Local Three-dimensional Shape-preserving ... - Semantic Scholar
Local Three-Dimensional Shape-Preserving Smoothing without Shrinkage Chaohuang Zeng
Mila,n Sonka
Department of Electrical Engineering Stanford University Stanford, CA 94305
Dept. of Electrical and Comp. Engineering The University of Iowa Iowa City, IA 52242
Abstract
used to smooth three-dimensional human spine curve [7]. The method is not applicable to two-dimensional or three-dimensional smoothing. More directly related t o our work, Oliensis designed a filter F(w) that has several desirable properties: 1) virtually no shrinkage; 2) approximate reproducibility; 3) locality [2]. However, sufficient evidence was not provided to show that the filter f ( t ) falls off at large t more quickly than any power o f t and is local. A brief proof of this important concept is provided below as it serves as a basis for the design of our shape-preserving filter - the locality of the filter F(w) is analyzed and a new local two-dimensional filter F(w,E ) is designed. Our filter not only preserves all the advantageous properties of the 1D filter F ( w ) ,but offers more flexibility and is easy to extend to n dimensions. Our new surface representation is based on decomposition of a regular 3D surface in two 2D functions that are smoothed independently.
Design of a novel three-dimensional ( 3 0 ) shapepreserving smoothing approach is described. Threedimensional surfaces are smoothed without shrinkage artifacts typical for m a n y other approaches. Using o u r n e w representation of t h e 3 0 surface, t h e process of smoothing p e r f o r m s substantially f a s t e r t h a n direct convolution in t h e spatial d o m a i n . T h e approach shows good smoothing results t h a t preserve 3 0 shape. T h e design can be easily extended t o n - d i m e n s i o n a l filtering.
1 Introduction Many methods have been developed for curve smoothing. Gaussian filtering is the most commonly used one having the minimum window size in which time and frequency are analyzed simultaneously. This is known as the u n c e r t a i n t y principle [l]. Unfortunately, the Gaussian filtering and many other smoothing techniques suffer from the shrinkage problem. The reason for smoothing-related shrinkage is the decrease of low frequency component amplitudes representing the smoothed curve [2]. A new representation of closed curves was previously developed by Horn (31. His approach is based on the extended circular image and partially solves the shrinkage problem. However, his method is limited t o convex curves. Lowe proposed t o estimate the amount of shrinkage and compensate for it by inflating the curve accordingly [4].The method is not easily applicable t o 2D and 3D smoothing. Sapiro approached the shrinkage problem by deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety [5]. His assumption that the initial curve be closed, smooth enough and have no self-intersections is not always satisfied. Marzani proposed an approach for processing the optical flow in sequences on cardiac images using an adaptive Fourier smoothness constraint [6]. This approach is not applicable to surfaces. Dual Kriging interpolation was
2
One way t o represent a curve is to use the coordinates z(s), g ( s ) , where s represents the arc-length from some arbitrary starting point. Smoothing a curve may be achieved by filtering the functions z(s) and y(s) separately. To prevent shrinkage of a smoothed curve, the filter should ideally preserve the low frequency while attenuating or removing the high frequency part [2]. The ideal low-pass filter is a first choice for its convenience to implement if its locality is not required. However, the locality is often preferred because a specific curve point does not typically get influenced by other curve points located relatively far on the curve. The filter F(w) given by Oliensis [2] has a faster decay than any power of s and is local. The explicit form of F(w) is given in Eq. 1, where the U , L and C are adjustable constants that control the transition sharpness between 1 and 0. Fig. 1 shows the response
393 0-8186-8183-7/97 $10.000 1997 IEEE
Smoothing and1 local low-pass filtering of curves
t
F(0)
open. If such parallel plane cannot be found, the surface is defined as irregular. In fact, most actual objects have a regular surface. Smoothing of irregular surfaces is briefly considered in Discussion.
Frame n Figure 1: Frequency response of the filter F(w) (Eq. 1). of filter F(w) in the frequency domain. - Baseline
Here, we provide a brief proof of the locality of the filter since it is an important concept that serves as a guideline for the design of our 2D local filters. The following equation describes the spatial response of the filter F(w).
