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Journal of Medical Systems

, 41:26 | Cite as

Characterizing Architectural Distortion in Mammograms by Linear Saliency

  • Fabián Narváez
  • Jorge Alvarez
  • Juan D. Garcia-Arteaga
  • Jonathan Tarquino
  • Eduardo RomeroEmail author
Systems-Level Quality Improvement
Part of the following topical collections:
  1. Systems-Level Quality Improvement

Abstract

Architectural distortion (AD) is a common cause of false-negatives in mammograms. This lesion usually consists of a central retraction of the connective tissue and a spiculated pattern radiating from it. This pattern is difficult to detect due the complex superposition of breast tissue. This paper presents a novel AD characterization by representing the linear saliency in mammography Regions of Interest (ROI) as a graph composed of nodes corresponding to locations along the ROI boundary and edges with a weight proportional to the line intensity integrals along the path connecting any pair of nodes. A set of eigenvectors from the adjacency matrix is then used to extract discriminant coefficients that represent those nodes with higher salient lines. A dimensionality reduction is further accomplished by selecting the pair of nodes with major contribution for each of the computed eigenvectors. The set of main salient lines is then assembled as a feature vector that inputs a conventional Support Vector Machine (SVM). Experimental results with two benchmark databases, the mini-MIAS and DDSM databases, demonstrate that the proposed linear saliency domain method (LSD) performs well in terms of accuracy. The approach was evaluated with a set of 246 RoI extracted from the DDSM (123 normal tissues and 123 AD) and a set of 38 ROI from the mini-MIAS collections (19 normal tissues and 19 AD) respectively. The classification results showed respectively for both databases an accuracy rate of 89 % and 87 %, a sensitivity rate of 85 % and 95 %, and a specificity rate of 93 % and 84 %. Likewise, the area under curve (A z ) of the Receiver Operating Characteristic (ROC) curve was 0.93 for both databases.

Keywords

Breast spiculated lesions Architectural distortion Mammography Linear saliency 

Introduction

Breast cancer is the most frequently diagnosed form of cancer and the second largest cause of death for women worldwide [1]. When it is detected in its early stages, there are therapeutical alternatives to manage and fully cure this form of cancer. Currently, mammography is considered the most cost-effective method for breast cancer detection in the early stages [10, 27]. From a mammography an expert radiologist is able to detect abnormalities such as calcifications, bilateral asymmetry, masses and Architectural Distortion (AD) [36]. Despite the widespread use and proven effectiveness of mammographies, screening mammography programs showed a high intra and inter-observer variability in the early 90s of last century with undetected cancer rates of between 10 % and 30 % [33]. Previous studies have reported rates of between 10 % and 25 % [6, 38]. This high rate is partly due to the fact that identifying radiologic signs for mammographic interpretation remains a complex visual task because of the blurriness originating from the overlaying of a complex superposition of breast tissues. Among the most important mammographic findings, AD has been reported as the most commonly missed abnormality in false-negative cases [18]. An AD lesion is any change of the spatial distribution of the breast connective tissue, in general with no visible mass. In such mammographic finding, the typical retraction of tissues is variable and gradual, commonly showing radial spiculations. Focal retraction or distortion at the boundary of the parenchyma without a visible or palpable mass may also be present. The set of radiological signs of an AD has been collected and formalized within the Breast Imaging Reporting and Database Systems (BI-RADS) standard [36]. Figure 1 illustrates the linear patterns in both AD and normal tissue, manually selected by a radiologist.
Fig. 1

a Mammogram showing AD. b A 128×128 AD ROI showing spiculations radiating from a central point. c Normal 128×128 ROI tissue portion. The ROIs shown in (b) and (c) were both extracted from the mammogram shown in (a)

