Learning vector quantization inference classifier in breast abnormality classification

Abstract

The Mammographic image is a tool for observing breast cancer. Analyzing difficulties include shape, size variety, nearby tissue, and noise. In this paper, we propose a method to classify mammogram abnormalities based on learning vector quantization inference classifier (LVQIC) with fuzzy co-occurrence matrix (FCOM) textural features. The system is implemented on the Mini-MIAS data set with a 5-class problem, i.e., the classification of architectural distortion (AD), spiculated mass (SPIC), calcification (CALC), well-defined/circumscribed masses (CIRC), and normal (NORM). The implementation is also on a 2-class problem consisting of AD-vs-All, SPIC-vs-All, CALC-vs-All, CIRC-vs-All, and NORM/abnormal. The best blind test result is from the 5-class problem with features from fuzzy co-occurrence matrix (FCOM) with 4 clusters, co-occurrence distance d = 2, and 16 prototypes per class. The best classification result is 100% correct classification with 0.03, 0.04, 0.06, and 0.02 false positive rate for AD, SPIC, CALC, and CIRC, respectively.

Keywords

Mammograms breast cancer breast abnormality detection neuro fuzzy feature selection

1 Introduction

Breast cancer is one of the most occurring diseases in women. It is also one of the fatal diseases. However, detecting the cancer at the earlier stage can help easing the curing process [1]. The Beast Imaging-Reporting and Data System (BI-RADS) [1], a quality assurance tool in mammography, is a form of reported results. There are several factors in BI-RADS including architectural distortion (AD), asymmetry (ASYM), calcification (CALC), well-defined/circumscribed masses (CIRC), other ill-defined masses (MISC), and spiculated mass (SPIC). There are several research works involving the abnormality detection without categorizing the abnormality types [2 –13]. There are some research works involving with abnormality types detection either detecting one type, i.e., the CALC [14 –20] or detecting two to five types [21 –26]. Most the existing works utilized a traditional method including feature generation and classification. However, there are many feature extraction methods. It might not be easy to find a good feature set for this problem. Also, it might be easy for human to understand the system if the system can provide rules for the detection.

Hence, in this paper we develop a mammogram abnormality detection model using the learning vector quantization inference classifier (LVQIC) [27] and the fuzzy co-occurrence matrix (FCOM) [26, 28]. The FCOM is utilized to generate a feature set for the problem. The LVQIC is utilized to select good features and generate rule set for the detection.

2 Methods and materials

2.1 Fuzzy co-occurrence matrix

In this paper, we use the fuzzy co-occurrence matrix (FCOM) [26, 28] as our feature generation method. The FCOM is a novel texture features based on the incorporation of the fuzzy set theory into the Gray Level Co-occurrence Matrix (GLCM) [29]. First, the fuzzy C-means (FCM) [30] with m = 2 is implemented on an original gray scale image I. Then each pixel is assigned to a cluster with the highest membership value in order to create the FCOM. The number of clusters (C) and the number of directions are used to indicate the number of FCOM planes. In our experiment, we set direction (θ) to 0, 45, 90, and 135 degrees. Hence, there are C × 4 FCOM planes. We set d = 1 or 2 in the experiment. The summary of fuzzy co-occurrence matrix algorithm is as follows:

-For each direction (θ)

-For each pixel (p)

-Find pixel (q) that is d apart from p

-Find the assigned cluster of pixel q

(Suppose assigned cluster of p is k and assigned cluster of q is i)

-Set all FCOMs to zero

-For each cluster (i)

-FCOM(i, k, l) = FCOM(i, k, l) + u_pi + u_qi

(where u_pi and u_qi are the membership values of pixels p and q in cluster i)

-End For i

-End For p

-End For θ

In the experiment, we implement the FCM with the number of clusters (C) of 4 and 8. Then for 4 clusters, each FCOM is with the size of 4×4 creating 16 FCOM planes, whereas that with the size of 8×4 creating 32 FCOM planes for 8 clusters. We compute 14 features according to Equations (1 to 15). Hence there are 14×16 = 224 and 14×32 = 448 dimensions for 4 clusters and 8 clusters, respectively. The mean and standard deviation of all directions for each of 14 features are also utilized. Hence there are 14×24 = 336 and 14×48 = 672 dimensions for 4 clusters and 8 clusters, respectively.

