Spatially Localized Sparse Representations for Breast Lesion Characterization

2.1. Sparse Representation Classification

Conventional sparse representations classification is a classification technique that utilizes sparse matrices (matrices with many zero entries). The nonzero entries of sparse matrices are of great importance as they represent the more relevant features of an image. Sparse representation classification has been utilized in the areas of image processing [26–28], face recognition [29–31], digit recognition [32], texture classification [33] and biomedical image classification [11, 34, 35]. Sparse representation techniques typically follow a two stage system, 1: dictionary learning (learning the dictionary to find approximations of signals) and 2: sparse coding (computing the sparse representation coefficients of a test signal). SRC first forms a dictionary matrix D from the training set.

with s_i data samples for the ith class, l is the dimension of the sample vector, and k is the number of classes. The total number of samples is s such that s = Σ_i s_i.

Given a dictionary of patterns D, corresponding to the object classes, sparse coding aims to solve for an unknown vector x₀ that represents the sample y as a linear combination of the atoms in D:

The solution vector x₀ is assumed to be sparse; the sparsity of x₀ ultimately ensures that signal recovery is possible. Sparsity is measured by the l₀ norm. If the solution x₀ is sparse enough, it is approximated by the solution $\widehat{x}$ of the l¹-minimization problem:

2.2. Integrative Ensemble Sparse Analysis

Our Integrative Ensemble Sparse Analysis technique regularizes the mathematical optimization problem of sparse representation by reducing the dimensionality and improving the spatial localization. Conventional SRC techniques generally cannot handle patterns in high dimensions and a small number of training samples. We divide each ROI into nonoverlapping blocks and solve a sparse coding problem for each block. At the final stage, we combine the output of all blocks to make a joint decision. We display an outline of our method in Figure 1 and detail the main stages next.

A test sample y is first divided into blocks. For each block, say the jth block, a solution x^j is found. The solution x^j is determined from the l¹-minimization problem

where NB is the number of blocks and j = 1, 2, … , NB. Then the test sample block y^j is assigned to a class ${\omega }_i^j$ that yields the minimal approximation error. The class assignment is determined by

Our method then uses an ensemble learning approach for classifier predictions. The block-based maximum a posteriori decision function (BBMAP) provides a majority voting and the block-based log likelihood approximation decision function (BBLL-S) is an ensemble of log-likelihood discriminants.

2.3. Data Description

A widely adopted method for diagnosis and early prediction of breast cancer is the X-ray mammographic test [36, 37]. In general, there are two views for each breast: the craniocaudal (CC) view, this is view from top to bottom of breast; another view is mediolateral oblique (MLO) view, ML is from middle to side and LM is from side to middle view. Mammograms show the masses, calcifications, architectural distortion of breast tissue, and symmetries [6].

We evaluated our CAD techniques for separation of breast lesions into two classes: malignant and benign. The training and testing data were obtained from the Mammographic Image Analysis Society (MIAS) database that is available online [2, 5]. The resolution of the mammograms is 200 micron pixel edge that corresponds to about 0.7559 pixel resolution resulting in about 264.58 μm pixel size, and the size of each image is 1024 × 1024 px after clipping/padding. MIAS contains 322 MLO scans from 161 subjects. The data is categorized into groups of normal subjects, subjects with benign lesions, and subjects with malignant lesions. Figure 2 shows an example of a malignant and a benign lesion sample used from the MIAS dataset. Our goal is to characterize the lesion type, therefore we utilized 68 benign and 51 malignant mammograms for performance evaluation.

ROI selection is applied first, in order to prepare the data for block decomposition. We need to ensure that the majority of the blocks cover the lesion to improve the accuracy. Hence, we designed our system so that the lesion ROI sizes are greater than or equal to the analysis ROI size. Our method reads-in the centroid and radius of each mass from the provided radiological readings. It uses these two values to automatically determine a minimum bounding square ROI and to select the masses that satisfy the size criterion. We trained and tested all classifiers on these ROI patches centered at the mass centroid. In order to evaluate the classification performance with respect to the lesion size, we performed validation experiments on variable minimum ROI sizes, i.e., we selected subsets of the dataset that met the minimum lesion radius criteria described above. We used an ROI size criterion of 64×64, resulting in 36 benign and 37 malignant lesions. These ROIs contain sufficient visual information, while preserving a big part of the data samples. We performed 10-, 20- and 30-fold cross-validation to study the effect of the size of folds on performance. We expect that the classification performance will improve as the number of folds increases, because more samples will be available for training. This effect is more noticeable on small datasets such as MIAS.

3.1. Texture-based Classification

The goal of our first experiment was to separate benign from malignant masses using conventional texture-based classification. The texture feature set consists of fractal dimension, local binary patterns, discrete wavelet frames, Gabor filters, discrete Fourier and Cosine Transforms, statistical co-occurrence indices, edge histogram, and Law’s energy maps [38]. Next, we employed feature selection methods (FSM) using a correlation-based functional criterion optimized using a best-first search strategy (CFS-BF), or genetic algorithm optimization (CFS-GA). For classification we employed bagging methods using fast decision tree learners, Random Forests (RF), Bayes Network (BN), or Naïve Bayes (NB) techniques.