Figure 2: Three-dimensional surface representation using sequential curves. The new surface representation has been developed using z(s, z ) , y(s, z ) , where s is again the arc-length and z is the distance along the frame direction starting from the first frame (Fig. 2). Based on this representation, the three-dimensional surface can be decomposed in two 2D functions z ( s , z ) and y(s,z) that can be smoothed separately using the filter F ( w , [ ) given below. After smoothing, z ( s , z ) and y(s,z) are C" smooth due to the compact support of the filter F(w,[), hence the surface is also C" smooth. Since z(s, z ) and y(s, z ) are two-dimensional functions, a new 2D filter F ( w , [ ) was designed along the ideas of Eq. 1. The filter does not need the same cutoff and transition sharpness for the frequency coordinates w and E. Thus, the filter may be radially asymmetric. The explicit form of the filter is given in Eq. 3, where w u , tu, W L , ( 1 , CI,C, are adjustable constants which control the shape of the filter.
Because F ( w ) is C" (any order of its derivative exists) and any order of the derivative is bounded, it can be concluded that the filter falls off much faster than any power of s. In addition, since the filter F(w) has compact support, the filtered coordinates X ( w ) and Y ( w ) have compact support. As a result, z(s) and y(s) are C" smooth [8].
3
Three-dimensional surface smoothing
In this section, we mainly deal with the smoothing of the sequential curves, which in this case are composed of many frames with one curve in each frame as shown in Fig. 2. For a regular 3D surface, we can often choose a suitable parallel plane sampling t o represent the surface by a sufficient number of planes with one curve in each plane. All such curves are either closed or
(4)
The filter F ( w , [ ) is C" smooth and any order of the derivative is bounded. Hence, similar to the proof
394
given in the previous section, the filter has a faster decay for large r than that of any power of r and the filter is local. F ( w ,5) also ideally preserves the low frequencies of the surfaces. Therefore, it solves the shrinkage problem. Since this filter is radially asymmetric, the smoothing extent can be controlled separately for the two frequency coordinates w and 5. This property is very useful because unequal smoothing may be desirable in different applications. Fig. 2 shows an example in which the curve in each frame ( z ( s , z ) , and y(s,z) with z fixed) may possibly benefit from removing more high frequency components than for the contour curves along the z direction. In fact, the filter defined by Eq. 3 can be generalized
51 L1
C(WI,...,Wn) F(Wl,. . ' ,U,)
=
0
U(Wl,...,Un)
l+erdG(w,...,w,))
otherwise
Figure 3: 3D surface of atherosclerotic plaque auto(5Aatically segmented from a coronary artery. Note the surface noisiness.
(6)
where U(w1,w 2 , . . . ,wn) and L(w1,w 2 , . . . ,wn) represent two symmetric functions that are monotonically increasing for all wi 2 0 , l 5 i 5 n and are C" smooth. Further, C(w1,~ 2 . . ,. ,w n ) > U ( w 1 ,~ 2 . .,. ,w n ) . Typically, U ( w l , . . * , w n )=
w1
(-)
2
wu1 L(Wl,...,Wn)
w1 2
= (-)
Wll
where w,i 2 wli, 1 5 i
4
+-+(-)2
Wn
(7)
CJun
+...+(-)Wn WLn
2
(8)
5 n.
Figure 4: Smoothed 3D surface, the local shape properties as well as the quantitative properties were well preserved (see Table I). Parameters used: w1 = 3,51 = 20.
Results
During the experimental work, we focused on surfaces defined using sequential curves. To avoid interference of the sampling frequency with the frequency inherent in the geometric shape of the sequential curves, each curve was evenly resampled in each frame and the same distance was kept between every two adjacent frames. The resampling schemes mainly affect the high frequency spectrum of the surface and thus have little effect on smoothing. Bilinear interpolation was chosen for resampling. For faster speed, the number of resampling points of each curve and the number of surface curves (frames) were set equal to the power of two. There are six adjustable parameters in Eq. 3. Clearly, this is inconvenient from the practical view. After some experiments and referring to [2], the following parameters were selected as generally applicable: C1 = C2 = 81,wu = 2 x w l , t U = 2 x ( 1 .
