Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

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4 Automated pit pattern classification 4.3.4 Centroid classification (CC) Similar to the best-basis method presented in section 4.2.1.1 this method is based on the calculation of the best wavelet packet basis too. But in contrast to the best-basis method this method is not based on features extracted from the subbands resulting from the wavelet packet decomposition but on the quadtrees resulting from the wavelet packet decomposition. The idea behind this method is the question whether images decomposed using the wavelet packets algorithm in conjunction with the best-basis algorithm result in decomposition trees which can be further used for classification. A perfect classification would then be possible if trees of one image class are more similar than trees resulting from images of different classes. In other words, the similarity between trees of the same class must be much higher than the similarity of trees of different classes to be able to perform some useful classification. The similarity of two quadtrees can be expressed by the distance between them as explained in section 4.3.1. It is worthwhile to mention that having a method to express the distances between two quadtrees we could easily use that distance measure for a k-NN classification. This is exactly what the feature extractor KNNTD (see section 4.3.3.3 above) does. Since we know the classes for all training images, for an unknown test image we create the respective bestbasis decomposition tree and calculate the quadtree distances to all trees of the training images. The class, which dominates over the k nearest training images is then assigned to the unknown training image. Now, having a method to express the distance between two quadtrees and a set of images I T used for training, first of all the distances between all possible pairs of quadtrees of images out of this set are calculated as follows d i,j = d(T i , T j ) i, j ∈ {1, . . . , n} (4.58) where T i and T j are the decomposition trees of the i-th and the j-th image respectively, n is the number of images in I T and d is the quadtree distance metric used. I Tc is the subset of I T which contains all images of class c. I T = I T1 ∪ I T2 ∪ · · · ∪ I TC (4.59) where C is the total number of classes of images. The trees T i and T j , as stated above, are the result of the best-basis algorithm and are the optimal basis for the respective images in respect to a certain cost function such as the ones listed in section 4.2.1.1. The distances d i,j obtained above are now used to construct a distance matrix D of size n × n with d Ti ,T i = 0 ∀i = 1, . . . , n (4.60) 54
4.3 Structure-based classification since d Ti ,T i is just the distance of a quadtree of an image to itself, which must by definition of a metric always be 0. Therefore D is ⎛ ⎞ 0 d T1 ,T 2 . . . d T1 ,T n d T2 ,T 1 0 . . . d T2 ,T n D = ⎜ ⎝ . . . .. ⎟ . ⎠ d Tn,T1 d Tn,T2 . . . 0 This matrix then contains the quadtree distances (similarities) between all possible pairs of images out of the training image set. For each class c the matrix contains a submatrix D c which contains the distances between all images belonging to class c. Therefore D can also be written as ⎛ ⎞ 0 d T11 ,T 12 . . . d T11 ,T 1n . . . d T11 ,T C1 . . . d T11 ,T Cn d T12 ,T 11 0 . . . d T12 ,T 1n . . . d T12 ,T C1 . . . d T12 ,T Cn . . . .. . . . . . d D = T1n ,T 11 d T1n ,T 12 . . . 0 . . . d T1n ,T C1 . . . d T1n ,T Cn . . . . . .. . . . d TC1 ,T ⎜ 11 d TC1 ,T 12 . . . d TC1 ,T 1n . . . 0 . . . d TC1 ,T Cn ⎝ . ⎟ . . . . . . .. . ⎠ d TCn ,T 11 d TCn ,T 12 . . . d TCn ,T 1n . . . d TCn ,T C1 . . . 0 where d Tij ,T kl is the distance between the j-th image (quadtree) of class i and the l-th image (quadtree) of class k. Figure 4.5(a) shows an example distance matrix for the two-class case. This example shows a perfect distance matrix with very small intra-class distances (black boxes) and very high inter-class distances (white boxes). In reality however, a distance matrix is more likely to look like the one illustrated in figure 4.5(b), since the trees for different images from a specific class hardly will be totally identical in real-world applications. As can been seen in figure 4.5 and following from equation (4.40) the distance matrices are all symmetric. Having calculated the matrix D the next task is to find the centroid of each class. A centroid of a class c in this context is the training image out of I Tc which has the smallest average distance to all other images of class c. In other words, it is now necessary to find the image of class c to which all other images of class c have the smallest distance in respect to the chosen quadtree distance metric. After the centroids of all classes have been found the classification process can now take place. To classify an arbitrary image first of all the image is decomposed using the wavelet packets transform to get the respective decomposition tree. It is important to note that for classification the same cost function has to be used for pruning the decomposition tree as was used during the training phase for all training images. Then the distances of the resulting decomposition tree to the decomposition trees of the centroids of all possible classes are 55
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Pit Pattern Classification in Colon
Page 5:
Preface Time is the school in which
Page 8 and 9:
Contents 4.3.1 A quick quadtree int
Page 10 and 11:
1 Introduction endoscope represents
Page 12 and 13: 1 Introduction Pit pattern type I I
Page 14 and 15: 1 Introduction 6
Page 16 and 17: 2 Wavelets (a) Haar (b) Mexican hat
Page 18 and 19: 2 Wavelets For the discrete wavelet
Page 20 and 21: 2 Wavelets Now, that we have define
Page 22 and 23: 2 Wavelets 2.5 Pyramidal wavelet tr
Page 24 and 25: 2 Wavelets Logarithm of energy (Log
Page 26 and 27: 2 Wavelets While the best-basis alg
Page 28 and 29: 2 Wavelets Euclidean distance D(p a
Page 30 and 31: 2 Wavelets 22
Page 32 and 33: 3 Texture classification image retr
Page 34 and 35: 3 Texture classification By approxi
Page 36 and 37: 3 Texture classification where E i
Page 38 and 39: 3 Texture classification After the
Page 40 and 41: 3 Texture classification and ⃗x i
Page 42 and 43: 3 Texture classification 34
Page 44 and 45: 4 Automated pit pattern classificat
Page 70 and 71: 5 Results (a) Pit Pattern I (b) Pit
Page 72 and 73: 5 Results (a) Canvas002 (b) Carpet0
Page 74 and 75: 5 Results Pit Pattern Type I II III
Page 76 and 77: 5 Results Feature Feature vector le
Page 80 and 81: 5 Results 5.2.2 Best-basis method b
Page 82 and 83: 5 Results (a) All classes (b) Pit P
Page 84 and 85: 5 Results (a) All classes (b) Non-n
Page 86 and 87: 5 Results (a) All classes (b) Class
Page 88 and 89: 5 Results classes case all the entr
Page 90 and 91: 5 Results the behaviour described a
Page 92 and 93: 5 Results (a) All classes (b) Non-n
Page 94 and 95: 5 Results (a) All classes (b) Class
Page 96 and 97: 5 Results (a) 2 classes (b) 6 class
Page 100 and 101: 5 Results and 85%, respectively. Th
Page 102 and 103: 6 Conclusion correctly. But also in
Page 104 and 105: List of Figures 96
Page 106 and 107: List of Tables 98
Page 108 and 109: Bibliography [13] Huang C.R., Sheu
Page 110: Bibliography [40] Naoki Saito. Clas
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Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

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