11 Chapter Feature Detection 2.1 Introduction Feature detection is an essential early step in image processing and analysis algorithms It is particularly important in 3-D data analysis to detect image features such as step (or jump) edges and crease (or fold) edges because they often correspond to boundaries of objects 3-D data is commonly provided as a range image (or a depth map) Step edges correspond to pixels where the depth values are discontinuous, while crease edges are where the surface normals are discontinuous It should be noted that crease edges include both ridge and valley edges It is straightforward to detect step edges in a range image because a standard gradient-based operator may be applied, as in the case of intensity images However, a special operator needs to be designed for detecting crease edges since the depth discontinuities in their vicinity are slight In this chapter, we study methods for detecting crease edges that can be employed for analyzing maxillofacial image data We conduct comparative studies of existing methods and propose a novel technique for crease edge detection Subsequently, we apply the technique to the extraction of tubular structures in 3-D space and compare with eigenvalue analysis Various approaches have been proposed for the detection of crease edges, such as (a) residual analysis [29]−[31], (b) contour map analysis [32], [33], (c) morphological methods [34]−[37], (d) surface-fitting methods [38], [39], (e) first-derivative-based 12 methods [40], [41], (f) second-derivative-based methods [42], [43], and (g) surface normal analysis (SNA) [44]−[47] Residual analysis uses the difference between an input image and its smoothed image to detect edge points [29] This is essentially equivalent to the use of the high-frequency component of an input image Hence, the method will respond strongly to step edges, but not to crease edges In fact, this technique was originally devised for detecting step edges [31], and may not be well suited to crease edge detection The next three methods (b)−(d) are designed specifically for detecting crease edges However, since they are dependent on the shape and direction of a crease edge, their applications are limited First-derivativebased methods detect crease edges by finding a pixel whose first derivative is zero or nearly zero We examine this approach using synthetic and real range images and demonstrate its inherent problem in Section 2.2.1 Second-derivative-based methods detect crease edges by searching for the local maximum of the second derivative of an input image We study this approach using the same test images and summarize its properties in Section 2.2.2 SNA is a direct method for detecting crease edges by finding discontinuities in the surface normal We compute the discontinuities in the surface normal using a simple linear operation and describe its performance in Section 2.2.3 In addition to studying existing methods, we propose a novel method for detecting crease edges in an image in Section 2.2.4 The method, which we call GOA, detects crease edges by finding discontinuities in gradient orientation and is closely related to SNA The difference between the two methods is that GOA is based on gradient orientation whereas SNA is based on the surface normal We compare these two 13 methods in Section 2.2.5 Both GOA and SNA are successfully used for detecting crease edges in Chapter Up to now, we have dealt with range images that contain 3-D information of an object, but in fact, they are only 2-D matrices from a mathematical point of view In medical image data analysis, however, 3-D volumetric data sets are recently becoming more common due to the increasing use of CT and magnetic resonance imaging (MRI) Feature detection in 3-D space can be approached from two aspects: the detection of the variations in voxel intensity and orientation The former can be achieved rather straightforwardly by extending an existing gradient-based operator to 3-D, for example, as described in [48] Then, we may detect 3-D edges that are essential to extract features in 3-D space