Energy: $f 1_{l} = \sum_{i, j} {FCOM (l, i, j)}^{2}$ (1)

Contrast: $f 2_{l} = \sum_{i, j} (i - j)^{2} FCOM (l, i . j)$ (2)

Correlation: $f 3_{l} = \sum_{i, j} \frac{(i - μ_{i}^{l}) (j - μ_{j}^{l}) FCOM (l, i, j)}{σ_{i}^{l} σ_{j}^{l}}$ (3)

Variance: $f 4_{l} = \sum_{i, j} (i - μ_{i}^{l}) (j - μ_{j}^{l}) FCOM (l, i, j)$ (4)

Homogeneity: $f 5_{l} = \sum_{i, j} \frac{FCOM (l, i, j)}{1 + | i - j |}$ (5)

Sum Average: $f 6_{l} = \sum_{i, j} (ij) FCOM (l, i, j)$ (6)

Sum Variance: $f 7_{l} = \sum_{i, j} (ij - f 6_{l})^{2} FCOM (l, i, j)$ (7)

Sum Entropy: $\begin{matrix} f 8_{l} = - \sum_{i, j} {FCOM}_{x + y} (l, i, j) \\ log {{FCOM}_{x + y} (l, i, j)} \end{matrix}$ (8)

Entropy: $f 9_{l} = - \sum_{i, j} FCOM (l, i, j) log {FCOM (l, i, j)}$ (9)

Difference Variance: $f 10_{l} = var {{FCOM}_{x - y} (l)}$ (10)

Difference Entropy: $\begin{matrix} f 11_{l} = - \sum_{i, j} {FCOM}_{x - y} (l, i, j) \\ log {{FCOM}_{x - y} (l, i, j)} \end{matrix}$ (11)

Information Measure of Correlation 1: $f 12_{l} = \frac{f 9_{l} - HXY 1}{max {HX, HY}}$ (12)

Information Measure of Correlation 2: $f 13_{l} = {1 - exp [- 2 (HXY 2 - f 9_{l})]}^{\frac{1}{2}}$ (13)

Maximal Correlation Coefficient: $f 14_{l} = (second largest eigenvalue of Q)^{\frac{1}{2}}$ (14) where, for 1 ≤ l ≤ C, $\begin{matrix} {FC}_{x}^{l} (i) = \sum_{j} FCOM (l, i, j) \\ {FC}_{y}^{l} (j) = \sum_{i} FCOM (l, i, j) \\ μ_{i}^{l} = \sum_{i} {iFC}_{x}^{l} (i) \\ μ_{j}^{l} = \sum_{j} {jFC}_{y}^{l} (j) \\ σ_{i}^{l} = \sum_{i} (i - μ_{i}^{l})^{2} {FC}_{x}^{l} (i) \\ σ_{j}^{l} = \sum_{j} (j - μ_{j}^{l})^{2} {FC}_{y}^{l} (j) \\ \begin{matrix} {FCOM}_{x + y} (l) = \sum_{i, j} \underset{i + j = k}{FCOM} (l, i, j), \\ for k = 2, 3, \dots, 2 C \end{matrix} \\ \begin{matrix} {FCOM}_{x - y} (l) = \sum_{i, j} \underset{| i - j | = k}{FCOM} (l, i, j), \\ for k = 0, 1, \dots, C - 1 \end{matrix} \\ HX = entropy of {FC}_{x}^{l} \\ HY = entropy of {FC}_{y}^{l} \\ HXY 1 = - \sum_{i, j} FCOM (l, i, j) log {{FC}_{x}^{l} (i) {FC}_{y}^{l} (j)} \\ HXY 2 = - \sum_{i, j} {FC}_{x}^{l} (i) {FC}_{y}^{l} (j) log {{FC}_{x}^{l} (i) {FC}_{y}^{l} (j)} \\ Q (i, j) = \sum_{k} \frac{FCOM (l, i, k) FCOM (l, j, k)}{{FC}_{x}^{l} (i) {FC}_{y}^{l} (k)} \end{matrix}$ (15)

2.2 Learning vector quantization inference classifier

After generating feature set mentioned in the previous section, the learning vector quantization inference classifier (LVQIC) [27, 31] is used to select feature and classify. The LVQIC is a neuro-fuzzy model based on learning vector quantization (LVQ) [32]. It has ability of classification, feature selection, and rule extraction. Figure 1 shows the structure of the LVQIC.

Fig.1

The structure of LVQIC.

Layer 1: This layer simply contains input features. Each feature is normalized to 0 to 1.

