An Efficient Algorithm for Fractal Analysis of Textures - Decom

More documents

Recommendations

Info

B. SFTA extraction algorithm After applying the Two Threshold Binary Decomposition to the input gray level image, the SFTA feature vector is constructed as the resulting binary images’ size, mean gray level and boundaries’ fractal dimension. The fractal measurements are employed to describe the boundary complexity of objects and structures segmented in the input image. The regions’ boundaries of a binary image Ib(x, y) are represented as a border image denoted by ∆(x, y) and computed as follows: ⎧ ⎪⎨ ∆(x, y) = ⎪⎩ 1 if ∃(x ′ , y ′ ) ∈ N8[(x, y)] : Ib(x ′ , y ′ ) = 0 ∧ Ib(x, y) = 1, 0, otherwise. where N8[(x, y)] is the set of pixels that are 8-connected to (x, y). ∆(x, y) takes the value 1 if the pixel at position (x, y) in the corresponding binary image Ib(x, y) has the value 1 and having at least one neighboring pixel with value 0. Otherwise, ∆(x, y) takes the value 0. Hence, one can realize that the resulting borders are one-pixel wide. The fractal dimension D is computed from each border image using the box counting algorithm described in section II. The mean gray level and size (pixel count) complement the information extracted from each binary image without significantly increasing the computation time. Thus, the SFTA feature vector dimensionality corresponds to the number of binary images obtained by TTBD multiplied by three, since the following measurements are computed from each binary image: fractal dimension, mean gray level and size. Figure 3 illustrates SFTA extraction algorithm. Figure 4 illustrates the three measurements that are extracted from each binary image. Algorithm 1 summarizes SFTA feature extraction process. VSFTA denotes the resulting feature vector. In line 1 the set of thresholds is computed using the multi-level Otsu algorithm. In line 2 all pairs of contiguous thresholds in T are added to TA. In line 3 the pairs of thresholds {ti, nl}, where nl corresponds to the maximum gray level in I(x, y), are added to TB. Line 5 iterates over all pairs of threholds in TA ∪ TB. Each pair of thresholds is used in line 6 to compute the two threshold segmentation as described in Equation 6. Lines 7-10 corresponds to fractal dimension, mean gray level and region area computation. Since fractal dimension can be efficiently computed in linear time by the box counting algorithm proposed in [18], SFTA extraction algorithm asymptotic complexity is O(N · |T |), where N is the number of pixels in the grayscale image I, and |T | is the number of different thresholds resulting from the multi-level Otsu algorithm. IV. EXPERIMENTS We evaluated our proposed method (SFTA) for the tasks of CBIR and image classification using three different datasets, Lung CT ROIs, KTH-TIPS and Textured Surfaces. The SFTA performance was compared to that of the following feature vectors: FFS, Haralick, Gabor, Histogram, Basic Texture and (7) Input Image Enhanced Binary Stack Decomposition I B1 I B2 I Bn Features Feature Vector ... ... D1 v1 A1 D2 v2 A2 Dn vn An Fig. 3. SFTA extraction diagram taking as input a grayscale image. First the input image is decomposed into a set of binary image by the TTBD algorithm. Then, the SFTA feature vector is constructed as the resulting binary images’ size, mean gray level and boundaries’ fractal dimension. Binary Image I B Border Finding A 1 = v 1 D 1 Area = Mean Fractal = Dimension Fig. 4. Features extracted from each binary image resulting from the TTBD. Area and mean gray level are computed directly from the binary image. Fractal dimension is computed from the border image (Equation 7). Combined. The extraction process for each feature vector is summarized in Table II. All extraction algorithms were implemented in Matlab. SFTA requires the user to set the parameter nt that defines the number of thresholds that will be employed in the input image decomposition. In section IV-E we provide an analysis showing how SFTA performance for image classification changes for different number of thresholds. Considering experimental results, we have set the nt parameter to 8. To evaluate the extraction algorithms for the task of CBIR, we employed precision and recall (P&R) curves. A rule of