Previous work

Nowadays, despite of development of Computer Assisted Diagnosis Systems (CAD) in mammography has proved to be effective in finding calcifications and masses [27], CAD systems have not been as effective in the detection of AD with adequate levels of accuracy (about 49 % of precision) [4, 22]. Several methods have been introduced to characterize AD connective tissue radiating from a center which have the appearance of radiolucent lines or stellate patterns. In these methods linear structures are enhanced to highlight information related to orientation, scale, or strength of lines, facilitating the characterization process. Several techniques have detected areas with potential stellate patterns, among others linear detectors [42], steerable filters [35], oriented field analysis by phase portraits maps [32], spicules enhancement by image transformation (Gabor and Radon spaces) [3, 5, 34] or fractal functions [14, 39]. Karssemeijer et al. [19] detected stellate patterns, including spiculated masses and AD using a multi scale method that classified the output of three-directional second order Gaussian operators and reported a sensitivity of about 90 % with one false-positive per image (FP/image). Zwiggelaar et al. [41] evaluated four distinct methods for detecting and classifying linear structures in synthetic images. The reported area under the curve (A z ) using line operators, Orientated Bins, Gaussian Derivatives and Ridge Detectors were of 0.94, 0.91, 0.9 and 0.82, respectively. The type of linear structure detector and the parameter selection depends on prior knowledge of the lesion being imaged. Sampat et al. [34, 35] proposed a transformed image and radial spiculation filters in the Radon space to detect spiculated masses and AD. The reported sensitivity was of 80 % and 14 FP/image for AD and the reported sensitivity was of 91 % and 12 FP/image for spiculated masses. Guo et al. [14] applied five methods to estimate a Region of interest (ROI) of a Fractal Dimension that was used to train a support vector machine classifier (SVM). The authors reported an A z of 0.875 when differentiating masses and AD from the normal parenchyma. Ichikawa et al. [17] used a concentration index of linear structures obtained by the mean curvature of the image, resulting in a sensitivity of 68 % and 3.4 FP/image. Nemoto et al. [26] proposed the likelihood of spiculation and a modified point convergence index weighted by the likelihood to enhance AD, establishing a sensitivity of 80 % and 0.80 FP/image. Biswas et al. [7] introduced a generative model with a bank of filters that learned “texton” patterns using a subset of images from the Mammographic Image Analysis Society (mini-MIAS) dataset [37] (19 AD and 21 normal tissues). Evaluation was performed under a leave-one-out scheme, obtaining 3.6 FP/image and a sensitivity of 81.3 % with a ROI radius of 5 mm (the minimum radius of an AD region in mini-MIAS). Kamra et al. [18] use a combination of Spatial Gray Level Co-occurrence Matrix (SGLCM), fractal and Fourier power spectrum based features to characterize the AD lesion, quantified in four directions 𝜃=0,45,90,135. This strategy was evaluated on a subset of ROIs from mini-MIAS [37] and the Digital Database for Screening Mammography (DDSM) [15] datasets with different size of ROI and a subset of fixed ones, the reported accuracy, sensitivity, specificity of the fixed ROI versions of DDSM was 92.9 %, 93.3 %, 92 %, for mini-MIAS was 97.2 %, 85.7 %, 95.3 % respectively. Matsubra et al. [21] applied directional and background filters to analyze the linear structure in eight directions. The detection sensitivity and the FP/image reported was 81 % and 2.5 respectively. The dataset used in this study contains 174 AD and 580 normal diagnosed control cases. Several studies have used the phase portrait maps to capture the outward radiating linear structure, a strategy based on Gabor filters that highlight linear patterns at certain orientations [2, 5, 30, 31] with significant results. Banik et al. [5] detected the AD using the Gabor filters and phase portrait analysis. For each detected ROI, the fractal dimension, the entropy of the angular power spread and 14 Haralick’s features were computed. Results showed an A z of 0.76 with the Bayesian classifier, 0.75 with Fisher linear discriminant analysis, and 0.78 with a single-layer feed-forward neural network. These approaches face the AD detection problem as a classification task of ROIs of the entire mammogram. The major drawback is then the need of choosing a large number of parameters to construct linear detectors, Gabor filters, steerable filters, linear filters or features in both spatial and transform image domain to measure spicules. Additionally, the intended detection of a node-pattern, representing radiating-out linear structures often fails because the spicules, in general, do not fully describe a well-defined converging pattern but instead spicules mostly make an incomplete “star” shape with missing parts, making the texture orientation highly noise sensitive.