We applied the new smoothing filter to surfaces resulting from an automated 3D segmentation of intravascular ultrasound images of coronary arteries (Fig. 3) [9]. The 3D surfaces consisted of 200 frames each with the same distance interval between the adjacent frames. Because th.e frame interval is constant, no interpolation was performed. Instead, the frame number was increased to 512 by extrapolation. After smoothing, only the 200 original frames were kept. In every frame, the contour curve was evenly resampled t o 1024 points. Fig. 4 shows the resulting smoothed surface. For the original surface and the smoothed surfaces with different parameters wl, , the surface- enclosed volumes were calculated. Table 1 lists the absolute
395
the spatial responses of the filter can be calculated and truncated at a suitable length. Then, smoothing may be done by convoluting each point of the surface with the truncated filter. Obviously, problems with surface representation and filter parameter selection must be solved.
Table 1: Volume enclosed by the original and smoothed surface.
6
L
20 30 Original volume:
4.48577e+06 4.48640ef06 4.49465e+06
99.80 99.82
I
100
Conclusion
Design of a novel local two-dimensional shapepreserving smoothing filter and its application to 3D surface smoothing were presented. The filter facilitates smoothing of regular 3D surfaces without shrinkage and can be generalized to n-dimensional filtering. Combined with the presented representation of the 3D surface, the smoothing approach can be implemented with high efficiency and offers high-speed performance.
n
Acknowledgment This work was supported in part by the NSF grant IRI 96- 16747. Help of Weidong Liang is gratefully acknowledged.
and relative volumes of the original and smoothed surfaces. As can be seen from Table 1, the volumes are nearly identical for the original and smoothed surfaces. Clearly, our smoothing approach produces virtually no shrinkage artifacts.
5
References [l] C K Chui. Introduction to wavelets. Academic Press,
Inc., San Diego, 1992. [2] J Oliensis. Local reproducible smoothing without shrinkage. IEEE Trans. Pattern Anal. and Machine Intelligence, 15(3):307-312, 1993. [3] B K P Horn and E J Weldon Jr. Filtering closed curves.
Discussion
In the practical experiments, the contour functions z ( s , z ) and y(s,z) are smoothed using the filter given by Eq. 3. Because s takes the arc-length starting from some arbitrary contour starting point, all the starting points need to be aligned to a reference contour along the z coordinate in all frames. The reference contour is called a baseline (Fig. 2). Sometimes, the baseline may be difficult t o determine. In our experiments, the starting points forming the baseline were selected as follows: First, the centroid (z,y) was calculated for each curve and the rightmost curve point ,,,z(, y) was marked as the starting point. After the appropriate baseline is obtained, the curve is evenly resampled. Unfortunately, each curve usually has a different total length L , and thus the resampling interval distance among different curves is not equal. It would seem that the unequal resampling intervals may result in an undesirable representation of the 3D surface and have significant negative effect on smoothing. However, if smoothing is considered a process of recovering the object’s shape from that corrupted by noise, i.e. assuming that the uncorrupted object is smooth, the resampling intervals are approximately the same in several adjacent frames. Importantly, the designed filter has a good property of locality and thus solves the problem of unequal resampling successfully. For irregular 3D surfaces, smoothing may have to be implemented in the spatial domain. From Eq. 5,
IEEE Trans. Pattern Anal. and Machine Intelligence,
[4]
[5]
[6]
[7]
[8]
[9]
396
PAMI-8(5):665-669, 1986. D G Lowe. Organization of smooth image curves at multiple scales. Int. J. of Comp. Vision, 3:119-130, 1989. G Sapiro and A Tannenbaum. Area and length preserving geometric invariant scale-spaces. IEEE Trans. Pattern Anal. and Machine Intelligence, 17(1):67-72, 1995. F Marzani, L Legrand, and L Dusserre. An adaptive smoothness constraint for computing optical flow on sequences of cardiac images. In Computers in Cardiology 1995, pages 569-572, Los Alamitos, CA, 1995. IEEE. B Andre, F Trochu, and J Dansereau. Approach for the smoothing of three-dimensional reconstructions of the human spine using dual kriging interpolation. Medical d biological engineering d computing, 34:185-191, 1996. G Battle. A block spin construction of ondelettes. Gommun. Math. Phys., 110:601-615, 1987. M Sonka, W Liang, X Zhang, S DeJong, S M Collins, and C R McKay. Three-dimensional automated segmentation of coronary wall and plaque from intravascular ultrasound pullback sequences. In Computers in Cardiology 1995, pages 637-640, Los Alamitos, CA, 1995. IEEE.