Layer 2: The closest prototype and the closest feature are determined. The closest prototype is calculated from:

for each prototype j, compute $D_{j} = ∥ {\vec{X}}_{n} - {\vec{G}}_{j} ∥,$ (16) and $j_{win} = \underset{j}{arg min} {D_{j}} .$ (17)

Then we need to find a winning prototype of each feature by,

for each feature i, compute $D_{ji} = | x_{i} - g_{ji} |, j = 1, \dots, Q,$ (18) and $j_{win}^{(i)} = \underset{j}{arg min} {D_{ji}} .$ (19)

Layer 3: There are 2 types of prototype closeness and feature closeness, i.e.,

1. Hard method (winner-takes-all) $μ G (x) = {\begin{matrix} 1 if x = D_{min} \\ 0 else \end{matrix},$ (20) where x ∈ {D_j} , D_min ≤ D_j ∀ j. $μ F_{i} (x) = {\begin{matrix} 1 if x = D_{min}^{(i)} \\ 0 else \end{matrix},$ (21) where $x \in {D_{ji}}, D_{min}^{(i)} \leq D_{ji} \forall j$ .

2. Soft method (winner-takes-most) $μ G (x) = \frac{D_{max} - x}{D_{max} - D_{min}},$ (22) where x ∈ {D_j} , D_max ≥ D_jand D_min ≤ D_j, ∀ j. $μ F_{i} (x) = \frac{D_{max}^{(i)} - x}{D_{max}^{(i)} - D_{min}^{(i)}},$ (23) where $x \in {D_{ji}}, D_{max}^{(i)} \geq D_{ji} and D_{min}^{(i)} \leq D_{ji}, \forall j$ .

Both closeness values are accounted as the output of this layer as follow: $h_{ji} = μ G_{j} \times μ F_{ji},$ (24) where μG_j = μG (D_j) , μF_ji = μF_i (D_ji).

Then the output vector of this layer will be ${\vec{H}}_{j} = {[h_{ji}]}_{i = 1}^{F}$ .

Layer 4: The output of this layer is calculated as $y_{k} = \sum_{j = 1}^{P} \sum_{i = 1}^{F} w_{kji} h_{ji},$ (25) where h_ji ∈ [0, 1], y_k ∈ [0, 1], w_kji ∈ [0, 1], $\sum_{j = 1}^{P}$ $\sum_{i = 1}^{F} w_{kji} = 1$ . The class label of the input vector is calculated by selecting the output node with the maximum output value, i.e., $class = l where y_{l} \geq y_{k} \forall k .$ (26)

The training process is based on the concept of the learning vector quantization and the gradient descent. Please be noted that the details of the training process of the LVQIC are shown in [30].

The output weight values are utilized in the feature selection process as follows:

-Find ${FR}_{i} = \sum_{k = 1}^{M} \sum_{j = 1}^{Q} w_{kji}$ for all i.

-Initialize topc = 0, slf = 0, count = 0, FS = () ##no. selected features.

-Repeat

-Pick ith feature from the remaining features which FR_i is maximum.

-Keep ith feature in FS and set count = count + 1.

-Set c to the classification rate of the training data set with features in FS

-If c > topc then topc = c, slf = count.

-Until no remaining feature.

-Set FS' = {k|k is a feature in the first selected slf features in FS}.

-Set FS = FS'. ##FS contains only the first selected slf features.

-Initialize topc = 0, slw = 0, count = 0, WS = () ## no. selected weights.

##Consider only weights w_kji connecting to the first selected slf

##features in FS.

-Repeat

-Pick w_kji connecting to the remaining features in the FS.

-Keep weight w_kji in WS and set count = count + 1.

-Set c to the classification rate of the training data set with weights in WS

-If c > topc then topc = c, slw = count.

-Until no remaining output weight.

-Set WS' = {l|l is a weight in the first selected slw weights in WS}.

-Set WS = WS' ##WS contains only the first selected slw weights.

-Return slf, slw, FS, WS.

The rule extraction is determined according to the prototype features and the output weights. There are 2 rule categories, i.e., crisp rule and fuzzy rule. We only consider weights w_kji that are in WS. For input feature i (x_i) and prototype g_ji connecting to w_kji, the generated rule can be

Crisp rules: IF (x_i is CLOSE TO g_ji) THEN class k

Fuzzy rules: IF w_kji×(x_i is CLOSE TO g_ji) THEN class k

2.3 Dataset

The Mammographic Image Analysis Society called mini-MIAS [33] is utilized in the experiment. There are 322 images of architectural distortion (AD), asymmetry (ASYM), calcification (CALC), well-defined/circumscribed masses (CIRC), other ill-defined masses (MISC), and spiculated mass (SPIC) and normal (NORM), each with the size of 1024×1024. Examples are shown in Fig. 2. We, however, only detect AD, SPIC, CALC, and CIRC in the experiment. The signature library is created from cropped images with size of 64×64 on the target regions selected from 8 AD, 9 SPIC, 6 CALC, 8 CIRC, and 42 NORM mammograms. At the end, the signature library contains 42 AD, 42 SPIC, 21 CALC, 21 CIRC, and 42 NORM images. Examples are shown in Fig. 3. The system is trained from the images in the signature library with 7-fold cross validation. The best model is selected to test with the blind test data set. The remaining of AD, SPIC, CALC, CIRC, and NORM mammograms are used as a blind test data set. There are 220 mammograms in total for the blind test data set.