Contribution

Unlike previous approaches, this work is focused on the search of the linear saliency of a stellate pattern in the spatial domain, aiming to strength the classification process. The proposed Linear Saliency Domain method (LSD) represents a ROI as an (initially) fully connected graph with nodes corresponding to the ROI boundary and edge weights calculated using line integrals of the intensities along the path connecting any pair of nodes. By doing this, the edges corresponding to the most salient lines are assigned a larger weight in the graph. A centrality measure based on the graph’s adjacency matrix’s eigenvectors is used to identify the most relevant nodes. A further dimensionality reduction is accomplished by selecting only the most salient edge originating from relevant nodes. The set of salient lines is then assembled as a feature vector used to train an SVM classifier. Experimental results in two benchmark databases, the DDSM and mini-MIAS databases, demonstrate that LSD outperforms the baseline techniques [2, 31] in accuracy and speed with a precision rate of 89 % and 87 %, a sensitivity rate of 85 % and 95 %, a specificity rate of 93 % and 84 % respectively.

The rest of this article is organized as follows: Section “Method” explains in detail the methodology and steps of the LSD. In Section “Datasets” the datasets used to test the method are presented. In Section “Results”, results of the validation experimentation, are shown. Section “??” presents the discussion, conclusions and possible future work directions are explored.

Method

The LSD method consists of five consecutive steps: Image Enhancement, Graph Construction, Central Node Detection, Detection of Salient Edges and Classification. The pipeline of the whole process is presented in Fig. 2 and each step is explained in detail in the following sub-sections.
Fig. 2

Pipeline of LSD

Image enhancement

The ROIs of the images selected as input for LSD are smoothed using a non-linear median filter [8] aiming to remove the image noise. Spicular details are enhanced by mapping each image’s dynamical range to the maximum and minimum gray level values of the interval ([0, 255]), followed by a contrast limited adaptive histogram equalization approach (CLAHE) [29].

Graph construction

The perimeter of a squared ROI is divided into 32 segments per border for a total of N=4×32=128 segments. The segments’ width, 4 pixels in the mini-MIAS database and 32 pixels in the DDSM database, were chosen to approach the typical width of a single spicula, i.e. approximately 200 μ m [3]. A graph G(V,E) is built with the N segments corresponding to nodes V. The weights of the edges E are computed as the integral of the ROI intensities along the lines connecting every pair of nodes, i.e. the value of the integral between node V i and node V j corresponds to edge E i j = E j i . Notice that when nodes lie on the same border, e.g. two nodes on the top border, the thickness of the line and the integral result are both zero, making G is a partially connected graph.

Graph relations are summarized as an adjacency matrix A, where each position [i,j] corresponds to the weight w i j = w j i of the edge between two vertices (V i ,V j ). An illustrated example of the construction of the graph may be seen in Fig. 3 and its adjacency matrix is shown in Eq. 1.
$$ A = \left( \begin{array}{ccccc} 0 & w_{1,2} & w_{1,3} & {\cdots} & w_{1,j} \\ w_{2,1} & 0 & w_{2,3} & {\cdots} & w_{2,j} \\ w_{3,1} & w_{3,2} & 0 & {\cdots} & w_{3,j} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ w_{i,1} & w_{i,2} & w_{i,3} & {\vdots} & 0 \end{array} \right) $$
(1)
Fig. 3

The ROI boundary is divided into N segments as illustrated in the left panel. The center of each segment corresponds to a node V i . The weight w of the edges connecting any pair of nodes corresponds to the value of the line integrals

Central node detection

A Markov chain may be associated to the graph G(V,E) by normalizing the weights of the edges. Nodes can then be interpreted as the states of the Markovian process, while the weights are equivalent to transition probabilities. Under the appropriate assumptions, such as irreducibility due to strong connectivity, this chain tends to a stationary probability distribution that represents the frequency of node visits that a random walker would make if allowed to walk forever. In the present problem, this probability distribution would naturally be concentrated at nodes that have higher linear intensities since transitions into subgraphs containing them is more likely.