On the other hand, a well-known technique that uses gradient orientation is eigenvalue analysis [49]−[51] The first step of eigenvalue analysis is the construction of the local gradient structure tensor (GST), and the second step is principal component analysis (PCA) of the GST By analyzing the three eigenvalues obtained by PCA, the technique can extract a 3-D object of a particular structure, such as, a blob, a line, and a plane We study the performance of eigenvalue analysis using synthetic volume data in Section 2.3.1 In Section 2.3.2, we apply GOA to a volumetric data set by extending the algorithm to 3-D Since GOA is capable of detecting the points where gradient orientations are discontinuous, it may be effective for detecting a 3-D line that is highly discontinuous in orientation We compare the performance of 3-D GOA with that of eigenvalue analysis using synthetic volume data corrupted by Gaussian noise in Section 2.3.3 In 14 Chapter 4, 3-D GOA is effectively used for enhancing tubular structures in the jawbone 2.2 Crease Edge Detection 2.2.1 First-derivative-based Methods The methods described here attempt to locate a crease point by searching for a pixel whose first derivative is zero or nearly zero The first derivative is computed in a direction that maximizes the second directional derivative The following is a brief description of Haralick’s method that is also known as the height condition [40] The Hessian matrix H at a point in an image g may be expressed with the second partial derivatives as ⎛ g xx H =⎜ ⎜ g yx ⎝ g xy ⎞ ⎟, g yy ⎟ ⎠ (2.1) where ⋅ ≡ ∫∫ ⋅ dxdy (2.2) Γ indicates the spatial integration over a small local region Γ g xx , g xy , g yx , and g yy denote ∂ ∂x∂x , ∂ ∂x∂y , ∂ ∂y∂x , and ∂ ∂y∂y , respectively The spatial integration is aimed at preventing over-detection (defined on page 15) and will be discussed later Let λ1 ≥ λ2 be the eigenvalues of the smoothed Hessian matrix H , and v1 , v be the corresponding eigenvectors A crease point is a pixel whose first derivative in the direction of v1 is zero 15 Instead of the Hessian matrix, the use of the GST has been suggested [41] The GST T at a point in an image g is defined by the dyadic product of the gradient vector with itself as ⎛ gx ⎞ T = ⎜ ⎟( g x ⎜gy ⎟ ⎝ ⎠ gy ) ⎛ gx ⎜ = ⎜ g g ⎝ y x gx g y ⎞ ⎟, gy ⎟ ⎠ (2.3) where g x and g y denote ∂g ∂x and ∂g ∂y , respectively Each tensor element is averaged over a small local region to obtain a smooth tensor Let λ1 ≥ λ2 be the eigenvalues of the GST T , and v1 , v be the corresponding eigenvectors A crease point is a pixel whose first derivative in the direction of v1 is zero Thus, this method is identical to Haralick’s method (above) except for the step for determining the direction for computing the first derivative It is claimed that the height condition shows more over-detection than the GST-based method on planar slopes [41] The methods in this category often suffer from the so-called over-detection problem in which an excessive number of false edges are detected, especially on planar surfaces We demonstrate this problem using a synthetic range image of size 240 by 240 pixels (Figure 2.1(a)) and a real range image of size 300 by 300 pixels (Figure 2.2(a)) In both images, each pixel has an 8-bit gray level The step edges of the range images are shown in red in Figures 2.1(b) and 2.2(b), while crease edges are indicated in blue in Figures 2.1(c) and 2.2(c) Some of the edges are considered to be both step and crease edges because they exhibit discontinuities in both gradient magnitude and orientation at the same time 16 Figure 2.1: (a) A synthetic range image (b) Step edges (red) (c) Crease edges (blue) Figure 2.2: (a) A real range image (b) Step edges (red) (c) Crease edges (blue) 17 For simplicity, the magnitudes of the first derivative of the images, M = g x + g , y are computed using the Sobel operator We observe the points whose first derivatives are very small by changing the threshold from to (Figures 2.3(a)-(d)) and also from 10 to 40 (Figures 2.4(a)-(d)) White pixels indicate points where the magnitude of