47 Chapter Analysis of Surface Data 3.1 Introduction The accurate segmentation of the individual teeth from the digitized dental model is an important component in computer-based algorithms for orthodontic feature detection and measurement, and in the simulation of orthodontic procedures such as tooth rearrangement. Previous work for tooth segmentation is not designed to deal with severe malocclusions (incorrect bites), but is only targeted at the mild cases [4]−[6], [14]. In this chapter, we present an automated method for segmenting the teeth of dental study models (plaster casts of the dentition) exhibiting a variety of malocclusions. Figure 3.1 shows the overall flow diagram of the method. We digitize the model by surface scanning with a laser (Section 3.2.1). After aligning the acquired image to a standard orientation (Section 3.2.3), we generate a plan-view range image (Section 3.2.3). Orthodontic features are extracted in this range image (Sections 3.3.1 and 3.3.2) for the determination of the dental arch (Section 3.3.2). Using the arch as the reference, we generate a panoramic range image that provides a complete outer view of the cast, i.e., the entire buccal3 surfaces of the teeth (Section 3.3.3). The interstices between the teeth are detected separately in the two range images (Sections 3.4 and 3.5). The two results are subsequently combined to determine the position and the orientation of the tooth interstices (Section 3.6). Finally, the teeth are separated The outer surface of the teeth, i.e., tooth surfaces that are next to the cheeks (opp. lingual). 48 from the gums in the panoramic range image (Section 3.7). A validation test confirms that the method works well with dental casts representing a variety of malocclusions. Dental Cast Digitization (3.2.1) Alignment (3.2.3) Generation of Plan-view Range Image (3.2.3) Ridge Detection (3.3.1) Dental Arch Detection (3.3.2) Generation of Panoramic Range Image (3.3.3) First Tooth-interstice Detection (3.4) Second Tooth-interstice Detection (3.5) Integration (3.6) Tooth&Gum Separation (3.7) Segmented Teeth Figure 3.1: Flow chart giving an overview of the segmentation procedure. 49 3.2 Data Acquisition and Processing 3.2.1 Laser Scanner A variety of methods for acquiring digital data of the dentition have been reported in the literature. The 3-D profile of wax imprints can be obtained using a depth-fromabsorption technique [4]−[6]. Range scanners have been used to scan dental study models for display and storage purposes [60]−[62]. A different approach that reconstructs the 3-D model of the patient’s dental occlusion by integrating a sequence of images captured by an intraoral video camera is reported by S.M. Yamany et al. in [63]. We digitize the dental study models with a commercially available laser scanner (Cyberware 3030R/HIREZ/MM [1]) that is particularly suited to digitizing small, highly detailed objects where resolution and accuracy are important factors (Figure 3.2). A vertical stripe of laser light is projected onto the plaster cast that is placed on the motion platform. As the object moves laterally, it is scanned by the stationary laser beam, and this is repeated for different rotated positions. Range information is computed from each scan based on triangulation. The bundled software produces the complete 3-D representation of the model by merging the range data from different scans. Spatial resolution is 0.15~1.00 mm horizontally and 0.30 mm vertically. Depth resolution depends on the surface quality of the object, but is typically 0.05~0.20 mm. 50 Figure 3.2: Laser scanner (Cyberware 3030R/HIREZ/MM Model). 3.2.2 Data Format Figure 3.3 shows a tooth digitized by the laser scanner. The surface is represented with numerous small triangular patches. We save the 3-D image data as a VRML4 file, in which an object is described as a list of vertices and a list of face indices (Figure 3.4). The list of vertices contains the 3-D coordinates of each vertex, while the list of face indices defines which three vertices form a single triangular patch. VRML stands for “Virtual Reality Modeling Language” and is a file format for describing 3-D objects and worlds interactively. 51 Figure 3.3: Tooth surface digitized by the Cyberware scanner. x Vertices y z Faces 1st vertex 2nd vertex 3rd vertex 4th vertex . . . . . . Figure 3.4: VRML format. The first face is drawn by connecting vertices 1, 2, and 3. 