This type of representation has been shown to be equivalent to the eigenvector centrality analysis described in [12], where the node importance is calculated in a self-referential way, i.e. a node’s importance is larger if it is adjacent to other important nodes. Different artificial vision strategies have represented the image relationships by a graph structure. Gopalakrishnan et al. [13], for example, have proposed that Markov random walks may represent the global or local image properties by performing the centrality analysis on either the complete or sparse k-regular graphs. From this information, the most salient foreground and background objects may be set and labeled.

The most important nodes may be extracted from the eigenvectors sorted using the eigenvalues of A. The first eigenvector represents the steady state of the associated induced Markov chain and includes the nodes with the most important connections. Subsequently, the n t h eigenvector corresponds to the most important nodes except those in the m t h eigenvector, for all m<n.

In each eigenvector the largest element \(\overrightarrow {V}_{k}\) corresponds to the position of the most relevant node v m :
$$ v_{m}=max(\overrightarrow{V}_{k}) \; \; \; \forall k \in [1, \dots, N] . $$
(2)

Detection of salient edges

Once the central nodes have been detected, it is necessary to select the most salient edges that radiate from them. This is done by locating the highest weights in the original A adjacency matrix. As stated before, higher values correspond to the more radiolucent lines. In particular, each relevant node stands for a column of the adjacency matrix A, as illustrated in Fig. 4.
Fig. 4

In the adjacency matrix A each column corresponds to a node in the graph and its connections

The highest value of the column corresponds to the most important connection of that node, as shown in the Fig. 5. This connection is then the most salient pattern for the selected central node.
Fig. 5

The highest value of the n t h column corresponds to the most salient edge radiating from the n t h node

ROIs of 128×128 pixels, such as the mini-MIAS database, correspond to a graph G of 32×32 nodes with at least 32 lines representing the saliency for any of the possible directions, i.e., a particular side is connected with each of three possible sides of the ROI.

Classification

The binary AD classification problem is approached by an SVM classifier [11]. This strategy is applied to datasets with complex separation boundaries between classes and is based on maximizing the separating margin between the two classes in an alternative space in which data relationships are linear. In this case two classes are defined: normal breast tissue and ADs [14].

The proposed method trains the classifier model using radial basis functions (RBF) as a kernel. Positive values correspond to AD and negative values to control cases. The optimal hyperplane is found by solving the discriminating function:
$$ f(x)=sign \left( \sum\limits_{i=1}^{n} y_{i}a_{j} \phi^{T}{(x_{i})}.\phi{(x_{j})} \right) $$
(3)
where sets of labels y i module the Lagrange multipliers a j and the kernel function K(x i ,x j ) is a similarity measure defined by computing the inner-product between the feature vectors as
$$ \phi^{T}{(x_{i})}.\phi{(x_{j})} = e^{-\left( \frac{1}{2\sigma^{2}}\lVert x-x_{i} \lVert^{2}\right) .} $$
(4)

The kernel parameters were optimized by cross validating on the synthetic data (\(\gamma = \frac {1}{5128} = 0.00019\) and σ=1).