Mila,n Sonka
Department of Electrical Engineering Stanford University Stanford, CA 94305
Dept. of Electrical and Comp. Engineering The University of Iowa Iowa City, IA 52242
Abstract
used to smooth three-dimensional human spine curve [7]. The method is not applicable to two-dimensional or three-dimensional smoothing. More directly related t o our work, Oliensis designed a filter F(w) that has several desirable properties: 1) virtually no shrinkage; 2) approximate reproducibility; 3) locality [2]. However, sufficient evidence was not provided to show that the filter f ( t ) falls off at large t more quickly than any power o f t and is local. A brief proof of this important concept is provided below as it serves as a basis for the design of our shape-preserving filter - the locality of the filter F(w) is analyzed and a new local two-dimensional filter F(w,E ) is designed. Our filter not only preserves all the advantageous properties of the 1D filter F ( w ) ,but offers more flexibility and is easy to extend to n dimensions. Our new surface representation is based on decomposition of a regular 3D surface in two 2D functions that are smoothed independently.
Design of a novel three-dimensional ( 3 0 ) shapepreserving smoothing approach is described. Threedimensional surfaces are smoothed without shrinkage artifacts typical for m a n y other approaches. Using o u r n e w representation of t h e 3 0 surface, t h e process of smoothing p e r f o r m s substantially f a s t e r t h a n direct convolution in t h e spatial d o m a i n . T h e approach shows good smoothing results t h a t preserve 3 0 shape. T h e design can be easily extended t o n - d i m e n s i o n a l filtering.
1 Introduction Many methods have been developed for curve smoothing. Gaussian filtering is the most commonly used one having the minimum window size in which time and frequency are analyzed simultaneously. This is known as the u n c e r t a i n t y principle [l]. Unfortunately, the Gaussian filtering and many other smoothing techniques suffer from the shrinkage problem. The reason for smoothing-related shrinkage is the decrease of low frequency component amplitudes representing the smoothed curve [2]. A new representation of closed curves was previously developed by Horn (31. His approach is based on the extended circular image and partially solves the shrinkage problem. However, his method is limited t o convex curves. Lowe proposed t o estimate the amount of shrinkage and compensate for it by inflating the curve accordingly [4].The method is not easily applicable t o 2D and 3D smoothing. Sapiro approached the shrinkage problem by deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety [5]. His assumption that the initial curve be closed, smooth enough and have no self-intersections is not always satisfied. Marzani proposed an approach for processing the optical flow in sequences on cardiac images using an adaptive Fourier smoothness constraint [6]. This approach is not applicable to surfaces. Dual Kriging interpolation was
2
One way t o represent a curve is to use the coordinates z(s), g ( s ) , where s represents the arc-length from some arbitrary starting point. Smoothing a curve may be achieved by filtering the functions z(s) and y(s) separately. To prevent shrinkage of a smoothed curve, the filter should ideally preserve the low frequency while attenuating or removing the high frequency part [2]. The ideal low-pass filter is a first choice for its convenience to implement if its locality is not required. However, the locality is often preferred because a specific curve point does not typically get influenced by other curve points located relatively far on the curve. The filter F(w) given by Oliensis [2] has a faster decay than any power of s and is local. The explicit form of F(w) is given in Eq. 1, where the U , L and C are adjustable constants that control the transition sharpness between 1 and 0. Fig. 1 shows the response
393 0-8186-8183-7/97 $10.000 1997 IEEE
Smoothing and1 local low-pass filtering of curves
t
F(0)
open. If such parallel plane cannot be found, the surface is defined as irregular. In fact, most actual objects have a regular surface. Smoothing of irregular surfaces is briefly considered in Discussion.
Frame n Figure 1: Frequency response of the filter F(w) (Eq. 1). of filter F(w) in the frequency domain. - Baseline
Here, we provide a brief proof of the locality of the filter since it is an important concept that serves as a guideline for the design of our 2D local filters. The following equation describes the spatial response of the filter F(w).
