Abstract:
A method for modeling a 2-dimensional tubular structure in a digitized image includes providing a digitized image of a tubular structure containing a plurality of 2D balls of differing radii, initializing a plurality of connected spline segments that form an envelope surrounding the plurality of 2D balls, each the spline segment S i  being parameterized by positions of the i th  and i+1 th  balls and contact angles α i , α i+1  from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment S i , each the α i  affecting spline segment S i  and S i−1 , and updating the angles by minimizing an energy that is a functional of the angles, where the updating is repeated until the energy is minimized subject to a constraint that the envelope is tangent to each ball at each point of contact, where the envelope is represented by the contact angles.

Description:
CROSS REFERENCE TO RELATED UNITED STATES APPLICATIONS 
     This application claims priority from “Variational Ball Skinning for Geometric Modeling of Tubular Structures”, U.S. Provisional Application No. 60/864,853 of Slabaugh, et al., filed Nov. 8, 2006, the contents of which are herein incorporated by reference. 
    
    
     TECHNICAL FIELD 
     This disclosure is directed to computing an interpolating envelope of an ordered set of 2D balls. 
     DISCUSSION OF THE RELATED ART 
     The geometric question of ball skinning is the computation of an interpolating envelope of a set of balls. An example ball skinning is shown in  FIGS. 1(   a )-( b ). Given an ordered sequence of balls, shown in  FIG. 1(   a ), one can produce in envelope that optimally interpolates the balls, as shown in  FIG. 1(   b ). This envelope has two splines, an inner  11  and an outer  12  spline that are computed using, differential equations. Ball skinning envelopes can be either one-sided or two-sided envelopes. A one-sided envelope is a contour that rests on one side of a collection of balls such as that portrayed in  FIG. 2 , while a two-sided envelope defines an interpolating region that has an inside and outside, as demonstrated in  FIGS. 1(   a )-( b ). 
     Ball skinning arises in numerous applications, including character skinning, molecular surface model generation, and modeling of tubular structures. The balls can have different radii, be configured in different positions, and may or may not overlap. In one formulation of the problem, the envelopes are required to touch each ball at a point of contact, and be tangent to the ball at the point of contact, as illustrated in  FIG. 1 . The envelope then forms a “skin” that rests on and interpolates the underlying balls. 
     The question of skinning appears in various contexts. In computer graphics and animation, often an articulated object or character is constructed by forming a layered representation consisting of a skeletal structure and a corresponding geometric skin. The skeleton has fewer degrees of freedom and is simpler to adjust by an animator. Given a new skeletal pose, the skinning algorithm is responsible for deforming the geometric skin to respond to the motion of the underlying skeleton. 
     The question of ball skinning appears frequently in the context of chemistry and molecular biology, when generating surface meshes for molecular models. Several algorithms exist to skin a molecular model to produce a C 1  continuous surface that is tangent smooth and has high mesh quality. These methods are typically either based on Delaunay triangulation or by finding the isosurface of an implicit function. While the surfaces generated by these methods are tangent to the balls and have smoothness at the point of tangency, these methods do not provide an optimally smooth envelope. 
     One application concerns modeling the geometry of a blood vessel that has been identified using a 2D variant of a ball packing algorithm, which places numerous balls of different radii so that they fit snugly inside an imaged blood vessel. Given these balls, one would like to find an smooth, C 1  envelope that smoothly interpolates the balls. This surface can then be used for visualization of the blood vessel as well as measurements such as volume or surface area. For a given configuration of balls, there are an infinite number of possible solutions to this question. 
     SUMMARY OF THE INVENTION 
     Exemplary embodiments of the invention as described herein generally include methods and systems for modeling the envelope as using a angle for each ball. An exemplary envelope is a C 1  spline, which, by construction, touches each ball at a point of contact and be tangent to ball at the point of contact. To formulate the question so that it is well-posed, one seeks the envelope that has minimal arc length and/or curvature. This can be achieved by deriving two differential equations that minimize an energy based on this constrained variational problem, one for deforming this constrained spline to minimize its arc length, and a second for minimizing its curvature. These differential equations can then be solved to update a given spline to its optimal position. Given an initial envelope, the envelope&#39;s parameters are evolved using the differential equations until convergence occurs. This produces the envelope that has minimal length and/or curvature, touches each hall at a point of contact, and is tangent to the ball at the point of contact. In this sense, the method provides an optimal constrained interpolation of the balls. Experimental examples are presented of how these differential equations are used perform optimally generating interpolating envelopes of balls of different sizes and in various configurations. 
     According to an aspect of the invention, there is provided a method for modeling a 2-dimensional tubular structure in a digitized image, the method including providing a digitized image of a tubular structure, the image comprising a plurality of intensities associated with a 2 dimensional (2D) grid of voxels, the tubular structure containing a plurality of 2D balls of differing radii, initializing a plurality of connected spline segments S i  that form an envelope surrounding the plurality of 2D balls, each the spline segment S i  being parameterized by positions of the i th  and i+1 th  balls and contact angles α i , α i+1  from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment S i , each the α i  affecting spline segment S i  and S i+1 , and updating the angles by minimizing an energy E, where E is an energy that is a functional of the angles equal to E=(1−k)E α +kE c , where k is a constant, E α  is an arc-length term and E c  is a curvature term, where the updating is repeated until the energy E is minimized subject to a constraint that the envelope is tangent to each ball at each point of contact, where the envelope is represented by the contact angles. 
     According to a further aspect of the invention, the angles are updated according to α n+1 =α n −Δt∇E α     n   (α n ) where Δt is a time step, superscript n is a time step index for the angles, and 
     