52 While the VRML format is adequate for visualization and data storage, it is not suitable for analysis because of its irregular 3-D structure. There is also the inherent added complexity of working in the 3-D domain compared to 2-D. Thus, instead of processing the VRML data directly, we devise algorithms that are applied to the range images (or depth maps) generated from the 3-D data. Since the range image is a 2-D matrix (though it has 3-D information encoded in the intensity values), the complex 3D tooth segmentation problem is significantly simplified. For instance, the regular structure of the range image makes mathematical operations such as convolution readily applicable. We employ both plan-view and panoramic range images to provide the 3-D information necessary for tooth segmentation. It should also be noted that the image size is about 100 KB, which contrasts with 10 MB for the original mesh data. 3.2.3 Plan-view Range Image In preparation for computing the plan-view range image, the digitized dental model has to be aligned to a standard orientation. We first specify the occlusal plane (the imaginary surface at which the upper and lower teeth touch) by manually identifying four reference points (Figure 3.5): the disto5-buccal cusps of the first molars (labeled and 4) and the buccal cusps of the first premolars (2 and 3). A reference point need not be precisely located as the program searches for the highest point in the vicinity of the manually selected point. The cast is aligned such that the occlusal plane is parallel to the x-y plane and the two disto-buccal cusps have the same y-coordinates. Posterior, i.e. towards the back of the mouth. (opp. mesio-) 53 Figure 3.5: Digitized dental cast after alignment and four reference points (1, 4: distobuccal cusps of the first molars, 2, 3: buccal cusps of the first premolars). A plan-view range image is obtained by mapping the vertices of the 3-D data onto a regularly spaced 2-D array that is parallel to the x-y plane (Figure 3.6). The x and y coordinates of each vertex determine the location of the corresponding pixel in the 2-D array, whereas the z coordinate of the vertex is used for computing the height of the vertex from the 2-D array, i.e., the reference surface. When two or more vertices are mapped onto one pixel in the 2-D array, the highest vertex is selected because it is the only point visible in the plan view. The computed heights are saved as the pixel values (gray levels) of the 2-D array. Figure 3.7 shows a generated plan-view range image, comprising 300 by 300 pixels with each pixel corresponding to a spatial separation of 1/3 mm. The highest 15-mm of the 3-D data is quantized into 256 gray levels (i.e., 8-bit unsigned char), giving a height resolution of 0.059 mm per gray level. It is pertinent to note that the spatial and 54 height resolution of the range image are comparable to those of the scanner, thus ensuring the preservation of fine details in the conversion process. Surface of a 3-D model Regularly spaced 2-D array 3-D model y x Figure 3.6: Mapping of the vertices onto a 2-D array. Figure 3.7: Plan-view range image of the digitized dental cast. 55 3.3 Determination of the Dental Arch In orthodontics, a complete description of the form of the dental arch is essential whenever changes due to growth or orthodontic therapies are under investigation [19]. Orthodontists need to know the maxillary (upper jawbone) and mandibular (lower jawbone) arches for assessing the degree of malocclusion [6]. Apart from its clinical use, the dental arch plays an important role in our tooth-segmentation algorithm. Current computer-assisted methods of determining the arch form require the user to manually select several feature points in the cast image [2] or interactively define the arch [65], [66]. We propose to improve on this with an automated method to determine the tooth-based dental arch. 3.3.1 Ridge Detection From the plan-view range image, we extract orthodontic features such as the incisal edges and cusps. These features appear as local peaks that form ridges (roof edges). Existing detection methods [38], [67] are greatly dependent on the gradient of the ridge and may fail if the slope is gentle, which is often the case with the cusps of the posterior (back) teeth. We use a new approach, GOA, to detect both gentle and sharp ridges. Discontinuities in gradient