Method Name TABLE II FEATURE EXTRACTION METHODS USED IN THE EXPERIMENTS. Description Number of Components SFTA SFTA feature vector using 8 thresholds. 48 FFS FFS feature vector using 8 thresholds. 8 Haralick Gabor Variance, entropy, uniformity, homogeneity, 3rd order moment, inverse of variance and step statistics from the image’s GLCMs. The GLCMs are computed using 4 angles, 5 displacements and 8 gray levels. Mean and standard deviation of the response of each filter from Gabor filter bank. The Gabor filter bank is generated using 6 orientations and 4 scales. Histogram 16 bin histogram computed by re-quantizing the input image to 16 gray levels. 16 Basic Texture Mean, contrast, skewness, kurtosis, entropy and standard deviation of the input image’s gray level distribution. Combined Combination of Haralick, Histogram, Basic Texture and Zernike moments [19]. 418 Algorithm 1 SFTA extraction algorithm. Require: Grayscale image I and number of thresholds nt. Ensure: Feature vector VSFTA. 1: T ← MultiLevelOtsu(I, nt) 2: TA ← {{ti, ti+1} : ti, ti+1 ∈ T, i ∈ [1..|T | − 1]} 3: TB ← {{ti, nl} : ti ∈ T, i ∈ [1..|T |]} 4: i ← 0 5: for {{tℓ, tu} : {tℓ, tu} ∈ TA ∪ TB} do 6: Ib ← TwoThresholdSegmentation(I, tℓ, tu) 7: ∆(x, y) ← FindBorders(Ib) 8: VSFTA[i] ← BoxCounting(∆) 9: VSFTA[i + 1] ← MeanGrayLevel(I, Ib) 10: VSFTA[i + 2] ← PixelCount(Ib) 11: i ← i + 3 12: end for 13: return VSFTA thumb to understand P&R curves is: the closer the curve is to the top of the graph, the better the technique’s retrieval performance is [20]. In P&R curves, precision and recall are defined as follows: number of relevant images retrieved precision = number of images retrieved number of relevant images retrieved recall = number of relevant images in dataset Each image was used as query center, returning the k most similar images using k-nearest neighbor (kNN) queries and the Euclidean distance function. An image was considered relevant when its class was the same as that of the query image. P&R curves were built by averaging the resulting curve of each query. As for the classification experiments, we make use of an SVM (Support Vector Machine) classifier, built on a polynomial kernel using the SMO (Sequential Minimal Optimization) (8) (9) algorithm [21] We chose SVM due to its effectiveness and wide exploitation in texture classification [8], [9]. A. CBIR, Lung CT ROIs Dataset The Lung CT ROIs dataset is composed of 3258 ROIs from lung computed tomography (CT) scans. Each ROI corresponds to an image of size 64 × 64 pixels classified either as normal or as one of the following lung disease patterns: consolidation, emphysema, thickening, honeycombing and ground glass. This dataset was used in [16] to evaluate the FFS extraction algorithm. Figure 6 shows a sample ROI from each class of the Lung CT ROIs dataset. 140 Consolidation Emphysema Thickening Normal Honeycombing Ground Glass Fig. 5. Sample ROI from each class of the Lung CT ROIs dataset Figure 6 shows the precision and recall graph obtained for the Lung CT ROIs dataset. SFTA corresponds to the black curve with ∗ marks, presenting the highest image retrieval precision for all recall levels. Gabor starts as the second best feature vector but its precision degrades for recall levels higher than 0.15 and is surpassed by Histogram. A possible explanation for this result is the problem of the curse of dimensionality. That is, the high dimensionality of the Gabor feature vector contributes to reduce its precision. 48 6
Page 1 and 2: An Efficient Algorithm for Fractal
Page 3: into a set of binary images. To dec
Page 7 and 8: Table III shows classification accu

An Efficient Algorithm for Fractal Analysis of Textures - Decom

Create successful ePaper yourself

Delete template?

Save as template?