       
         
           
             
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     According to a further aspect of the invention, each spline segment is a C 1  cubic spline segment modeled as S i =A i t 3 +B i t 2 +C i t+D i , where tε[0,1] and where each the curve satisfies the constraints 
                   S   i     ⁢       ❘       t   =   0         =     p   i       ,     
     ⁢           ⅆ     S   i         ⅆ   t       ⁢     ❘     t   =   0         =     t   i       ,     
     ⁢         S   i     ⁢     ❘     t   =   1         =     p     i   +   1         ,     
     ⁢           ⅆ     S   i         ⅆ   t       ⁢     ❘     t   =   1         =       t     i   +   1       .             
where p i  is the contact point on the ith ball
 
               p   i     =       c   i     +     [             r   i     ⁢   cos   ⁢           ⁢     α   i                   r   i     ⁢   sin   ⁢           ⁢     α   i             ]             
where c i  is the center of the ball and r i  is the radius, and
 
               t   i     =     [             -     a   i       ⁢   sin   ⁢           ⁢     α   i                   a   i     ⁢   cos   ⁢           ⁢     α   i             ]           
is the starting direction of the curve, where α i  is a stiffness factor.
 
     According to a further aspect of the invention, each α i  is fixed to be half the distance between the next and previous ball centers, and where for the first and last balls, the α i  is fixed to the distance between the ball center and its neighbor ball center. 
     According to a further aspect of the invention, minimizing the arc-length comprises minimizing 
               E   a     =       ∑     i   =   1     N     ⁢     ∫            S   t   ′          ⁢     ⅆ   t                 
using derivatives with respect to the angles α i , where the prime indicates a derivative with respect to t and the sum is over all the curves, and where
 
     
       
         
           
             
               
                 
                   
                     
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               E   c     =         ∑     i   =   1     N     ⁢     ∫         κ   i   2     ⁡     (   t   )       ⁢     ⅆ   t     ⁢           ⁢   where   ⁢           ⁢     κ   i           =           〈       S   i   ′     ,     JS   i   ′′       〉         〈       S   i   ′     ,     S   i   ′       〉       3   2         ⁢           ⁢   and   ⁢           ⁢   J     =     [         0       1             -   1         0         ]               
using derivatives with respect to the angles α i , where the primes indicate derivatives with respect to t and the sum is over all the curves, and where
 
     
       
         
           
             
               
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     According to a further aspect of the invention, coefficients A i , B i , C i , D i  are defined as
 
 A   i =−2 p   i+1 +2 p   i   +t   i   +t   i+1  
 
 B   i =3 p   i+1 −3 p   i −2 t   i   −t   i+1 ,
 
C i =t i  
 
D i =p i  
 
and where
 
     
       
         
           
             
               
                 
                   
                     
                       