Figure 2: Three-dimensional surface representation using sequential curves. The new surface representation has been developed using z(s, z ) , y(s, z ) , where s is again the arc-length and z is the distance along the frame direction starting from the first frame (Fig. 2). Based on this representation, the three-dimensional surface can be decomposed in two 2D functions z ( s , z ) and y(s,z) that can be smoothed separately using the filter F ( w , [ ) given below. After smoothing, z ( s , z ) and y(s,z) are C" smooth due to the compact support of the filter F(w,[), hence the surface is also C" smooth. Since z(s, z ) and y(s, z ) are two-dimensional functions, a new 2D filter F ( w , [ ) was designed along the ideas of Eq. 1. The filter does not need the same cutoff and transition sharpness for the frequency coordinates w and E. Thus, the filter may be radially asymmetric. The explicit form of the filter is given in Eq. 3, where w u , tu, W L , ( 1 , CI,C, are adjustable constants which control the shape of the filter.
Because F ( w ) is C" (any order of its derivative exists) and any order of the derivative is bounded, it can be concluded that the filter falls off much faster than any power of s. In addition, since the filter F(w) has compact support, the filtered coordinates X ( w ) and Y ( w ) have compact support. As a result, z(s) and y(s) are C" smooth [8].
3
Three-dimensional surface smoothing
In this section, we mainly deal with the smoothing of the sequential curves, which in this case are composed of many frames with one curve in each frame as shown in Fig. 2. For a regular 3D surface, we can often choose a suitable parallel plane sampling t o represent the surface by a sufficient number of planes with one curve in each plane. All such curves are either closed or
(4)
The filter F ( w , [ ) is C" smooth and any order of the derivative is bounded. Hence, similar to the proof
394
given in the previous section, the filter has a faster decay for large r than that of any power of r and the filter is local. F ( w ,5) also ideally preserves the low frequencies of the surfaces. Therefore, it solves the shrinkage problem. Since this filter is radially asymmetric, the smoothing extent can be controlled separately for the two frequency coordinates w and 5. This property is very useful because unequal smoothing may be desirable in different applications. Fig. 2 shows an example in which the curve in each frame ( z ( s , z ) , and y(s,z) with z fixed) may possibly benefit from removing more high frequency components than for the contour curves along the z direction. In fact, the filter defined by Eq. 3 can be generalized
51 L1
C(WI,...,Wn) F(Wl,. . ' ,U,)
=
0
U(Wl,...,Un)
l+erdG(w,...,w,))
otherwise
Figure 3: 3D surface of atherosclerotic plaque auto(5Aatically segmented from a coronary artery. Note the surface noisiness.
(6)
where U(w1,w 2 , . . . ,wn) and L(w1,w 2 , . . . ,wn) represent two symmetric functions that are monotonically increasing for all wi 2 0 , l 5 i 5 n and are C" smooth. Further, C(w1,~ 2 . . ,. ,w n ) > U ( w 1 ,~ 2 . .,. ,w n ) . Typically, U ( w l , . . * , w n )=
w1
(-)
2
wu1 L(Wl,...,Wn)
w1 2
= (-)
Wll
where w,i 2 wli, 1 5 i
4
+-+(-)2
Wn
(7)
CJun
+...+(-)Wn WLn
2
(8)
5 n.
Figure 4: Smoothed 3D surface, the local shape properties as well as the quantitative properties were well preserved (see Table I). Parameters used: w1 = 3,51 = 20.
Results
During the experimental work, we focused on surfaces defined using sequential curves. To avoid interference of the sampling frequency with the frequency inherent in the geometric shape of the sequential curves, each curve was evenly resampled in each frame and the same distance was kept between every two adjacent frames. The resampling schemes mainly affect the high frequency spectrum of the surface and thus have little effect on smoothing. Bilinear interpolation was chosen for resampling. For faster speed, the number of resampling points of each curve and the number of surface curves (frames) were set equal to the power of two. There are six adjustable parameters in Eq. 3. Clearly, this is inconvenient from the practical view. After some experiments and referring to [2], the following parameters were selected as generally applicable: C1 = C2 = 81,wu = 2 x w l , t U = 2 x ( 1 .