orientation will indicate the presences of local peaks. Let h( x, y ) be a plan-view range image. The gradient of h( x, y ) is written as ⎡ p( x, y )⎤ ⎡∂h( x, y ) / ∂x ⎤ ∇h( x, y ) = ⎢ ⎥=⎢ ⎥. ⎣ q( x, y ) ⎦ ⎣∂h( x, y ) / ∂y ⎦ The gradient orientation at a point ( x, y ) is defined as (3.1) 56 ⎧ tan −1 (q( x, y ) / p( x, y ) ) p( x, y ) ≠ ⎪ θ ( x, y ) = ⎨π p( x, y ) = 0, q( x, y ) > , ⎪− π p( x, y ) = 0, q( x, y ) < ⎩ (3.2) where θ ( x, y ) lies in the range ( −π ,π ] . We ignore the pixels in flat regions where both p ( x, y ) and q ( x, y ) are approximately zero. Discontinuities in gradient orientation can be obtained by determining the partial derivatives of the trigonometric functions of θ ( x, y ) : ⎡ s x ( x, y ) ⎤ ⎡∂ (sin θ ( x, y ) ) / ∂x ⎤ ∇(sin θ ( x, y ) ) = ⎢ ⎥=⎢ ⎥, ⎣ s y ( x, y )⎦ ⎣∂ (sin θ ( x, y ) ) / ∂y ⎦ (3.3) ⎡c x ( x, y ) ⎤ ⎡∂ (cosθ ( x, y ) ) / ∂x ⎤ ∇(cosθ ( x, y ) ) = ⎢ ⎥=⎢ ⎥. ⎣c y ( x, y )⎦ ⎣∂ (cosθ ( x, y ) ) / ∂y ⎦ (3.4) The Sobel operator is used for computing the partial derivatives. The magnitude of the discontinuities in gradient orientation is obtained as D ( x, y ) = s x2 ( x, y ) + s y2 ( x, y ) + c x2 ( x, y ) + c y2 ( x, y ) . (3.5) At this point, both ridges and valleys are detected because they are equally discontinuous in gradient orientation. Ridges are distinguished from valleys by the sign of the Laplacian: ( ) ⎧ D( x, y ) sgn ∇ h( x, y ) < D1 ( x, y ) = ⎨ otherwise. ⎩ (3.6) Since we are not interested in the lower portion of the cast, we assign zeros to the points in D1 ( x, y ) where the gray levels of the corresponding plan-view range image h( x, y ) are below 128: ⎧ D ( x, y ) 128 ≤ h( x, y ) D ( x, y ) = ⎨ otherwise ⎩ (3.7) 68 especially when they are well aligned. To overcome this problem, we introduce another indicator that is obtained from the panoramic range image. 3.5 Tooth-interstice Detection in the Panoramic Range Image 3.5.1 Valley Detection We first employ SNA [44] to detect valleys and ridges (or roof edges) by finding the points where the surface normal changes direction abruptly. We apply SNA in the horizontal direction to detect the vertical valleys formed by the interstices. Let the panoramic range image be d (i, j ) . The gradient of d (i, j ) is defined by ⎡ p(i, j )⎤ ⎡∂d (i, j ) ∂i ⎤ ∇d (i, j ) = ⎢ ⎥. ⎥=⎢ ⎣q(i, j ) ⎦ ⎣∂d (i, j ) ∂j ⎦ (3.15) The Sobel operator is again used for estimating the partial derivatives. Denoting the unit surface normal at a 3-D point ( i, j , d (i, j ) ) on an object by n , we express the unit surface normal using surface gradient components. If we move a small distance ∂i in the i direction, the change of height is ∂d = p∂i . Thus, the vector [ 1, 0, p ]T is the tangent to the surface. Analogously, [ 0, 1, q ]T is also a tangent to the surface. The surface normal is perpendicular to both tangents and may be computed using the vector product ⎡ ⎢ ⎢ ⎢⎣ p (i, j ) ⎤ ⎡ ⎥×⎢ ⎥ ⎢ ⎥⎦ ⎢⎣q (i, j ) Hence, the unit surface normal n is ⎤ ⎡− p (i, j )⎤ ⎥ = ⎢− q (i, j ) ⎥ . ⎥ ⎥ ⎢ ⎥⎦ ⎥⎦ ⎢⎣ (3.16) 69 ⎡ni (i, j ) ⎤ ⎡− p (i, j )⎤ ⎢ ⎥ ⎢− q (i, j ) ⎥ . n = ⎢n j (i, j ) ⎥ = ⎢ ⎥ 2 + p (i, j ) + q (i, j ) ⎢ ⎢n (i, j )⎥ ⎥⎦ ⎣ ⎣ d ⎦ (3.17) We take the d component of the surface normal to be positive since only the surface facing the viewer is visible. Discontinuities in n will indicate the presence of valleys and ridges in the image. Since we are only interested in detecting vertical valleys here, we search in a horizontal direction (the i direction) for these discontinuities (1-D SNA), which are measured by N (i, j ) = (∂ni (i, j ) ∂i ) + (∂n j (i, j ) ∂i ) + (∂nd (i, j ) ∂i ) . (3.18) To separate valleys from ridges, we use the sign of ∇ d (i, j ) : ( ) ⎧ N (i, j ) sgn ∇ d (i, j ) > N valley (i, j ) = ⎨ otherwise. ⎩ (3.19) Finally, N valley (i, j ) is smoothed by 3×1 median filtering. Figure 3.15 shows N valley (i, j ) of the panoramic range image of Figure 3.10. Figure 3.15: Detected vertical valleys, N valley (i, j ) , for the image in Figure 3.10. 