Fig.2

Original mammograms.

Fig.3

Example of sub-image in the signature library.

3 Experiments

As mentioned in the previous section, there are 2 types of feature set used in the experiment, i.e., features from FCOM without means and standard deviations (called Feat1) and features from FCOM with means and standard deviations (called Feat2). To train the system, the number of prototypes per class is set to 4, 7, 11, and 16. The LVQIC models are trained with 7-fold cross validation using only the data in the signature library. Each model is trained 5 times. Tables 1 and 2 show the example of the validation set result for 5-class classification using Feat1 and Feat2, respectively. The feature set Feat1 with 8 clusters, d = 1, and 11 prototypes per class (Feat1C8D1P11) and set Feat2 with 4 clusters, d = 2, and 16 prototypes per class (Feat2C4D2P16) both yield 79.17% correct classification. The 5-trial average classification rate from Feat1C8D1P11 is 69.17% ±6.97%, whereas that of Feat2C4D1P11 is 64.17±9.13. Tables 3 10 show the example of the validation set result for 2-class classification using Feat1 and Feat2. For AD-vs-All case, Feat1 with 8 clusters, d = 2 and 16 prototypes per class (Feat1C8D2P16) and Feat2 with 4 clusters, d = 2 and 11 prototypes per class (Feat2C4D2P11) both provide 91.67% correct classification. Feat1C8D2P16 provides 5-trial classification rate of 83.33% ±5.1%, whereas Feat2C4D2P11 provides that of 85.00% ±4.75%. The best model for the SPIC-vs-All using Feat1 with 4 clusters, d = 2 and 4 prototypes per class (Feat1C4D2P4) and Feat2 with 4 clusters, d = 1 and 11 prototypes per class (Feat2C4D1P11) give 83.33% correct classification. While the 5-trial average classification rate of Feat1C4D2P4 and Feat2C4D1P11 are 76.67% ±3.73% and 78.33% ±4.56%, respectively. In CALC classification, Feat1 with 8 clusters, d = 1 and 7 prototypes per class (Feat1C8D1P7) and Feat2 with 8 clusters, d = 2 and 11 prototypes per class (Feat2C8D2P11) give 95.83% correct classification rate. The 5-trial average classification rate from Feat1C8D1P7 and Feat2C8D2P11 in this case are 90.00% ±3.73%. The CIRC classification rate in 2-class problem using Feat1 with 4 clusters, d = 2 and 11 prototypes per class (Feat1C4D2P11) and Feat2 with 4 clusters, d = 2 and 7 prototypes per class (Feat2C4D2P7) are 100%. The 5-trial average classification rate from Feat1C4D2P11 and Feat2C4D2P7 are 95.00% ±3.48% and 96.66% ±1.86%, respectively.

Table 1
Example of validation set result of 5-class classification using Feat1

Cluster, Distance, Prototype best trial 5 trials

#rules #feat. % avg±std of 7 folds best fold % avg±std of best fold

C4D2P4 383 24 39.88±13.15 62.50 54.17±6.59

C4D2P7 3047 88 53.57±12.08 79.17 70.00±6.85

C8D1P7 3451 236 53.57±11.66 75.00 66.67±5.10

C4D1P11 4289 104 52.38±12.37 75.00 65.83±6.18

C4D2P11 6418 163 53.57±7.19 62.50 60.00±2.28

C8D1P11 5289 105 55.36±16.02 79.17 69.17±6.97

C8D2P11 5861 117 52.98±10.85 75.00 67.50±6.85

C4D2P16 7298 123 54.17±11.14 70.83 65.83±4.57

C8D1P16 6026 86 47.62±13.34 75.00 69.17±6.32

C8D2P16 9389 289 54.76±11.23 70.83 65.83±3.49

Cluster, Distance, Prototype	best trial	5 trials
C4D2P4	383	24	39.88±13.15	62.50	54.17±6.59
C4D2P7	3047	88	53.57±12.08	79.17	70.00±6.85
C8D1P7	3451	236	53.57±11.66	75.00	66.67±5.10
C4D1P11	4289	104	52.38±12.37	75.00	65.83±6.18
C4D2P11	6418	163	53.57±7.19	62.50	60.00±2.28
C8D1P11	5289	105	55.36±16.02	79.17	69.17±6.97
C8D2P11	5861	117	52.98±10.85	75.00	67.50±6.85
C4D2P16	7298	123	54.17±11.14	70.83	65.83±4.57
C8D1P16	6026	86	47.62±13.34	75.00	69.17±6.32
C8D2P16	9389	289	54.76±11.23	70.83	65.83±3.49