the first derivative is lower than the threshold values, and they are supposed to correspond to crease edges In Figures 2.3(a) and 2.4(a), parts of the crease edges in the images are barely detected, while scattered false edges are also detected in the smooth surface areas The number of the false edges increases rapidly when the threshold values are further relaxed For instance, the occlusal (biting) surfaces of the posterior (back) teeth are almost all white and the dental features such as the cusps (high points or peaks) and fissures are not detected selectively (Figures 2.4(c), (d)) The backgrounds are also selected as crease points because they are completely flat and the magnitudes of the first derivative are zero The exclusion of the background, however, may be easily achieved by ignoring very dark regions in the range image The local integration of gradient vectors when the Hessian matrix or GST is constructed is intended to reduce the over-detection of false edges at the cost of image details However, as long as searching for a pixel whose first derivative is zero or nearly zero is performed, many false edges will be inevitably detected in flat or very smooth areas in an image The selection of suitable threshold values for edge detection is frequently problematic because of the conflicting requirements of minimizing the number of false edges detected while simultaneously minimizing the number of true edges missed [27] Therefore, the first-derivative approach is not suitable when the detection of smooth crease edges is concerned 18 Figure 2.3: Crease edges detected in a synthetic range image by a first-derivativebased method (a) M < (b) M < (c) M < (d) M < Figure 2.4: Crease edges detected in a real range image by a first-derivative-based method (a) M < 10 (b) M < 20 (c) M < 30 (d) M < 40 19 2.2.2 Second-derivative-based Methods These methods attempt to detect crease edges using the second derivative of an image The second derivative can be obtained by applying the Laplacian operator to an image or applying a first-derivative operator twice [42] We summarize Khalifa’s approach, described in [42], here Given an input range image g ( x, y ) , the gradient of the image ∇g is computed using the Sobel operator: ⎡ ∂g ∇g = ⎢ ⎣ ∂x T ∂g ⎤ = [g x ∂y ⎥ ⎦ g y ]T (2.4) The gradient magnitude M and its direction θ are defined as M = ∇g = g x + g , y ⎛ gy ⎝ gx θ1 = tan −1 ⎜ ⎜ ⎞ ⎟ ⎟ ⎠ (2.5) (2.6) A first-derivative operator is again applied to each of the derivative images g x and g y to obtain g xx , g xy , g yx , and g yy The magnitude of the second derivative M and its direction are then given by 2 M = g xx + g xy + g + g , yx yy ⎛ g xy + g yy ⎜ g xx + g yx ⎝ θ = tan −1 ⎜ ⎞ ⎟ ⎟ ⎠ (2.7) (2.8) Both step and crease edges are detected using M and M , respectively However, it should be stressed that step edges, in general, have large magnitudes in both M and M In Figures 2.5 and 2.6, we demonstrate this using the range images of Figures 2.1(a) and 2.2(a) White pixels correspond to points where the magnitude of the second 20 derivative has large values The second derivative of the image is computed using the 3×3 Laplacian operator L [78]: ⎞ ⎛ ⎜ ⎟ L = ⎜ − 10 ⎟ ⎜ 5 ⎟ ⎝ ⎠ (2.9) Both step and crease edges can be seen in Figure 2.5(a) As the threshold value is raised from to 8, however, the crease edges start to vanish, leaving only step edges (Figures 2.5(b)−(d)) This clearly shows that the M magnitudes of step edges are greater than those of crease edges For the same reason, the method completely fails to detect crease edges in the real range images and the strong responses are seen only at the pixels where step edges are present (Figure 2.6) Therefore, in using the second derivative, we require an extra step for separating crease edges from step edges It is difficult to determine an appropriate threshold value because the magnitude of the second derivative of crease edges is not necessarily well separated from that of step edges Another potential problem is that computation of the second derivative is generally sensitive to noise It may be necessary