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     According to a further aspect of the invention, the energy E is minimized subject to a further constraint of one of that the envelope has a minimal arc-length, the envelope has a minimal curvature, or the envelope has a minimal arc-length and curvature. 
     According to a further aspect of the invention, the spline segments are initialized by choosing an angle α i  for each ball i that matches a ray orthogonal to a centerline connecting adjacent ball center points. 
     According to a further aspect of the invention, the method includes initializing a plurality of connected spline segments S i  that interpolate an inner boundary of the plurality of 2D balls, each the spline segment S i  being parameterized by positions of the i th  and i+1 th  balls and contact angles α i , α i+1  from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment S i , each the α i  affecting spline segment S i  and S i−1 , and updating the angles by minimizing an energy E, where E is an energy that is a functional of the angles equal to E=(1−k)E α +kE c , where k is a constant, E α  is an arc-length term and E c  is a curvature term, where the updating is repeated until the energy E is minimized subject to a constraint that the inner boundary is tangent to each ball at each point of contact, where the inner boundary is represented by the contact angles. 
     According to a further aspect of the invention, the tubular structure represents a blood vessel. 
     According to another aspect of the invention, there is provided a program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for modeling a 2-dimensional tubular structure in a digitized image. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1(   a )-( b ) illustrate an example of a two sided ball skinning, according to an embodiment of the invention. 
         FIG. 2  illustrates the goal of ball skinning, which is finding an envelope that interpolates the ordered set of balls. 
         FIG. 3  depicts a segment of an envelope, according to an embodiment of the invention. 
         FIGS. 4(   a )-( c ) depict a simple example of ball skinning, according to an embodiment of the invention. 
         FIGS. 5(   a )-( c ) depict the skinning of another set of balls, according to an embodiment of the invention. 
         FIGS. 6(   a )-( c ) show an example of generating an interpolating region for a collection of balls, according to an embodiment of the invention. 
         FIGS. 7(   a )-( c ) illustrate ball skinning for balls arranged on a sine wave and having a variable radius, according to an embodiment of the invention. 
         FIG. 8  illustrates an initialization of an envelope for a gradient descent minimalization of the envelope, according to an embodiment of the invention. 
         FIG. 9  is a flowchart of a method for computing an interpolating envelope of an ordered set of 2D balls, according to an embodiment of the invention. 
         FIG. 10  is a block diagram of an exemplary computer system for implementing a method for computing an interpolating envelope of an ordered set of 2D balls, according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Exemplary embodiments of the invention as described herein generally include systems and methods for computing an interpolating envelope of an ordered set of 2D halls. Accordingly, while the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that there is no intent to limit the invention to the particular forms disclosed, but on the contrary, the invention is to cover all modifications, equivalents, and alternatives falling, within the spirit and scope of the invention. 
     As used herein, the term “image” refers to multi-dimensional data composed of discrete image elements (e.g., pixels for 2-D images). The image may be, for example, a medical image of a subject collected by computer tomography, magnetic resonance imaging, ultrasound, or any other medical imaging system known to one of skill in the art. The image may also be provided from non-medical contexts, such as, for example, remote sensing systems, electron microscopy, etc. Although an image can be thought of as a function from R 2  to R, the methods of the inventions are not limited to such images. For a 2-dimensional image, the domain of the image is typically a 2-dimensional rectangular array, wherein each pixel or voxel can be addressed with reference to a set of 2 mutually orthogonal axes. The terms “digital” and “digitized” as used herein will refer to images or volumes, as appropriate, in a digital or digitized format acquired via a digital acquisition system or via conversion from an analog image. 
       FIG. 2  illustrates the goal of a ball skinning method according to an embodiment of the invention, which is computing an interpolating envelope of an ordered set of 2D balls. Each spline S i  is tangent to ball B i  at point p i  where the tangent is orthogonal to a ray extended from the centerpount c i  where the ray makes an angle α i  with a horizontal. It is desired to find an envelope S that satisfies several geometric criteria: 
     1. The envelope should be modeled by a point of contact with a ball. 
     2. The envelope should be tangent to a ball at the point of contact. 
     3. The envelope should minimize a functional composed of terms based on arc length and curvature. 
     An envelope according to an embodiment of the invention can be represented using the contact angle of each ball within the envelope. 
       FIG. 2  depicts a desirable envelope as a dotted line, which is a C 1  curve passing through the point of contact on each ball, where at each contact point, the envelope is tangent to the ball. A C 1  curve is a continuous curve with continuous first derivatives. The envelope S is composed of a set of envelope segments, S i , for i=1, . . . , N, where N is the total number of segments. 
     Segments 
       FIG. 3  depicts a segment of an envelope. As shown in  FIG. 3 , one can choose to model each segment S i  using a spline that starts at point p i  in direction t i , and ends at point p i+1  in direction t i+1 . The segment can be modeled using a cubic polynomial curve
 