Table 2

Example of validation set result of 5-class classification using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	2510	131	43.45±7.67	58.33	53.33±3.49
C4D2P7	3760	184	48.21±11.33	70.83	61.67±6.18
C8D1P7	2436	376	49.40±13.07	79.17	65.83±9.04
C4D1P11	5982	154	54.76±8.75	70.83	62.50±5.89
C4D2P11	6286	129	55.36±13.31	75.00	64.17±6.32
C8D1P11	16615	444	52.38±13.15	79.17	70.00±6.18
C8D2P11	8843	224	58.93±11.23	75.00	70.00±5.43
C4D2P16	5678	115	53.57±11.66	79.17	64.17±9.13
C8D1P16	11324	199	53.57±11.23	75.00	67.50±8.01
C8D2P16	12475	303	57.14±9.63	75.00	64.17±6.32

Table 3

Example of validation set result of 2-class (AD-vs-all) using Feat1

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	30	6	77.38±5.39	87.50	80.00±5.43
C4D2P7	58	5	76.19±2.92	79.17	77.50±2.28
C8D1P7	182	18	76.19±1.88	79.17	75.83±1.86
C4D1P11	141	12	78.57±4.69	87.50	80.83±4.75
C4D2P11	775	95	77.38±3.76	83.33	81.66±3.73
C8D1P11	170	10	77.38±3.04	83.33	80.00±3.48
C8D2P11	174	8	77.38±6.63	91.67	81.67±7.57
C4D2P16	1462	102	75.00±7.04	87.50	82.50±3.48
C8D1P16	1145	37	76.79±6.99	87.50	81.67±3.72
C8D2P16	357	24	78.57±10.31	91.67	83.33±5.10

Table 4

Example of validation set result of 2-class (AD-vs-all) using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	15	5	76.79±3.04	83.33	77.50±3.73
C4D2P7	54	4	77.38±3.04	83.33	81.67±2.28
C8D1P7	96	10	74.40±5.65	83.33	77.50±3.73
C4D1P11	118	9	77.98±1.88	79.17	79.17±0.00
C4D2P11	739	197	80.36±5.32	91.67	85.00±4.75
C8D1P11	672	32	75.00±4.98	83.33	77.50±3.73
C8D2P11	140	25	76.19±6.57	87.5	80.83±4.75
C4D2P16	48	5	75.60±7.53	87.5	84.16±1.86
C8D1P16	935	442	77.38±3.04	83.33	81.67±2.28
C8D2P16	433	18	75.00±7.39	91.67	86.67±4.56

Table 5

Example of validation set result of 2-class (SPIC-vs-all) using Feat1

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	28	4	76.19±2.92	83.33	76.67±3.73
C4D2P7	3	1	75.60±1.46	79.17	75.83±1.86
C8D1P7	1	1	74.40±1.46	75.00	75.00±0.00
C4D1P11	88	4	75.00±3.86	83.33	80.00±3.48
C4D2P11	136	36	76.19±2.92	83.33	79.17±2.95
C8D1P11	33	2	75.60±2.66	79.17	75.83±1.86
C8D2P11	59	7	75.60±1.46	79.17	76.67±2.28
C4D2P16	29	7	74.40±5.65	83.33	77.50±3.73
C8D1P16	357	13	75.60±3.47	83.33	78.33±4.56
C8D2P16	2470	81	76.19±1.88	79.17	78.34±1.86

Table 6

Example of validation set result of 2-class (SPIC-vs-all) using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	1	1	75.00±0.00	75.00	75.00±0.00
C4D2P7	42	3	76.19±2.92	83.33	78.33±3.49
C8D1P7	36	3	76.19±2.92	83.33	77.50±3.73
C4D1P11	226	13	76.79±4.37	83.33	78.33±4.56
C4D2P11	59	3	74.40±5.19	83.33	77.50±3.73
C8D1P11	204	170	75.60±1.46	79.17	76.67±2.28
C8D2P11	53	4	76.19±3.67	83.33	80.00±3.48
C4D2P16	804	73	73.21±6.63	83.33	79.17±2.95
C8D1P16	1572	62	74.40±3.47	79.17	75.83±1.86
C8D2P16	784	33	75.60±1.46	79.17	78.34±1.86