to apply a low-pass filter to a noisy input image at the cost of image details The Laplacian of Gaussian (LoG) is a wellknown solution to this problem [52] The computation of the second-derivative-based methods is highly efficient especially when the Laplacian operator is used because it requires only one convolution mask In addition, it is easy to identify roof and valley edges with the sign of the second derivatives of the image, given by sgn(∇ g ) > if valley sgn(∇ g ) < if ridge (2.10) 32 Figure 2.13: Two mapping functions for the gradient vector (SNA: blue, GOA: red) This slight difference between the two mapping functions characterizes each method Let us compare the two methods using a roof edge model that is described with a second-degree polynomial (or parabola) as y = − A ⋅ x2 + B ( A > 0) (2.23) The gradient of the roof edge is given by the first derivative of the function: y ′ = −2 A ⋅ x (2.24) In SNA, the gradient is transformed by Eq (2.21) as nx = 2A⋅ x , + A2 ⋅ x (2.25) and the discontinuity in these values is used as a measure for detecting crease edges The discontinuity, i.e the first derivative of this function, is written as n′ = x 2A (1 + A ⋅ x2 ) 3/ (2.26) 33 Figure 2.14 shows roof edges with various values of A with B is set to The corresponding discontinuity magnitudes are depicted in Figure 2.15 When A is small and the slope of the roof edge is gentle, the discontinuity magnitude also becomes small and broad This indicates that SNA may not be able to detect and locate very smooth crease edges accurately Therefore, the effectiveness of SNA depends on the sharpness of a crease edge, namely gradient magnitude In other words, since the gradient vectors are transformed by a hyperbolic-like function in SNA, the gradient vectors whose magnitude is close to zero have less contribution to the detection of crease edges On the other hand, since the gradient vector is transformed by the signum function in GOA, the discontinuity magnitude always has a delta-function-like peak at the place corresponding to the summit of a crease edge regardless of its sharpness (vertical black line at x = in Figure 2.15) Thus, all the gradient vectors (except zero gradient) contribute equally to the crease edge detection 34 Figure 2.14: Roof edges with various sharpness Figure 2.15: Responses of SNA (color-keyed to the function in Figure 2.14) and GOA (black) to the various roof edges of Figure 2.14 35 Figure 2.16 provides a further comparison of GOA with SNA The profiles of a step edge, a sharp roof edge and a smooth roof edge are shown, respectively, in Figures 2.16(a), (b) and (c) The magnitudes of the derivatives of the surface normal are shown, respectively, in Figures 2.16(d), (e) and (f) It is observed that SNA gives a strong response to the step edge as well as to the sharp roof edge, while the response to the smooth roof edge is lower than that of the step edge Figures 2.16(g), (h) and (i) show the magnitudes of the derivatives of the gradient orientations With GOA, the responses to both sharp and smooth roof edges are equal and exceed the response to the step edge Since GOA is not concerned with the gradient magnitude, it is invariant to the strength of crease edges Therefore, compared with SNA, GOA is more sensitive to a smooth crease edge, whose gradient magnitude is very small, while GOA and SNA work in a very similar manner when the gradient magnitude is large In the same context, we can interpret the second-derivative-based method as the approach in which the gradient vector is used directly without any mapping function This explains why the second-derivative-based method responds more strongly to step edges 36 Figure 2.16: Comparison of GOA with SNA (a) Step edge (b) Sharp roof edge (c) Smooth roof edge (d)-(f) Responses of SNA (g)-(i) Responses of GOA Let us summarize the four types of methods for detecting crease edges described above: see Table 2.2 The first-derivative method has an over-detection problem (Section 2.2.1) The key issues for obtaining good performance are correctly determining the direction for computing