Datasets

Clinical cases

Each AD ROI corresponded to a window with the size and location annotated in the database. In contrast, the reference tissue or control case was selected by and expert radiologist who manually set a window within the normal fibroglandular tissue, with a similar size to the marked AD regions. Two public mammography databases were used to test the method: DDSM and mini-MIAS. A ROI is the smallest square region containing the complete lesion, 128×128 for the mini-MIAS database and 1024×1024 for the DDSM collection.
  1. The DDSM database: the Digital Database for Screening Mammography database1 (DDSM) [15] is a mammography database widely used as an evaluation benchmark [16, 23, 24, 25, 40]. This open access database is constituted of digitized images of mammographic films with the corresponding technical and clinical information. The whole DDSM database contains a total of 2620 cases, each including four images obtained from Cranio-Caudal (CC) and Mediolateral-Oblique (MLO) views as well as their specific BI-RADS descriptions that were annotated by expert radiologists according to BI-RADS fourth edition. In this work, a subset of mammographies were chosen, the inclusion criteria were: 1) similar image quality, 2) balanced number of ROIs (control tissues and Architectural Distortion), 3) only a single lesion was assessed, i.e., this study excluded masses, calcifications or cases with the presence of clips or pencil marks in the lesion area. Mammographies were digitized either with a Lumisys laser film scanner at 50 μ m or a Howtek scanner at 43.5 μ m pixel resolution and at a dynamic range of intensities of 212= 4096 gray level tones. The AD location and size, associated to each case, were used to manually crop squared sub-images centered at the lesion. Specifically, RoIs were cropped to the bounding box of the lesions and rescaled to 1024×1024 pixels, preserving the aspect ratio when either width or height were greater than 1024. Otherwise the lesion is centered without scaling and preserving the background tissues. In consequence, a set of 246 ROIs were extracted from this DDSM database, distributed as 123 ROIs with normal tissue and 123 ROIs with Architectural Distortion.

     
  2. The mini-MIAS database: The collection published by the Mammography Image Analysis Society (MIAS)2 provides digitized mammograms from a screened population [37]. The X-ray films in the database have been carefully selected from the United Kingdom National Breast Screening Programme and digitized with a Joyce-Lobel scanning microdensitometer to a resolution of 50 μ m 2. Each pixel is 8-bit depth represented. The database contains left and right breast images from 161 patients, and radiologist’s truth-markings. For each film, experienced radiologists give the type, location, scale, and other useful information. The database provides a metadata file describing the lesions as well as their location and size. For evaluation, a set of 19 ROIs with Architecture Distortion were selected as well as 19 control RoIs. The AD location and size were taken from the metadata file. The cropping process was carried out as described for the DDSM database, except that the size was 128×128 pixels.

     

Synthetic data

A set of synthetic images was constructed to assess and fine tune the method’s parameters. These digital phantoms aim to emulate actual AD and therefore must keep the radiated pattern of linear structures around a focus. The method implemented was presented by Parr et al. [28] and consists of finding a representative element of a series of artificially constructed radiating patterns. Briefly, the method splits a square image of 81×81 pixels into tiles of 9×9 that serve to focus radiated patterns starting from the center of each tile. For a particular centered pattern, the same 9×9 representation grid is used but this time each tile stores the average angle of every line crossing the tile. From these images, the obtained 81 radiating patterns storing the mean angle for each tile, are then vectorized and used to feed a Principal Component Analysis strategy (PCA). The main twelve eigenvectors of the covariance matrix, that explain a 90 % of the variability, are set as the statistical representative elements. Afterwards, the weights of the six vectors that approximate 80 % of the variance are retrieved by simply subtracting the mean to the observation and projecting the result onto the matrix of twelve eigenvectors.

The weights of the principal directions b i are found then by centering the data x i and projecting onto the principal direction matrix P, as stated in Eq. 5 and illustrated in Fig. 6
$$ b_{i} = P^{T}(x_{i}-\mu) $$
(5)
Fig. 6

a Artificially generated radiating pattern. b Angular representation of a radiating pattern. c Final focal radiating phantom