Table 7

Example of validation set result of 2-class (CALC-vs-all) using Feat1

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	1	1	87.50±0.00	87.50	87.50±0.00
C4D2P7	1	1	87.50±0.00	87.50	87.50±0.00
C8D1P7	51	4	88.69±2.92	95.83	90.00±3.73
C4D1P11	79	4	87.50±3.86	95.83	90.00±3.73
C4D2P11	95	5	87.50±2.23	91.67	88.33±1.86
C8D1P11	7	1	86.31±4.84	91.67	88.33±1.86
C8D2P11	2	1	86.31±2.92	91.67	89.17±2.28
C4D2P16	1	1	87.50±0.00	87.50	87.50±0.00
C8D1P16	1	1	86.90±1.46	87.50	87.50±0.00
C8D2P16	241	8	86.90±2.66	91.67	90.00±2.28

Table 8

Example of validation set result of 2-class (CALC-vs-all) using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	1	1	87.50±0.00	87.50	87.50±0.00
C4D2P7	1	1	87.50±0.00	87.50	87.50±0.00
C8D1P7	54	4	88.10±2.66	91.67	88.33±1.86
C4D1P11	1	1	87.50±0.00	87.50	87.50±0.00
C4D2P11	510	92	86.90±3.47	91.67	88.33±1.86
C8D1P11	128	18	87.50±2.23	91.67	89.17±2.28
C8D2P11	36	2	87.50±3.86	95.83	90.00±3.73
C4D2P16	262	10	87.50±2.23	91.67	88.33±1.86
C8D1P16	1	1	87.50±0.00	87.50	87.50±0.00
C8D2P16	97	8	88.10±2.66	91.67	90.00±2.28

Table 9

Example of validation set result of 2-class (CIRC-vs-all) using Feat1

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	8	1	90.48±2.92	95.83	94.17±2.28
C4D2P7	31	11	91.67±3.15	95.83	93.33±2.28
C8D1P7	1	1	87.50±0.00	87.5	87.50±0.00
C4D1P11	143	57	89.88±4.37	95.83	91.67±2.95
C4D2P11	193	20	89.29±6.24	100.00	95.00±3.48
C8D1P11	155	13	86.90±2.66	91.67	89.17±2.28
C8D2P11	87	5	88.10±1.46	91.67	88.33±1.86
C4D2P16	128	11	91.67±2.23	95.83	95.83±0.00
C8D1P16	1133	64	89.29±3.04	95.83	92.50±1.86
C8D2P16	406	13	86.31±5.32	95.83	92.50±1.86

Table 10

Example of validation set result of 2-class (CIRC-vs-all) using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	83	11	89.29±4.37	100.00	95.00±3.48
C4D2P7	330	77	89.88±4.37	100.00	96.66±1.86
C8D1P7	86	7	87.50±2.23	91.67	88.33±1.86
C4D1P11	353	19	91.07±4.12	95.83	94.17±2.28
C4D2P11	127	7	90.48±3.67	95.83	95.00±1.86
C8D1P11	110	5	89.29±3.04	95.83	91.67±4.17
C8D2P11	4	1	88.10±1.46	91.67	90.00±2.28
C4D2P16	1555	176	92.26±3.47	95.83	95.00±1.86
C8D1P16	1294	91	89.88±5.83	100.00	93.34±3.73
C8D2P16	344	14	89.29±3.04	95.83	93.33±2.28

Another 2-class problem is when we consider AD, SPIC, CALC, and CIRC as one abnormal class. The best validation result of Feat1 and Feat2 are shown in Tables 11 and 12. Feat1 with 4 clusters, d = 1 and 11 prototypes per class (Feat1C4D1P11) and Feat2 with 8 clusters, d = 1 and 16 prototypes per class (Feat2C8D1P16) provide 100% correct classification. Whereas the 5-trial average classification rate are 94.17% ±3.72% for Feat1C4D1P11 and 93.34% ±3.73% for Feat2C8D1P16.