the first derivative and implementation of local integration The second-derivative method uses the first derivative of the gradient vector, and thus it responds more strongly to step edges than crease edges (Section 2.2.2) It is easy to identify a roof edge and a valley edge using the sign of the second derivative of an input image In SNA, crease edges are detected by finding discontinuities in the surface normal (Section 2.2.3) We use the first derivative of the unit surface normal for detecting the interstices between the teeth (Section 3.5.1) and also the separation of the teeth from the gums (Section 3.7) SNA is currently the most common method for crease edge detection because it responds to crease edges 37 sensitively In GOA, crease edges are detected by finding discontinuities in gradient orientation (Section 2.2.4) For this, we use the first derivative of the normalized (or unit) gradient vector for extracting roof edges (Section 3.3.1) A unique feature of this method is its capability of equally detecting both sharp and gentle crease edges Table 2.2: Summary of the four methods for crease edge detection Methods Features • No response to step edges First-derivative Second-derivative Problems • Over-detection problem in flat areas • Cannot separate roof and valley edges • Can separate roof and valley • Responds more strongly to edges step edges than to crease edges • Responds well to crease • Responds edges in general edges SNA also to step • Not sensitive to smooth crease edges • Cannot separate roof and valley edges GOA • Responds well to crease • Cannot separate roof and edges regardless of their valley edges gradient magnitudes 38 2.3 3-D Line Detection 2.3.1 Eigenvalue Analysis It is known that a 3-D object of particular shape can be extracted by analyzing the eigenvalues of the GST that is defined locally This technique, generally called eigenvalue analysis, is used not only for shape analysis of a 3-D object but also for motion detection using spatio-temporal images (or time-sequential images) [56], [57] They are mathematically identical because, in both cases, input data are given in the form of a 3-D matrix, either g ( x, y , z ) or g ( x, y , t ) where x, y are the coordinates within an image at z (depth) or t (time) The 3-D GST T for a local neighborhood g ( x, y , z ) is a 3×3 matrix: ⎛ gx ⎜ T = ⎜ gx g y ⎜ ⎜ gx gz ⎝ gx g y g2 y g y gz gx gz ⎞ ⎟ g y gz ⎟ ⎟ gz ⎟ ⎠ (2.27) Hence, the GST can be viewed as a correlation matrix of the gradient vector in principal component analysis (PCA) Let λ1 , λ2 , λ3 ( λ1 > λ2 > λ3 > ) be the eigenvalues of T Using these values, 3-D objects may be classified into three shape groups: ⎧ λ1 ≈ λ2 ≈ λ3 ⎪ ⎨ λ1 >> λ2 ≈ λ3 ⎪ λ ≈ λ >> λ ⎩ if isotropic if plane - like if line - like (2.28) 3-D lines can be extracted using the following measure [49], [50]: Cline = λ2 − λ3 λ2 + λ3 (2.29) 39 Cline is a normalized measure that indicates the “line-likeness” of the local region constituting T When the local region captures a 3-D line, the smallest eigenvalue λ3 may be close to zero and hence Cline will be close to one It should be noted that this technique is invariant to the contrast of an object because of normalization (Eq (2.29)) An underlying assumption of this technique is that local gradient orientations contain vital information about local structure [51] We demonstrate the performance of eigenvalue analysis in an ideal situation in which 3-D objects are perfectly defined without noise Figure 2.17 depicts a noise-free 3-D space of size 20×40×20 voxels that contains a line (red) and a plane (blue) Figure 2.18(a) shows a cross-section of the 3-D space (x-y plane) The plane is of one-voxel width while the line is of one-voxel diameter Figure 2.18(b) shows the distribution of the line-likeness Cline defined in Eq (2.29) Since the eigenvalues on a plane hold λ2 = λ3 = , Cline must be zero and there is no response to the plane Conversely, the eigenvalues around a line satisfy λ2 >> λ3 = and thus Cline should become one We can see a thick line with its center coinciding with the original line The