3.5.2 Second Tooth-interstice Indicator In the second stage, we obtain the vertical projection of N valley (i, j ) p (i ) = ∑ N valley (i, j ) . j (3.20) 70 The second interstice indicator f (i ) is p(i ) scaled to the range [ ] (Figure 3.16). The peaks in f (i ) can be used to identify the tooth interstices. The indicator f (i ) is especially effective when the teeth are well aligned and thus the interstices are nearly vertical. It is particularly suitable for detecting the interstices between the incisors because they are long and produce prominent peaks as the result of the projection. Conversely, f (i ) may not be very effective if the teeth are severely malaligned. In this case, however, the indicator f1 (i ) works well (as mentioned above). It should be emphasized that the two indicators complement each other. Figure 3.16: Scaled vertical projections, f (i ) . It is possible to use GOA in place of SNA here, but the latter is considered more reliable in this situation since it detects valleys independently of the viewpoint. Figure 3.17 shows the profile of a panoramic range image in which each arc indicates a tooth. 71 The slope increases monotonically to the left of the central arc and decreases monotonically to the right; GOA will thus fail to detect the tooth interstices in the two circles because there is no change in gradient orientation. Such as situation could arise if the dental arch does not fit the arrangement of the teeth well. Figure 3.17: Cross sectional profile of the panoramic range image. 3.6 Integration of the Two Tooth-interstice Indicators It is important to note that both tooth-interstice indicators are obtained from the same dental arch and share the same abscissa. The ith value in the depth graph and the ith column of the panoramic range image provide the depth information at the ith sample point on the dental arch. Hence it is straightforward to combine these two indicators to give a single measure that will serve as a robust indicator of tooth interstices. We require a composite indicator that yields a positive result should any one of the two indicators indicate the presence of an interstice. A simple yet effective measure is the average of f1 (i ) and f (i ) : f c (i ) = [ f1 (i) + f (i)] , (3.21) which gives a peak value if at least one of the two indicators successfully detects a tooth interstice. Figure 3.18 shows the composite indicator for the dental model of Figure 3.5. 72 Figure 3.18: Composite tooth-interstice indicator, f c (i ) . We determine the location and orientation of the tooth interstices as follows. Suppose we know the number of teeth N (normally 14, except for the wisdom teeth). To find the N − tooth interstices, we first select the highest N − peaks in the composite indicator f c (i ) and check their corresponding heights in f1 (i ) . If a peak value is large ( > 0.5 ), we determine the location and orientation of the tooth interstice from g1 (i ) and g (i ) , respectively. Local minima are searched for in the depth graph, g1 (i ) , in the vicinity of the corresponding peaks in the composite indicator f c (i ) . When the peak value is low ( ≤ 0.5 ) and f1 (i ) is not reliable, we determine the location of the tooth interstice using the second indicator f (i ) and its orientation is considered to be perpendicular to the dental arch. This alternative step is needed when the anterior teeth are well aligned because their interstices are often too small to be detected in the plan view. 73 With the knowledge about the locations of the two disto-buccal cusps of the first molars (Section 3.2.3), we can limit the number of tooth interstices to be detected along the dental arch between the two cusps to N − . Two more interstices are to be detected outside, but in the vicinity of, the two cusps. In this way, our method does not depend on whether the wisdom teeth are present. When there are only M ( < N − ) prominent peaks in the composite indicator, we will attempt to determine M tooth interstices and inform the user of the number of missing interstices, N − M − . Though we assume that N is known, it is a trivial matter to allow for missing teeth. A deep and wide valley in the depth graph or panoramic range image would signal clearly that one or more teeth are missing, with statistical knowledge of tooth sizes employed to determine the actual number. 