Table 11

Example of validation set result of 2-class (normal-vs-abnormal) using Feat1

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	28	18	83.93±5.19	95.83	93.33±3.72
C4D2P7	27	2	83.93±6.47	95.83	90.00±3.73
C8D1P7	53	25	83.93±7.85	95.83	93.33±3.72
C4D1P11	202	29	85.71±6.99	100.00	94.17±3.72
C4D2P11	42	2	83.93±7.19	100.00	93.33±4.75
C8D1P11	321	20	81.55±6.63	95.83	89.17±4.75
C8D2P11	488	23	84.52±8.83	95.83	91.67±2.95
C4D2P16	115	6	80.36±7.29	95.83	91.67±4.17
C8D1P16	9	8	83.93±8.46	100.00	92.50±5.43
C8D2P16	106	7	80.95±8.29	100.00	91.67±5.10

Table 12

Example of validation set result of 2-class (normal-vs-abnormal) using Feat2

Cluster, Distance, Prototype	best trial				5 trials
Cluster, Distance, Prototype	#rules	#feat.	% avg±std of 7 folds	best fold	% avg±std of best fold
C4D2P4	279	169	85.12±4.91	91.67	90.00±2.28
C4D2P7	103	8	84.52±5.77	91.67	88.33±1.86
C8D1P7	327	90	83.33±4.98	91.67	90.84±1.86
C4D1P11	21	18	85.12±5.83	91.67	90.84±1.86
C4D2P11	307	25	79.76±5.19	87.50	88.33±1.86
C8D1P11	141	7	83.93±7.19	95.83	89.17±4.75
C8D2P11	84	4	82.74±5.19	91.67	88.33±3.49
C4D2P16	348	12	82.74±5.19	91.67	91.67±0.00
C8D1P16	279	10	82.74±8.16	100.00	93.34±3.73
C8D2P16	609	22	82.14±4.84	91.67	90.00±2.28

After the best model is selected for each case, we test each model on the blind test data set. A window of 64×64 is scanned with step size of 16 pixels from top to bottom and left to right to generate features and classify object. The result from the system is put at the center of that window. To show the performance of the system, we create the receiver operating characteristics (ROC) curve. The curve shows the probability of detection against the false positive rate. The false positive rate is a ratio of the number of false regions to the number of regions of the other classes. We counted false positive as one false positive if all pixels are connected as 8-connected component. The same method is utilized to count the detection region. Figures 4 and 5 show the ROC curves for the 5-class problem with Feat1C8D1P11 and Feat2C4D2P16, respectively.

Fig.4

Blind test result of the 5-class problem with Feat1C8D1P11.

Fig.5

Blind test result of the 5-class problem with Feat2C4D2P16.

Figure 6 shows the blind test ROC curve of AD-vs-All with Feat1C8D2P16, SPIC-vs-All with Feat1C4D2P4, CALC-vs-All with Feat1C8D1P7, and CIRC-vs-All with Feat1C4D2P11. While Fig. 7 shows that of AD-vs-All with Feat2C4D2P11, SPIC-vs-All with Feat2C4D1P11, CALC-vs-All with Feat2C8D2P11, and CIRC-vs-All with Feat2C4D2P7. Figures 8 and 9 show the blind test ROC curves of normal-vs-abnormal with Feat1C4D1P11 and Feat2C8D1P16, respectively.

Fig.6

Blind test result of the 2-class problem with Feat1.

Fig.7

Blind test result of the 2-class problem with Feat2.

Fig.8

Blind test result of the 2-class problem (normal-vs-abnormal with Feat1C4D1P11.

Fig.9

Blind test result of the 2-class problem (normal-vs-abnormal with Feat2C8D1P16.

4 Discussion

Table 13 shows the best blind test result from each model. We can see that the best result is from 5-class problem with the Feat2C4D2P16 model. The detection result is as high as 100% correct classification. However, there are some false positives shown in the result.

Table 13
The best blind test result from each model

Problem Model AD SPIC CALC CIRC Abnormal

5-class Feat1C8D1P11 100% with 0.07 FPRs 100% with 0.05 FPR 100% with 0.07 FPR 100% with 0.02 FPR

Feat2C4D2P16 100% with 0.03 FPR 100% with 0.04 FPR 100% with 0.06 FPR 100% with 0.02 FPR −