Different linear patterns can be obtained from this representation by simply de-centering the data by one or two standard deviations, as formulated in Eq. 6 and illustrated in Fig. 7. The final phantom consists of a pattern superimposed to an actual mammogram region diagnosed as normal. In practice, this mix is achieved as follows: the average intensity value of the normal ROI is assigned to the lines of the previously generated pattern. This image is then added to the mammogram, producing the result illustrated in panel (b) of Fig. 8. The final dataset is composed of 162 phantoms, namely 81 focal that are de-centered as previously explained by one standard deviation to yield the 81 non-focal patterns.
$$ b_{i} = P^{T}(x_{i}-\mu + s) $$
(6)
Fig. 7

Effects of de-centering the focal line pattern

Fig. 8

Different phantom patterns. a background from a control MIAS Mammogram. b focal AD phantom. c focal AD phantom with Gaussian noise. d Non-focal regular AD phantom. e Non-focal regular AD phantom with Gaussian noise. Gaussian noise was set to μ=0 and σ=0.05

Table 1

Comparative results of the classification with synthetic images for each evaluated method

 

Acc

Sens

Spec

A z

 

P

P G

P

P G

P

P G

P

P G

LSD

0.98

0.92

1.0

0.87

0.96

0.96

1.0

0.96

Ayres et al.

0.54

0.48

0.66

0.33

0.42

0.62

0.52

0.52

Banik et al.

1.0

0.5

1.0

0.75

1.0

0.25

1.0

0.5

Column P shows the results for the initial synthetic image and P G for the Gaussian corrupted phantom test sets

Experimental setup

The performance of the LSD method was tested using series of normal and AD ROI either synthetically constructed or extracted from the two public datasets described above. Classification followed an SVM strategy with an RBF kernel using the LIBSVM library implementation [9]. The strategy herein implemented consists of 128 nodes that result in 32 salient lines from the graph introduced in previous section. The different metrics used to evaluate the results were: \(accuracy = \frac {TP + TN}{ TP + FP + FN + TN }\), \(sensitivity = \frac {TP}{ TP + FP}\) and \(specificity = \frac {TN}{ TN + FN}\), being TP the number of True Positives, TN the true negatives, FP the false positives and FN the false negatives, respectively. Performance is also evaluated using the ROC curves generated by setting thresholds as the membership.

Results

Comparison with baseline methods was performed by implementing two different strategies:
  • Firstly, the classic approach proposed by Ayres and Rangayyan [2]: This seminal work was included as a baseline as it is, to the best of our knowledge, the earliest example of an AD detection method in the Gabor space.

  • Secondly, the strategy proposed by Banik et al. [5] which analyzes both the node maps of the phase portraits and Haralick texture descriptors from the Gabor magnitude.

These two representations are based on projections to the Gabor space, but while Ayres and Rangayyan use the phase space orientation map, from which different features are extracted, Banik et al. include also some characteristics of the phase space magnitude. It is worth mentioning that these methods were implemented using exclusively the frequency and texture features reported by the authors to be significative.

Experiments with synthetic data

As introduced in Section “Synthetic data”, synthetic data emulates different AD focal and non-focal oriented patterns, corresponding to the pathological and control cases, superimposed to a regular distribution of normal tissue, as illustrated in Fig. 8. This background was chosen by an expert from a case which was randomly selected from the mini-MIAS set of control cases. The obtained synthetic dataset is composed of 162 ROIs, 81 focal and 81 no focal oriented patterns.

This evaluation tested not only the classification using pure phantoms, but also the quality of the result obtained when the phantom was contaminated with different levels of Gaussian noise. The added noise emulates the type of noise resulting from the contribution of many independent sources and associated to tomography and radiographic images [20]. The Gaussian noise was set to μ=0 and σ=0.05, a level for which the peak signal to noise ratio (PSNR) between the original image and the degraded one was of 19.6 d B. A total of 162 phantoms (81 focal and 81 regular patterns) were synthetically generated, and each of these subsets was randomly sorted out. These two sets were then split to a, roughly, 70 to 30 training to testing cross-validation ratio with 57 of each group used for training and the remaining 24 of each group used for evaluation. A first test was made on the pure phantoms and a second one over the original patterns corrupted with the Gaussian noise. This classification task was carried out using a conventional SVM classification with an RBF kernel. Comparative results obtained for the two baseline methods and LSD, using both the original and corrupted patterns, are shown in Table 1.

