paradigms: statistical learning theory,connectionism, and ensemble learning.Table 1 presents the results of the three employed shallow classifiers on theinitial collection of digit images (with the training and test pre-decided splits)and on the reduced set given by H and the accordingly encoded Htest. Theapproaches were implemented in R with the corresponding packages e1071,RSNNS and randomForest. The parameter configurations for the three ap-proaches, which are the result of manual tuning, are also outlined within Table1. The NN architecture for the initial data set was chosen as a trade-off betweentime and accuracy. The accuracy given by the RF is the result of 30 repeatedruns of the algorithm, given the stochastic nature of the learner.There are several conclusions that can be drawn from examining the accuracy-time results alongside the necessary parametrization requirements in Table 1:• The RF improves runtime when training on the compressed data H.• For the MLP, the encoding allows for a larger architecture to be trained.• The SVM, when using NMF, is much faster and reaches the RF accuracy.3.2 Experiment 2: Real-world Scenario of Histopathological ImagesThe second experiment investigates the potential of the NMF on a more difficultas well as highly informative image data from the medical field. The problemtackles automatic cancer grading from histopathological digitalized slides forcolorectal tissues [6]. The pathologist is supposed to establish the cancer stageby measuring the variation of the size, shape and texture of the glands andnuclei [7]. There are 96 G1 (grade 1) records, 99 G2 (grade 2) and 100 G3(grade 3) images. The collection comes from the Emergency Hospital of Craiova,Romania 1. The original slides are at resolution 800x600 and also contain benigninstances, obtained by cutting healthy portions from tissue images of ill patientsat the indication of the pathologist [8], since biopsy is usually performed onlywhen there is suspicion of cancer presence.For this experiment, the task is formulated as binary classification, i.e. todistinguish between G1 (cancer in an incipient stage) and G23 (advanced in-vasion). This discrimination is commonly necessary for prompt assessment bythe physician, in order to decide subsequent therapy. For a direct processing ofthe images within the NMF and subsequent learning, they are compressed at25%, i.e. having 200x150 pixels, and transformed to greyscale, thus resulting in30000-dimensional input vectors. In order to account for the stochastic nature ofclassifiers, 10 splits of repeated random sub-sampling cross-validation with 70%training (250 samples in Y ) - 30% test (107 examples in Ytest) are performed.Note that classification by a shallow approach of these initial samples can eitherbe restricted by memory limits or be intractable in terms of computational time,depending on the implementation platform.The NMF is applied to the training set with r = 100, which is close to themaximum imposed by the restriction r  min(n,m). The generated matrixH is then used for learning. The test set is obtained again by the encodingHtest = W+ Ytest, as mentioned before in subsection 3.1.2.The mean reached accuracy over the 10 trials is of 70%. The relativelyhigh number of errors is probably due to the loss of information resulting fromencoding 30000 pixels into only 100 features through the NMF. Additionally,histopathological slides are originally stained for a better visualization by thepathologist. The conversion to greyscale has certainly lost other important fea-tures regarding colour display between regions.1https://doi.org/10.6084/m9.figshare.4508672.v14 ConclusionsThe paper tailored a NMF methodology to dimensionality reduction for imagelearning tasks. The approach was first validated on a benchmark collection ofhandwritten digits. The features were visually inspected and the encoded imageversions were seen to share a great similarity with the initial ones. Learningonly the reduced H matrix resulting from the NMF led to a test classificationaccuracy into the 10 categories of 97.5% in at least 20 times less runtime.The second test instance, the histopathological data classification into lessand greatly invasive colorectal cancer, was a preliminary study of the applicationof NMF as an image encoder for a real-world medical image task. The obtainedresults are acceptable, in the light of a drastic dimensionality reduction of thedata from 30000 to 100 and the use of a greyscale transformation of otherwisestained images. These results highlight a critical issue when dealing with medicalimages: labelling is time-consuming and subjective, resulting in scarce and noisyinstances for supervised classification. From a mathematical viewpoint, the datamatrix has more rows than columns, thus limiting the number of basis imagesthat can be obtained. The next obvious step is to artificially increase the numberof slides, e.g. by partitioning a full 800x600 image into quarters. This will allowto increase the value of r within NMF resulting in a larger number of features,which should eventually lead to more accurate learning. Secondly, the use ofthe original RGB condition will probably comprise further important featuresin the discrimination process. Dealing with these three-dimensional matricescould be achieved by techniques of tensor factorization. Finally, the rank r ofthe factorization should be optimally set through a parameter tuning heuristic,independent of the size of the vectors under consideration for each problem.References[1] Nicolas Gillis. The Why and How of Nonnegative Matrix Factorization. In Regularization,Optimization, Kernels, and Support Vector Machines, pages 257–291. 2014.[2] C. Boutsidis and E. Gallopoulos. SVD based initialization: A head start for nonnegativematrix factorization. Pattern Recognition, 41(4):1350–1362, April 2008.[3] Imola Fodor. A survey of dimension reduction techniques. Technical report, 2002.[4] Yoshua Bengio. Learning deep architectures for AI. Foundations and Trends in MachineLearning, 2(1):1–127, 2009.[5] Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan, and Shun-Ichi Amari. NonnegativeMatrix and Tensor Factorizations. John Wiley & Sons, Ltd, Chichester, UK, 2009.[6] Stefan Postavaru, Ruxandra Stoean, Catalin Stoean, and Gonzalo Joya Caparros. Adap-tation of Deep Convolutional Neural Networks for Cancer Grading from HistopathologicalImages, pages 38–49. Springer International Publishing, Cham, 2017.[7] M.N. Gurcan, L.E. Boucheron, A. Can, A. Madabhushi, N.M. Rajpoot, and B. Yener.Histopathological image analysis: A review. IEEE Reviews in Biomedical Engineering,2:147–171, 2009.[8] Olcay Sertel, Jun Kong, Umit V. Catalyurek, Gerard Lozanski, Joel H. Saltz, and Metin N.Gurcan. Histopathological image analysis using model-based intermediate representationsand color texture: Follicular lymphoma grading. Journal of Signal Processing Systems,55(1):169–183, 2008.

Non-negative matrix factorization for medical imaging

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Non-negative matrix factorization for medical imaging

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