3.7 Separation of the Teeth from the Gums The teeth need to be separated from the gums in applications that require modeling or designing of tooth crowns [7]−[10], while the extracted gum portion would be of use in applications that require measurements to be made of the gum. The border between the teeth and the gums (the gum margin) appears as a valley at each column of the panoramic range image. The valley can be located by finding the pixel where discontinuity in surface normal is the largest at each column of the image. Since the sharpness of the valleys varies widely from the anterior teeth (shallow) to the posterior teeth (deep), we employ a family of derivative-operators such as [ 1, 0, 0, 0, − ]T , and [ 1, 0, 0, 0, 0, 0, − ]T surface normal (using the 1-D SNA). [ 1, 0, − ]T , for computing the discontinuities in the 74 In every column of the image, we obtain three candidate valley pixels as shown in Figure 3.19. They are ranked according to their reliability, i.e., the magnitude of the derivative. The gum border is initially drawn by connecting the most reliable candidates (blue) and its smoothness evaluated by s (i ) = t j (i − 1) − ⋅ t j (i ) + t j (i + 1) , (3.22) where i denotes the index of a column of the image and t j (i ) the row of the jth candidate (j=1, 2, 3) of the ith column. If s(i ) exceeds a preset value, this valley pixel is considered isolated from its neighbours and replaced with the second candidate (yellow) (and the third (red), if necessary) pixel of the column. In the unlikely event that all three candidates at a column fail to satisfy the smoothness criterion, we interpolate the border pixel. Figure 3.20 shows the detected gum border of Figure 3.19. The detected gum margin can be backprojected to the original 3-D model (Figure 3.21). 75 Figure 3.19: Three candidate points of the valley at each column (blue dots: first candidates, yellow: second candidates, red: third candidates). Figure 3.20: Detected gum margin. Figure 3.21: 3-D presentation of the detected gum margin. 76 3.8 Results and Discussion We have tested our tooth-segmentation algorithm on 34 dental casts with various types of malocclusions, namely, Class I7, Class I with bimaxillary proclination8, Class II division 19, and Class II division 210. Each class has eight casts comprising four maxillas and four mandibles. Two casts, a maxilla and a mandible, out of 34 are of unknown class. Visual inspection was used to evaluate the segmentation results. The method correctly detected all the 390 tooth interstices in 30 casts and missed six interstices of the anterior teeth in the remaining four (Figure 3.22), a failure rate of 1.4%. There are two causes of unsuccessful detection: • Limited resolution of the scanner. When the valley formed by two adjoining teeth is extremely fine (Figure 3.22(a)), it may not appear in both plan and panoramic views. • Severe overlap of two adjoining teeth. When two adjoining teeth overlap with each other (Figures 3.22(b), 3.22(c), and 3.22(d)), they not form a valley in the panoramic view, which renders the second indicator ineffective. If the contact surface between two adjoining teeth is highly curved, the straight inspection spoke cannot fit into the interstice at any orientation, thus causing the first indicator to be also ineffective. It is significant that the system does not give any false detections even in such situations, but will notify the user of the number of undetected interstices. Class I malocclusion has the normal molar relationship but the incorrect line of occlusion. Both the upper and lower incisors are proclined towards the lips. Class II, division 1, malocclusion has the lower molar placed behind the upper molar, and the upper incisors are inclined outward. 10 Class II, division 2, malocclusion has the lower molar placed behind the upper molar, and the upper incisors are inclined inward. 77 (a) (b) (c) (d) Figure 3.22: Detected dental arches and tooth interstices. (a) Mandible of Class I with bimaxillary proclination. (b) Mandible of Class I with bimaxillary proclination. (c) Mandible of Class II division 1. (d) Mandible of Class II division 1. 78 The orientations of the detected tooth interstices were evaluated by comparing with those specified by an orthodontist. The deviations in the two orientations are summarized in Table 3.1. The average deviation is about 5°, which is equal to the quantization error as we rotate the inspection spoke every 10°. In addition, it should be noted that there is some tolerance in defining the orientation of a tooth interstice (Figure 3.23). Table 3.1: Orientation deviations of the detected tooth interstices with respect to the ground truth. Anterior Teeth Posterior Teeth Average Maxilla 5.7° 3.6° 4.7° Mandible 6.2° 4.6° 5.4° Average 5.9° 4.1° 5.1° Ground Truth Tooth Interstice Tolerance Figure 3.23: Tolerance in interstice orientation. The