high response ( Cline = 1) is wider than the original line because of the spatial integration involved in the construction of T 40 Figure 2.17: Noisy-free 3-D space containing a line (red) and a plane (blue) z x Figure 2.18: Eigenvalue analysis in a noise-free 3-D space (a) Cross-sections of a line (upper line) and a plane (lower line) (b) Distribution of C line values 41 2.3.2 Gradient Orientation Analysis (3-D) We extend the algorithm of gradient orientation analysis to 3-D in this section Let g (x, y , z ) be the input 3-D data set Each component of the 3-D gradient vector of g (x, y , z ) is described by the partial derivative as ⎧ g x ( x, y , z ) = ∂g ( x, y , z ) ∂x ⎪ ⎨ g y ( x, y , z ) = ∂g ( x, y , z ) ∂y ⎪ g ( x, y , z ) = ∂g ( x, y , z ) ∂z ⎩ z (2.30) We employ the 3-D Sobel operator for computing the partial derivatives [58] The convolution mask is given by a set of three 3×3×3 masks, each of which is designed for obtaining a discrete approximation of the partial derivatives in the x, y, and z directions (Figure 2.19) -1 1 1 -2 0 -1 -1 -2 -1 -2 2 0 -3 0 0 0 -2 -2 -3 -2 0 -1 1 -1 -2 -1 -2 0 -2 -3 -2 -1 -1 -2 -1 -1 -2 -1 z y x 3-D convolution mask kx ky kz Figure 2.19: Convolution kernels for 3-D Sobel operator Since we are only interested in gradient orientation, we normalize the gradient vector as in Eq (2.19): 42 ⎧ n ( x, y , z ) = g ( x, y , z ) x ⎪ x ⎪ ⎨n y ( x , y , z ) = g y ( x , y , z ) ⎪ ⎪ n z ( x, y , z ) = g z ( x, y , z ) ⎩ where 2 2 2 g x ( x, y , z ) + g y ( x, y , z ) + g z ( x, y , z ) g x ( x, y , z ) + g y ( x, y , z ) + g z ( x, y , z ) , (2.31) g x ( x, y , z ) + g y ( x, y , z ) + g z ( x, y , z ) g x ( x, y , z ) + g y ( x, y , z ) + g z ( x, y , z ) ≠ As mentioned in Section 2.2.4, we exclude pixels where there is no gradient information, i.e g x = g y = g z = To find discontinuities in gradient orientations, we compute the first derivative of the normalized gradient vector: ⎧ d xx ( x, y, z ) = ∂n x ( x, y, z ) ⎪ d ( x, y , z ) = ∂n ( x, y , z ) x ⎪ xy ⎪ d xz ( x, y , z ) = ∂n x ( x, y , z ) ⎪ ⎪d yx ( x, y , z ) = ∂n y ( x, y , z ) ⎪ ⎨d yy ( x, y , z ) = ∂n y ( x, y , z ) ⎪ d ( x, y , z ) = ∂n ( x, y , z ) y ⎪ yz ⎪ d zx ( x, y , z ) = ∂n z ( x, y , z ) ⎪ d ( x, y , z ) = ∂n ( x, y , z ) z ⎪ zy ⎪ d zz ( x, y , z ) = ∂n z ( x, y , z ) ⎩ ∂x ∂y ∂z ∂x ∂y ∂z (2.32) ∂x ∂y ∂z The magnitude of discontinuities in gradient orientations D3 ( x, y , z ) may be expressed as D32 ( x, y , z ) = d xx ( x, y , z ) + d xy ( x, y , z ) + d xz ( x, y , z ) + d yx ( x, y , z ) + d yy ( x, y , z ) + d yz (x, y , z ) + (2.33) d zx ( x, y , z ) + d zy ( x, y , z ) + d zz (x, y , z ) We apply this technique to the same test data (Figure 2.20(a)) Figure 2.20(b) displays the magnitude of discontinuities in gradient orientations D3 Unlike eigenvalue analysis, GOA responds to the plane because the gradient orientation is discontinuous in the direction normal to the plane More importantly, however, GOA responds more strongly to the 3-D line because the gradient orientation is discontinuous in two 43 directions orthogonal to the line The discontinuity magnitude for the line is 1.42 times greater than that for the plane in this simulation In addition, the detection result is more localized as no spatial integration is involved in the computation of GOA z x Figure 2.20: 3-D gradient orientation analysis in a noise-free 3-D space (a) Cross2 sections of a line (upper line) and a plane (lower line) (b) Distribution of D3 values See Figure 2.17 2.3.3 Comparative Study In the noise-free case, eigenvalue analysis successfully extracts a 3-D line, ignoring a plane completely On the other hand, GOA responds to both the 3-D line and the plane, but the response to the line is 1.42 times stronger than that to the plane We further compare the two techniques in a noisy case Figure 2.21(a) shows cross-sections of a line (upper line) and a plane (lower line) defined in a noisy 3-D space of size 20×40×20 voxels Normally distributed random 44 numbers were added to the original noise-free space, and the SNR is about 22dB Figure 2.21(b) visualizes