2-class (AD-vs-All) Feat1C8D2P16 100% with 0.06 FPR − − − −

Feat2C4D2P11 80% with 0.06 FPR − − − −

2-class (SPIC-vs-All) Feat1C4D2P4 − 100% with 0.05 FPR − − −

Feat2C4D1P11 − 100% with 0.05 FPR − − −

2-class (CALC-vs-All) Feat1C8D1P7 − − 80% with 0.04 FPR − −

Feat2C8D2P11 − − 100% with 0.15 FPR − −

2-class (CIRC-vs-All) Feat1C4D2P11 − − N/A 100% with 0.05 FPR −

Feat2C4D2P7 − − − 100% with 0.06 FPR −

2-class (normal-vs-abnormal) Feat1C4D1P11 − − − − 100% with 0.06 FPR

Feat2C8D1P16 − − − − 100% with 0.02 FPR

Problem	Model	AD	SPIC	CALC	CIRC	Abnormal
5-class	Feat1C8D1P11	100% with 0.07 FPRs	100% with 0.05 FPR	100% with 0.07 FPR	100% with 0.02 FPR
Feat2C4D2P16	100% with 0.03 FPR	100% with 0.04 FPR	100% with 0.06 FPR	100% with 0.02 FPR	−
2-class (AD-vs-All)	Feat1C8D2P16	100% with 0.06 FPR	−	−	−	−
Feat2C4D2P11	80% with 0.06 FPR	−	−	−	−
2-class (SPIC-vs-All)	Feat1C4D2P4	−	100% with 0.05 FPR	−	−	−
Feat2C4D1P11	−	100% with 0.05 FPR	−	−	−
2-class (CALC-vs-All)	Feat1C8D1P7	−	−	80% with 0.04 FPR	−	−
Feat2C8D2P11	−	−	100% with 0.15 FPR	−	−
2-class (CIRC-vs-All)	Feat1C4D2P11	−	−	N/A	100% with 0.05 FPR	−
Feat2C4D2P7	−	−	−	100% with 0.06 FPR	−
2-class (normal-vs-abnormal)	Feat1C4D1P11	−	−	−	−	100% with 0.06 FPR
Feat2C8D1P16	−	−	−	−	100% with 0.02 FPR

Example of successful detection with some false positives of each model are shown in Figs. 10 15. We can also see that there are many false positive regions. One of the reasons might be because window size of 64×64 is smaller than that of many abnormalities areas. Some areas of AD, SPIC, CIRC, and CALC are bigger than the window size, hence, the LVQIC cannot classify those regions correctly.

Fig.10

Example of the mammogram result of the 5-class problem with Feat1C8D1P11.

Fig.11

Example of mammogram result of the 5-class problem with Feat2C4D1P11.

Fig.12

Example of the mammogram result of the 2-class problem with Feat1.

Fig.13

Example of the mammogram result of the 2-class problem with Feat2.

Fig.14

Example of the mammogram result of the 2-class problem (normal-vs-abnormal with Feat1C4D1P11.

Fig.15

Example of the mammogram result of the 2-class problem (normal-vs-abnormal with Feat2C8D1P16.

Table 14 shows the ability of our system that is comparable with the existing state-of-the-art methods. We can see that our method is comparable with the existing ones. However, there is no pre-processing or ROI selection in this proposed system.

Table 14

Existing and our purposed method detection result

Method	Best Detection Result						Pre-process	ROI selection
	AD	SPIC	CALC	CIRC	MISC	ASYM	(y/n)	(y/n)
Netprasat, et al. [21]	100% with 16 FPI	58.33% with 8 FPI	−	−	−	−	n	n
Kanadam and Chereddy [22]	100%	100%	100%	100%	100%	100%	y	y
Khouaja, et al. [23]	94.7%	89.40%	−	95.8%	86.6%	−	y	y
Tivatansakul and Uchimura [24]	−	77.50%	−	80.05%	82.70%	−	y	y
Anitha, et al. [25]	89.50%	84.20%	−	100%	100%	93.30%	y	y
Munklang, et al. [26]	100% with 9.46 FPI	90% with 13.72 FPI	100% with 3.39 FPI	81.25% with 18 FPI	−	−	n	n
Our purposed method	100% with 0.03 FPR	100% with 0.04 FPR	100% with 0.06 FPR	100% with 0.02 FPR	−	−	n	n

Note FPI stands for false positive per image.

5 Conclusion

This paper proposes a scheme of breast abnormalities detection in mammograms using fuzzy co-occurrence matrix (FCOM) to extract features and learning vector quantization inference classifier (LVQIC) to classify the abnormalities. We implement the system on 5-class problem. i.e., architectural distortion (AD), spiculated mass (SPIC), calcification (CALC), well-defined/circumscribed masses (CIRC), and normal (NORM). The system is also implemented on the 2-class problem, i.e., AD-vs-All, SPIC-vs-All, CALC-vs-All, CIRC-vs-All, and NORM/abnormal. The best blind test result is from the 5-class problem with features from FCOM with means and standard deviations with 4 clusters, d = 1, and 11 prototypes per class. The best classification result is 100% correct classification with 0.03, 0.04, 0.06, and 0.02 false positive rates for AD, SPIC, CALC, and CIRC.

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