proposed method successfully delineated the border between the teeth and gums in every cast. The overall result is noteworthy, considering the fact that we have used casts with malocclusions that are commonly encountered in orthodontic treatments. Figure 3.24 shows two sets of tooth segmentation results in both plan and panoramic 79 views. Figures 3.24(a) and 3.24(b) show the segmented upper teeth of a Class I malocclusion with bimaxillary proclination, and Figures 3.24(c) and 3.24(d) the lower teeth of a Class II division malocclusion. The small circular artifact in Figure 3.24(d) indicates a region that is far from the dental arch and was ignored when the panoramic range image was generated. The segmentation results can be backprojected to the digitized cast to give a 3-D presentation of the segmentation (Figure 3.25). (a) (b) Artifact (c) (d) Figure 3.24: Segmented teeth. (a) Segmented maxilla of Class I with bimaxillary proclination in the plan view. (b) Same cast in the panoramic view. (c) Segmented mandible of Class II division in the plan view. (d) Same cast in the panoramic view. 80 (a) (b) Figure 3.25: 3-D presentation of the segmented teeth. (a) Segmented maxilla of Class I with bimaxillary proclination (corresponding to the dental cast of Figures 3.24(a) and 3.24(b)). (b) Segmented mandible of Class I, division (corresponding to the dental cast of Figures 3.24(c) and 3.24(d)). We discuss below the features of our approach with respect to the two cases of wellaligned teeth and malaligned teeth. • Well-aligned case. When the teeth are well aligned, it is often difficult to detect the interstices between the incisors in the plan view, whereas they form clear vertical valleys in the panoramic view. Figure 3.26 demonstrates the importance of using both indicators. Three tooth interstices between the incisors are missing (or too low) in the first indicator (Figure 3.26(a)). In contrast, the second indicator shows the three peaks clearly (Figure 3.26(b)). These peaks are so prominent that the tooth interstices missing in the first indicator are recovered in the composite indicator (Figure 3.26(c)), leading to the successful detection of the interstices (Figure 3.26(d)). 81 (a) (b) (c) (d) Figure 3.26: Integration of the two tooth-interstice indicators in a well-aligned case. (a) First tooth-interstice indicator. (b) Second tooth-interstice indicator. (c) Composite tooth-interstice indicator. (d) Detected tooth interstices. • Malaligned case. When the teeth are malaligned, their interstices not always form valleys in the panoramic view, resulting in lower reliability of the second indicator. Conversely, the first indicator is effective because interstices between malaligned teeth generally appear as deep valleys in the plan view. This situation can be seen in Figure 3.27, where the 14 teeth result in 13 tooth interstices. There is one missing peak in the second indicator (Figure 3.27(b)) caused by severe tooth malalignment. However, the 82 first indicator has the highest peak at the same position (Figure 3.27(a)), and 13 peaks can be clearly seen in the composite indicator (Figure 3.27(c)). The tooth interstices in this case are successfully detected (Figure 3.27(d)). (a) (b) (c) (d) Figure 3.27: Integration of the two tooth-interstice indicators in a malaligned case. (a) First tooth-interstice indicator. (b) Second tooth-interstice indicator. (c) Composite tooth-interstice indicator. (d) Detected tooth interstices. As confirmed by our experiments, the first indicator (obtained from the plan view) could be unreliable in detecting interstices between well-aligned incisors, but is quite tolerable to malaligned teeth. The second indicator (obtained from the panoramic view), on the other hand, is rather unreliable when the teeth are malaligned, but works 83 very well for well-aligned teeth. Therefore, these two indicators work in a complementary manner by covering for each other’s weakness. Consequently, the method has achieved a high success rate in segmenting teeth with a variety of malocclusions. [...]... 3. 20 shows the detected gum border of Figure 3. 19 The detected gum margin can be backprojected to the original 3- D model (Figure 3. 21) 75 Figure 3. 19: Three candidate points of the valley at each column (blue dots: first candidates, yellow: second candidates, red: third candidates) Figure 3. 20: Detected gum margin Figure 3. 