The baseline method proposed by Banik et al. [5], for the original test P shows a perfect performance for the set of generated patterns, demonstrating its effectiveness for detection of focal patterns. Likewise, LSD showed an accuracy of 98 %, a sensitivity of 100 % and a specificity of 96 %, evidencing a comparable performance with these simulated lesions. These two methods outperform the implementation of the Ayres et al. proposal [2]. In the corrupted phantoms tests LSD outperformed the two baseline methods showing comparable figures to what was observed with the uncorrupted patterns. The baseline methods completely lose the discriminative power shown with the uncorrupted dataset.

Figure 9 presents the obtained results as ROC curves that depict the classification in terms of sensitivity vs 1−s p e c i f i c i t y. Panel (a) in Fig. 9 shows a perfect performance for the proposed method and the baseline proposed by Banik, yielding an A z of about 1.0 in both case. In contrast the classical Ayres method shows a very low performance, with an A z of 0.52. Interestingly, when a Gaussian noise is added to the synthetic phantoms, the proposed approach results remarkably more resistant than the baseline approaches, as illustrated in panel (b) of Fig. 9. The A z of the tested methods were LSD =0.96, Ayres =0.52 and Banik =0.5.
Fig. 9

ROC curves after the evaluation of each method for both experimental groups. Results of the original synthetic images are shown in panel (a) and the results of the corrupted ones are shown in panel (b)

Experiments with DDSM and mini-MIAS databases

Given the small number of AD cases (19) available in the mini-MIAS database, cross-validation tests were carried out using a leave-one-out scheme.

The baseline method proposed by Banik et al. showed an accuracy of 89 %, a sensitivity of 84 % and a specificity of 89 %, evidencing a good performance for the mini-MIAS set. The proposed method showed for the same database an accuracy of 87 %, a sensitivity of 95 % and a specificity of 84 %, a slightly better specificity with the Banik’s method. This result may be attributed to a trained model with very few examples, 18 in this case. These two methods strongly outperforms the Ayres et al. strategy. Results are shown in Table 2.
Table 2

Comparative classification results with the DDSM and mini-MIAS datasets for each evaluated method

 

Acc

Sens

Spec

 

DDSM

mini-MIAS

DDSM

mini-MIAS

DDSM

mini-MIAS

LSD

0.89±0.07

0.87±0.09

0.85±0.13

0.95±0.12

0.93±0.13

0.84±0.1

Ayres et al.

0.59±0.13

0.65±0.13

0.33±0.15

0.68±0.13

0.68±0.14

0.63±0.15

Banik et al.

0.76±0.11

0.89±0.12

0.75±0.13

0.84±0.13

0.76±0.15

0.89±0.14

Accuracy, Sensitivity and Specificity are reported

The number of AD cases in the DDSM dataset allowed a k-fold cross validation scheme with k=10 folds to be used. In this test, LSD showed an accuracy of 89 %, a sensitivity of 85 % and a specificity of 93 %: a much better performance than the one observed with Banik et al. that yielded an accuracy of 76 %, a sensitivity of 75 % and a specificity of 76 %. The classical method proposed by Ayres et al. showed an even lower performance. These results support the previous statement regarding a minimal number of cases with sufficient variability. In this case, the presented method is more robust, the data base is much bigger and the variability is larger in terms of different types of lesions.