 S   i   =A   i   t   3   +B   i   t   2   +C   i   t+D   i ,  (1)
 
since the four constraints require four degrees of freedom. For the i th  segment, A i , B i , C i , and D i  are coefficients, and tε[0,1] is a time variable that parameterizes the curve.
 
     To determine the coefficients for a segment, the following constraints should be satisfied: 
                   S   i     ⁢     ❘     t   =   0         =     p   i       ,     
     ⁢           ⅆ     S   i         ⅆ   t       ⁢     ❘     t   =   0         =     t   i       ,     
     ⁢         S   i     ⁢     ❘     t   =   1         =     p     i   +   1         ,     
     ⁢           ⅆ     S   i         ⅆ   t       ⁢     ❘     t   =   1         =       t     i   +   1       .             
With these constraints, and the derivative of the segment,
 
                       ⅆ     S   i         ⅆ   t       =       3   ⁢     A   i     ⁢     t   2       +     2   ⁢     B   i     ⁢   t     +     C   i               (   2   )               
one can obtain a system of four equations for the four coefficients:
 
D i =p i ,
 
C i =t i ,
 
 A   i   +B   i   +C   i   +D   i   =p   i+1 ,
 
3 A   i +2 B   i   +C   i   =t   i+1 ,
 
which can be solved, yielding:
 
 A   i =−2 p   i+1 +2 p   i   +t   i   +t   i+1  
 
 B   i =3 p   i+1 −3 p   i −2 t   i   −t   i+1  
 
C i =t i  
 
D i =p i  
 
Endpoints
 
     Now that there is a way to model each segment of the envelope, one can determine the endpoints p i , p i+1  and their respective tangents, t i , t i+1  of each segment. As shown in  FIG. 2 , the point of contact p i  on the i th  ball can be represented as 
                       p   i     =       c   i     +     [             r   i     ⁢   cos   ⁢           ⁢     α   i                   r   i     ⁢   sin   ⁢           ⁢     α   i             ]         ,           (   4   )               
where r i  is the radius of a ball, c i  is its center, and α i  is an angle. In addition, the tangent can be represented as
 
                       t   i     =     [             -     a   i       ⁢   sin   ⁢           ⁢     α   i                   a   i     ⁢   cos   ⁢           ⁢     α   i             ]       ,           (   5   )               
where α i  is a stiffness term that controls the influence of the tangential constraint. Each α i  can be fixed to be half the distance between the next and previous ball centers (for the first and last halls, it is the distance between the hall center and its neighbor ball center). Note that both the point p i  and the tangent t i  are only a function of the angle α i , since the radius of the ball is fixed.
 
     There is now a way to represent the envelope S as a set of segments S i , where each segment S i  interpolates between the points of contact p i , p i+1  with balls B i , B i+1 , subject to tangent conditions t i , t i+1  respectively. 
     By construction of the problem, the angle α i  affects only the segment S i  as well as the segment S i−1 , as can be easily seen in  FIG. 2 . Finally, the envelope is fully parameterized by the balls and the spline angles α i . Since the balls are fixed, the objective will be to compute the angles α i  that form the optimal envelope. 
     Energy Minimization 
     There are an infinite number of envelopes that are modeled by a contact point on each ball and have a direction tangent to the bail at the point of contact. To further constrain the problem, the envelope can be required to have minimal arc length and/or be smooth. This can be achieved by finding the angles α i  that optimize an energy functional. First, equations are derived to compute the envelope with minimal arc length, then curvature is considered. 
     Arc Length Minimization 
     Minimization of the arc length results in the shortest envelope that satisfies the geometric constraints imposed by the ball representation. That is, it is desired to find angles α i  that minimize
 
 E   α   =∫|S′|dt,  
 
where S′ is the derivative of S with respect to t. Since the envelope is represented as a set of segments, this is equivalent to
 