21: 3- D presentation of the detected gum margin 76 3. 8 Results and Discussion... Segmented mandible of Class II division 1 in the plan view (d) Same cast in the panoramic view 80 (a) (b) Figure 3. 25: 3- D presentation of the segmented teeth (a) Segmented maxilla of Class I with bimaxillary proclination (corresponding to the dental cast of Figures 3. 24(a) and 3. 24(b)) (b) Segmented mandible of Class I, division 1 (corresponding to the dental cast of Figures 3. 24(c) and 3. 24 (d) ) We discuss... and the structuring element B is a 3 by 3 matrix of ones Figure 3. 8(a) shows the ridges extracted from the plan-view range image of Figure 3. 7 It should be noted that, unlike standard methods of detecting ridges, we use neither the magnitude of the gradient nor the surface normal directly GOA is capable of detecting ridges irrespective of their gradient magnitude because it focuses solely on the detection... inclined outward 10 Class II, division 2, malocclusion has the lower molar placed behind the upper molar, and the upper incisors are inclined inward 8 77 (a) (b) (c) (d) Figure 3. 22: Detected dental arches and tooth interstices (a) Mandible of Class I with bimaxillary proclination (b) Mandible of Class I with bimaxillary proclination (c) Mandible of Class II division 1 (d) Mandible of Class II division... (Figure 3. 13( b)) 6 Anterior, i.e forward or front of the mouth (opp disto-) 66 (a) (b) Figure 3. 13: Designed finite impulse response (FIR) band-pass filter (BPF) (a) Pointspread function of the filter (b) Its frequency response 67 3. 4 .3 First Tooth-interstice Indicator The depth graph after band-pass filtering is scaled to the range [0, 1 ] and denoted by g ′ (i ) (Figure 3. 14) BPF Figure 3. 14: Depth... column of the image, we obtain three candidate valley pixels as shown in Figure 3. 19 They are ranked according to their reliability, i.e., the magnitude of the derivative The gum border is initially drawn by connecting the most reliable candidates (blue) and its smoothness evaluated by s (i ) = t j (i − 1) − 2 ⋅ t j (i ) + t j (i + 1) , (3. 22) where i denotes the index of a column of the image and t j... (c) (d) Figure 3. 9: Various forms of dental arches (a) Malaligned upper teeth with outwardly inclined incisors (b) Severely malaligned upper teeth (c) Malaligned lower teeth (d) Asymmetrically aligned lower teeth 3. 3 .3 Panoramic Range Image We generate a panoramic range image by computing the distance between the buccal surface of the teeth and the reference surface that is defined by extending the detected... curves and do not introduce a bias in the assessment of the dental arches Figure 3. 9 shows various forms of dental arches determined by the proposed method 59 (a) (b) (c) (d) Figure 3. 8: Two-step curve fitting technique for determining the dental arch (a) First curve fitting to the extracted ridge pixels (b) Inspection spokes (c) Detected local peaks (d) Second curve fitting to the local peaks (the dental... be detected in the plan view  73 With the knowledge about the locations of the two disto-buccal cusps of the first molars (Section 3. 2 .3) , we can limit the number of tooth interstices to be detected along the dental arch between the two cusps to N − 3 Two more interstices are to be detected outside, but in the vicinity of, the two cusps In this way, our method does not depend on whether the wisdom... the row of the jth candidate (j=1, 2, 3) of the ith column If s(i ) exceeds a preset value, this valley pixel is considered isolated from its neighbours and replaced with the second candidate (yellow) (and the third (red), if necessary) pixel of the column In the unlikely event that all three candidates at a column fail to satisfy the smoothness criterion, we interpolate the border pixel Figure 3. 20 shows . Panoramic Range Image (3. 3 .3) First Tooth-interstice Detection (3. 4) Dental Arch Detection (3. 3.2) Ridge Detection (3. 3.1) Second Tooth-interstice Detection (3. 5) Integration (3. 6) Segmented Teeth Alignment. (Section 3. 2 .3) , we generate a plan-view range image (Section 3. 2 .3) . Orthodontic features are extracted in this range image (Sections 3. 3.1 and 3. 3.2) for the determination of the dental arch. visualization and data storage, it is not suitable for analysis because of its irregular 3- D structure. There is also the inherent added complexity of working in the 3- D domain compared to 2 -D. Thus,