Figure 10 presents the results of SVM classification as a ROC curves. Panel (a) shows a high performance on the DDSM for LSD, with an A z of 0.93 while the baseline proposed by Banik et al. yield 0.83 and the classical method proposed by Ayres et al. show an A z of 0.55.
Fig. 10

a ROC curves for the DDSM and mini-MIAS databases obtained from a classical 10-fold cross validation for the set of ROIs. b The ROC curve corresponds to a leave one out scheme

RoIs were selected estimating that the averaged AD size was 1024×1024 for the DDSM database. However, some lesions are actually displaced within the RoI, usually those observed in the Cranio-Caudal view. Others, close to the nipple, are also displaced since in such a case the lesion is not symmetrical and must be displaced for capturing the entire set of spiculae. In total, the number of of displaced cases was 28. The trained model was assessed using only these cases and the method correctly classified 26, an accuracy of 89.28 %.

Discussion and conclusions

This work introduced a new form of characterizing AD on mammograms by representing ROI tissue distribution as a graph that captures the salient directions of linear structures within a particular ROI. The AD saliency is highlighted using a graph structure that stores the intensity linear information in the weight of the edges connecting the borders of a ROI. This external border undergoes a dyadic partition that is driven by the maximal width of an AD spicula, in the present investigation four pixels for the mini-MIAS database and 32 for the DDSM.

The intrinsic AD characteristics contained in the graph are quantified using the spectral decomposition of the adjacency matrix. This allows the identification of linear clusters, represented by the set of ordered eigenvectors, where each eigenvector stands for a different degree of relevance or saliency. The final descriptor is constructed by concatenating the lines associated to the most important nodes, i.e., the largest value of each eigenvector. This descriptor was assessed and compared against two baselines, outperforming both a classic and a state-of-the-art approach.

The former, proposed by Ayres and Rangayyan, set a precedent by using directional and distribution information from curvilinear structures present in the breast parenchyma. The latter is a recent improvement proposed by Banik et al. uses node map information combined with texture features such as Haralick’s or the Fractal Dimension. The presented work was compared against this method using Haralick texture features because is a feature being commonly and widely used.

Results using synthetic images, which can be viewed as an “idealized case” of an AD, showed almost perfect classification results. This indicates that, even though it does not explicitly try to extract linear or spiculated patterns, LSD does highlight the image information permitting a consistent detection under ideal conditions. The main advantage of the method, however, is its robustness to noise. This is evidenced by its continued good performance under noisy conditions, opposed to the notable drop in performance of the method proposed by Banik. The method proposed by Ayres shows a very low discriminative power which is only slightly better than chance when noise is added (A z =0.52).

When evaluated with open databases, such as DDSM, the robustness of LSD is evidenced by showing similar results to the synthetic datasets and outperforming both the Banik and the Ayres methods. Although the features proposed by Banik are more rapidly learned, they are unable to cope with the larger biological variability introduced in the dataset.

As expected, the results are less concluding in case of the mini-MIAS database, probably due to a much smaller population (barely nineteen cases). Still, for the mini-MIAS basically the Banik strategy and LSD are equivalent, with very similar A z (LSD =0.93 and Banik =0.9).

Future work includes evaluation of the whole mammography as a screening approach. Interestingly, the graph structure might be used to compare different lesions in topological terms and to establish a similarity metric between lesions.

It should be noted that the method is dependent on parameters such as the cross sectional width of a typical spicule. This is a possible source of error that might be mitigated by fine tuning this parameter for a particular set of mammograms, e.g. by varying this number of pixels between nodes. This limitation is clearly common to any method that attempts to search spiculated lesions [3, 5, 18, 19, 21, 26, 34, 42].

Footnotes

Notes

Acknowledgments

This work was partially supported by the Ecuadorian government through ”La Secretaría de Educación Superior, Ciencia, Tecnología e Innovacion (SENESCYT)”, [Grant number: 20110958, 2011].

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Fabián Narváez
    • 1
  • Jorge Alvarez
    • 1
  • Juan D. Garcia-Arteaga
    • 1
  • Jonathan Tarquino
    • 1
  • Eduardo Romero
    • 1
    Email author
  1. 1.Computer Imaging and Medical Applications Laboratory - Cim@labFaculty of Medicine - Universidad Nacional de ColombiaBogotáColombia

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