               E   a     =       ∑     i   =   1     N     ⁢           ⁢     ∫            S   i   ′          ⁢       ⅆ   t     .                 
Next, the derivative of the energy is taken with respect to the angle α i . As stated above, the i th  angle only affects the segments S i−1  and S i . Therefore,
 
                       ∂     E   a         ∂     α   i         =         ∂     ∂     α   i         ⁢     (     ∫            S     i   -   1     ′          ⁢     ⅆ   t         )       +       ∂     ∂     α   i         ⁢       (     ∫            S   i   ′          ⁢     ⅆ   t         )     .                 (   6   )               
Second Term
 
     Considering the second term of EQ. (6) first, propagating the derivative with respect to α i  through the integral, it is easy to show that 
                   ∂     ∂     α   i         ⁢     (     ∫            S   i   ′          ⁢     ⅆ   t         )       =       ∫       ∂     ∂     α   i         ⁢       〈       S   i   ′     ,     S   i   ′       〉       1   2       ⁢     ⅆ   t         =     ∫         〈       S   i   ′     ,     S   i   ′       〉       -     1   2         ⁢     〈       S   i   ′     ,       ∂     S   i   ′         ∂     α   i           〉     ⁢     ⅆ   t             ,         
where &lt; &gt; denotes an inner product. Next, an expression for the
 
               ∂     S   i   ′         ∂     α   i             
terms can be derived using EQ. 2, yielding
 
                 ∂     S   i   ′         ∂     α   i         =       3   ⁢     t   2     ⁢       ∂     A   i         ∂     α   i           +     2   ⁢   t   ⁢       ∂     B   i         ∂     α   i           +         ∂     C   i         ∂     α   i         .             
The derivatives
 
                 ∂     A   i         ∂     α   i         ,       ∂     B   i         ∂     α   i         ,     and   ⁢           ⁢       ∂     C   i         ∂     α   i                 
can be derived using EQ. 3, as
 
                         ∂     A   i         ∂     α   i         =       2   ⁢       ∂     p   i         ∂     α   i           +       ∂     t   i         ∂     α   i             ⁢     
     ⁢         ∂     B   i         ∂     α   i         =         -   3     ⁢       ∂     p   i         ∂     α   i           -     2   ⁢       ∂     t   i         ∂     α   i               ⁢     
     ⁢         ∂     C   i         ∂     α   i         =       ∂     t   i         ∂     α   i                   (   7   )               
Finally, the derivatives
 
                 ∂     p   i         ∂     α   i         ⁢           ⁢   and   ⁢           ⁢       ∂     t   i         ∂     α   i               
can be derived from EQS. 4 and 5 as
 
                 ∂     p   i         ∂     α   i         =     [             -     r   i       ⁢   sin   ⁢           ⁢     α   i                   r   i     ⁢   cos   ⁢           ⁢     α   i             ]                     ∂     t   i         ∂     α   i         =     [             -     a   i       ⁢   cos   ⁢           ⁢     α   i                   -     a   i       ⁢   sin   ⁢           ⁢     α   i             ]           
All the derivatives needed to compute the second term in EQ. 6 have now been derived.
 
First Term
 
     Now consider the first term of EQ. 6, which has a very similar derivation. Propagating the derivative with respect to α i  through the integral yields 
                 ∂     ∂     α   i         ⁢     (     ∫            S     i   -   1     ′          ⁢     ⅆ   t         )       =     ∫         〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       -     1   2         ⁢     〈       S     i   -   1     ′     ,       ∂     S     i   -   1     ′         ∂     α   i           〉     ⁢     ⅆ   t               
As before, an expression can be derived for the
 
               ∂     S     i   -   1     ′         ∂     α   i             
terms using EQ. 2, yielding
 
               ∂     ∂     α   i         =       3   ⁢     t   2     ⁢       ∂     A     i   -   1           ∂     α   i           +     2   ⁢   t   ⁢       ∂     B     i   -   1           ∂     α   i           +         ∂     C     i   -   1           ∂     α   i         .             
Next, the derivatives
 
                 ∂     A     i   -   1           ∂     α   i         ,       ∂     B     i   -   1           ∂     α   i         ,     and   ⁢           ⁢       ∂     C     i   -   1           ∂     α   i                 
can be derived using EQ. 3, as
 
                         ∂     A     i   -   1           ∂     α   i         =         -   2     ⁢       ∂     p   i         ∂     α   i           +       ∂     t   i         ∂     α   i             ,     
     ⁢         ∂     B     i   -   1           ∂     α   i         =       3   ⁢       ∂     p   i         ∂     α   i           -       ∂     t   i         ∂     α   i             ,     
     ⁢         ∂     C     i   -   1           ∂     α   i         =   0.             (   8   )               
All the derivatives needed to compute the first term of EQ. 6 have been derived.
 