In Table 1 we display texture-based classification results computed for lesions with 64 × 64 pixels minimum ROI size that performed better than the other ROI sizes. The feature dimensionality in this experiment is 451. The best accuracy with 10-fold cross-validation was 71.2% and corresponding area under the curve was 69.8% for 64 × 64 ROI size, obtained by no feature reduction and Random Forest classification.

3.2. Convolutional Neural Networks (CNNs)

In this part of our experiments we implemented CNN classification using the AlexNet [24] and Googlenet [39] architectures with transfer learning. Both networks were pre-trained on Imagenet that is a database of 1.2 million natural images.

We applied transfer learning to each network in slightly different ways. To adjust Alexnet to our data, we replaced the pre-trained fully connected layers with three new fully connected layers. We set the learning rates of the pre-trained layers to 0 to keep the network weights fixed, and we trained the new fully connected layers only. In the case of Googlenet, we set the learning rates of the bottom 10 layers to 0, we replaced the top fully connected layer with a new fully connected layer, and we assigned a greater learning rate factor for the new layer.

To provide the networks with additional training examples, we employed data resampling using randomly-centered patches, followed by data augmentation by rotation, scaling, horizontal flipping, and vertical flipping. Finally, we applied hyperparameter tuning using Bayesian optimization to find the optimal learning rate, mini-batch size and number of epochs.

We first applied 10-, 20-, and 30-fold cross-validation to 64 × 64px ROIs. Because deep networks are able to extract information from the edges of the lesion, we also decided to test our method on 256 × 256px ROIs of all lesions (66 benign and 51 malignant) to improve the classification performance. We report the results of our cross-validation experiments in Table 2. We note that Alexnet produces the top ACC of 67.65% for 30-fold cross-validation and for all ROIs.

3.3. Integrative Ensemble Sparse Analysis

In this experiment we validated our block-based ensemble classification system. We performed 10-, 20- and 30-fold cross-validation on these samples. We present results on characterization of lesions with minimum ROI size of 64 × 64 pixels that forms a dataset of 36 benign and 37 malignant lesions.

Table 3 contains the classification rates produced by our cross-validation experiments for multiple block sizes. In the first row of this table, the results were obtained from a single block that is equivalent to conventional SRC analysis [28]. We note that the accuracy increases when the number of folds increases for the same ROI size. The highest accuracy by using 10-fold cross-validation is 70.00% for 16×16 block size using BBMAP and 8 8 block size using BBLL. The largest area under the curve for 10-fold CV ×is 71.42% for 8×8 block size using BBLL. For 20-fold cross-validation, the best accuracy is 76.67% and AUC is 81.20% for 8×8 block size. The best overall performance is obtained for 30-fold cross validation. The highest accuracy is 86.67% for 16 × 16 block size, and the largest area under the curve is 89.10% for 8 × 8 block size. There are 2 or 3 test samples in each fold when k = 30. Fig. 3 displays the ROC graphs. These graphs lead to the same observations that we made from Table 3.

We estimated the AUC values of BBLL with optimized threshold ${\tau }_{LLS}^{*}$ and BBMAP by applying DeLong’s statistical test between the ROCs produced by BBMAP and BBLL as well for k-fold cross-validation. The p-values for block sizes of 32 × 32, 16 × 16, 8 × 8 and 4 × 4 were 0.78, 0.49, 0.24 and 0.21 respectively for 10-fold cross-validation, and p-values for 30-fold cross-validation were 0.72, 0.54, 0.16 and 0.0086 respectively.

Texture-based Classification.

Our experiments on MIAS 64 × 64 ROIs have shown the greatest classification accuracy with 10 fold CV using RF classifiers. The best classification accuracy achieved was 71.2% and the top area under the curve was 69.8%. Our Integrative Ensembles Sparse Analysis method outperforms texture based classification methods for CV folds of 20 and 30 and achieves similar best performance for 10-fold CV.

Convolutional Neural Networks.

The CNN models show moderate classification performances for this dataset despite extensive work on data resampling and augmentation and applying multiple transfer learning techniques. The small size of this dataset is the main limiting factor for CNN approaches. This is mainly because there are not enough training examples for the network to learn, especially when the ROIs are small. This effect may explain the improved performance of all ROIs relatively to the 64 × 64px ROIs.

Integrative Sparse Classification.

We note that the BBLL method using 8 × 8 block size improved ACC by 28.1% on average, relative to the traditional SRC method. Traditional SRC is applied when the block size and the ROI size are equal. This indicates that the block decomposition and sampling combined with classifier decision fusion yields more accurate solutions than SRC. We selected block sizes for our experimentation starting with a block size of 64 (the entire ROI) and decreasing block size by a factor of 2 until reaching a block size of 8×8. Block sizes smaller than 8×8 would contain a small amount of image information, thus making it more challenging to produce accurate classification predictions in addition to requiring more computational time. Figure 3 compares the ROC curves for both decision functions for block size of 32×32, 16×16, and 8×8. There is an observable increase in AUC between the 32×32 block size and the smaller block sizes of 16×16 and 8×8.