We applied the new smoothing filter to surfaces resulting from an automated 3D segmentation of intravascular ultrasound images of coronary arteries (Fig. 3) [9]. The 3D surfaces consisted of 200 frames each with the same distance interval between the adjacent frames. Because th.e frame interval is constant, no interpolation was performed. Instead, the frame number was increased to 512 by extrapolation. After smoothing, only the 200 original frames were kept. In every frame, the contour curve was evenly resampled t o 1024 points. Fig. 4 shows the resulting smoothed surface. For the original surface and the smoothed surfaces with different parameters wl, , the surface- enclosed volumes were calculated. Table 1 lists the absolute
395
the spatial responses of the filter can be calculated and truncated at a suitable length. Then, smoothing may be done by convoluting each point of the surface with the truncated filter. Obviously, problems with surface representation and filter parameter selection must be solved.
Table 1: Volume enclosed by the original and smoothed surface.
6
L
20 30 Original volume:
4.48577e+06 4.48640ef06 4.49465e+06
99.80 99.82
I
100
Conclusion
Design of a novel local two-dimensional shapepreserving smoothing filter and its application to 3D surface smoothing were presented. The filter facilitates smoothing of regular 3D surfaces without shrinkage and can be generalized to n-dimensional filtering. Combined with the presented representation of the 3D surface, the smoothing approach can be implemented with high efficiency and offers high-speed performance.
n
Acknowledgment This work was supported in part by the NSF grant IRI 96- 16747. Help of Weidong Liang is gratefully acknowledged.
and relative volumes of the original and smoothed surfaces. As can be seen from Table 1, the volumes are nearly identical for the original and smoothed surfaces. Clearly, our smoothing approach produces virtually no shrinkage artifacts.
5
References [l] C K Chui. Introduction to wavelets. Academic Press,
Inc., San Diego, 1992. [2] J Oliensis. Local reproducible smoothing without shrinkage. IEEE Trans. Pattern Anal. and Machine Intelligence, 15(3):307-312, 1993. [3] B K P Horn and E J Weldon Jr. Filtering closed curves.
Discussion
In the practical experiments, the contour functions z ( s , z ) and y(s,z) are smoothed using the filter given by Eq. 3. Because s takes the arc-length starting from some arbitrary contour starting point, all the starting points need to be aligned to a reference contour along the z coordinate in all frames. The reference contour is called a baseline (Fig. 2). Sometimes, the baseline may be difficult t o determine. In our experiments, the starting points forming the baseline were selected as follows: First, the centroid (z,y) was calculated for each curve and the rightmost curve point ,,,z(, y) was marked as the starting point. After the appropriate baseline is obtained, the curve is evenly resampled. Unfortunately, each curve usually has a different total length L , and thus the resampling interval distance among different curves is not equal. It would seem that the unequal resampling intervals may result in an undesirable representation of the 3D surface and have significant negative effect on smoothing. However, if smoothing is considered a process of recovering the object’s shape from that corrupted by noise, i.e. assuming that the uncorrupted object is smooth, the resampling intervals are approximately the same in several adjacent frames. Importantly, the designed filter has a good property of locality and thus solves the problem of unequal resampling successfully. For irregular 3D surfaces, smoothing may have to be implemented in the spatial domain. From Eq. 5,
IEEE Trans. Pattern Anal. and Machine Intelligence,
[4]
[5]
[6]
[7]
[8]
[9]
396
PAMI-8(5):665-669, 1986. D G Lowe. Organization of smooth image curves at multiple scales. Int. J. of Comp. Vision, 3:119-130, 1989. G Sapiro and A Tannenbaum. Area and length preserving geometric invariant scale-spaces. IEEE Trans. Pattern Anal. and Machine Intelligence, 17(1):67-72, 1995. F Marzani, L Legrand, and L Dusserre. An adaptive smoothness constraint for computing optical flow on sequences of cardiac images. In Computers in Cardiology 1995, pages 569-572, Los Alamitos, CA, 1995. IEEE. B Andre, F Trochu, and J Dansereau. Approach for the smoothing of three-dimensional reconstructions of the human spine using dual kriging interpolation. Medical d biological engineering d computing, 34:185-191, 1996. G Battle. A block spin construction of ondelettes. Gommun. Math. Phys., 110:601-615, 1987. M Sonka, W Liang, X Zhang, S DeJong, S M Collins, and C R McKay. Three-dimensional automated segmentation of coronary wall and plaque from intravascular ultrasound pullback sequences. In Computers in Cardiology 1995, pages 637-640, Los Alamitos, CA, 1995. IEEE.