the distribution of Cline values The result is very obscure and the 3-D line is hardly detected This result indicates that eigenvalue analysis is highly sensitive to noise Figure 2.21: Eigenvalue analysis in a noisy 3-D space (SNR=22dB) (a) Cross-sections of a line (upper line) and a plane (lower line) (b) Distribution of Cline values Figure 2.22(a) shows cross-sections of a line (upper line) and a plane (lower line) defined in a noisy 3-D space of size 20×40×20 voxels Normally distributed random numbers were again added to the original noise-free space, and the SNR is about 22 dB Figure 2.22(b) visualizes the distribution of D3 values The result is noisy, yet the 3-D line is still well represented 45 Figure 2.22: 3-D gradient orientation analysis in a noisy 3-D space (SNR=22dB) (a) Cross-sections of a line (upper line) and a plane (lower line) (b) Distribution of D3 values Table 2.3 summarizes the quantitative comparison of the two methods Prior to the comparison, the response of each method is scaled to the range of and 255 The values in the table denote the mean responses to the line, plane, and also the background of the noisy 3-D space used for the test, while the values in parentheses are their variances Eigenvalue analysis appears to respond strongly to the line relative to the plane and background but its large variance indicates that the method fails to detect the line consistently, which is obvious in Figure 2.21(b) By contrast, 3-D GOA shows a much larger response to the line and much less variance, indicating that the method detects the line successfully in noisy 3-D space The response to the plane is also relatively large because the gradient orientation around a plane is also discontinuous to some extent However, the response is distinguishably lower than that for the line The responses and variances to the background of GOA are larger than 46 those of eigenvalue analysis as GOA sensitively responds to bright and dark spots (see Table 2.1) As the false responses appear like a random pattern without any particular structure, it is possible to suppress these spurious responses using a low-pass filter Table 2.3: Mean responses and variances to the line, plane, and background of eigenvalue analysis and 3-D gradient orientation (3-D GOA) Eigenvalue Analysis 3-D GOA Line 132.8 (4890.9) 223.0 (744.6) Plane 48.6 (2305.5) 187.5 (643.7) Background 32.1 (800.7) 84.6 (1326.5) The simulation results clearly show that 3-D GOA is far more robust to noise than eigenvalue analysis This difference may be attributed to the different level of condition required for a target object of each method Eigenvalue analysis requires that three eigenvectors locally computed are all correct, which is not guaranteed when a target object is not well defined or corrupted by noise In this sense, the requirement of eigenvalue analysis is quite demanding On the contrary, the requirement of 3-D GOA about the target object is only the discontinuity in gradient orientation, which is hardly affected by noise Thus, the requirement is not demanding, and GOA is more robust to noise In Chapter 4, we apply 3-D GOA to the detection of nerve canals in the human jaw (Section 4.4.2) The nerve canal is a hollow channel and its tubular structure can be considered to be a 3-D line Eigenvalue analysis will not be suitable for the task because the nerve canal has many holes and is not well defined The situation is similar to the detection of 3-D lines in noisy 3-D data ... approximation of the partial derivatives in the x, y, and z directions (Figure 2. 19) -1 1 1 -2 0 -1 -1 -2 -1 -2 2 0 -3 0 0 0 -2 -2 -3 -2 0 -1 1 -1 -2 -1 -2 0 -2 -3 -2 -1 -1 -2 -1 -1 -2 -1 z y x 3- D convolution... 22 dB Figure 2. 22( b) visualizes the distribution of D3 values The result is noisy, yet the 3- D line is still well represented 45 Figure 2. 22: 3- D gradient orientation analysis in a noisy 3- D. .. edges • Cannot separate roof and valley edges GOA • Responds well to crease • Cannot separate roof and edges regardless of their valley edges gradient magnitudes 38 2 .3 3 -D Line Detection 2 .3. 1