Curvature Minimization
 
     In order to minimize the curvature, note that curvature can be positive or negative. Thus, the squared curvature is minimized by finding the angles α i  that minimize
 
 E   c =∫κ 2 ( t ) dt,  
 
where κ(t) is the curvature of S at point t. Since the envelope is represented as a set of segments, this is equivalent to
 
                 E   c     =       ∑     i   =   1     N     ⁢     ∫         κ   i   2     ⁡     (   t   )       ⁢     ⅆ   t             ,         
where κ(t) is the curvature at point t along segment S i . Next, the derivative of the energy is taken with respect to the angle α i . As stated above, the i th  angle only affects the segments S i−1  and S i . Therefore,
 
                       ∂     E   c         ∂     α   i         =         ∂     ∂     α   i         ⁢     (     ∫         κ     i   -   1     2     ⁡     (   t   )       ⁢     ⅆ   t         )       +       ∂     ∂     α   i         ⁢       (     ∫         κ   i   2     ⁡     (   t   )       ⁢     ⅆ   t         )     .                 (   9   )               
Recall that the curvature is given by
 
                 κ   i     =              S   i   ′     ×     S   i   ″                     S   i   ′          3         ,         
which can be re-expressed as
 
                 κ   i     =       〈       S   i   ′     ,     JS   i   ″       〉         〈       S   i   ′     ,     S   i   ′       〉       3   2           ,     
     ⁢   where               J   =     [         0       1             -   1         0         ]           
is a 90 degree rotation matrix and &lt; &gt; denotes an inner product. Using these equations, EQ. 9 becomes
 
                       ∂     E   c         ∂     α   i         =         ∂     ∂     α   i         ⁢     (     ∫         [       〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉         〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       3   2         ]     2     ⁢     ⅆ   t         )       +       ∂     ∂     α   i         ⁢       (     ∫         [       〈       S   i   ′     ,     JS   i   ″       〉         〈       S   i   ′     ,     S   i   ′       〉       3   2         ]     2     ⁢     ⅆ   t         )     .                 (   10   )               
Second Term
 
     To derive the second term of EQ. 10, the derivative with respect to α i  is propagated through the integral to obtain 
                 ∂     ∂     α   i         ⁢     (     ∫         [       〈       S   i   ′     ,     JS   i   ″       〉         〈       S   i   ′     ,     S   i   ′       〉       3   2         ]     2     ⁢     ⅆ   t         )       =     ∫       2   [       〈       S   i   ′     ,     JS   i   ″       〉         〈       S   i   ′     ,     S   i   ′       〉       3   2         ]     ⁢     (           ∂     ∂     α   i         ⁢     〈       S   i   ′     ,     JS   i   ″       〉           〈       S   i   ′     ,     S   i   ′       〉       3   2         -       3   2     ⁢         〈       S   i   ′     ,     JS   i   ″       〉     ⁢     ∂     ∂     α   i         ⁢     〈       S   i   ′     ,     S   i   ′       〉           〈       S   i   ′     ,     S   i   ′       〉       5   2             )     ⁢       ⅆ   t     .               
For this, one needs the derivatives
 
                 ∂     ∂     α   i         ⁢     〈       S   i   ′     ,     JS   i   ″       〉     ⁢           ⁢   and   ⁢           ⁢     ∂     ∂     α   i           ⁣       〈       S   i   ′     ,     S   i   ′       〉     .           
These derivatives can be shown to be
 
                     ∂     ∂     α   i         ⁢     〈       S   i   ′     ,     JS   i   ″       〉       =       〈         ∂     S   i   ′         ∂     α   i         ,     JS   i   ″       〉     +     〈       S   i   ′     ,     J   ⁢       ∂     S   i   ″         ∂     α   i             〉         ,     
     ⁢         ∂     ∂     α   i         ⁢     〈       S   i   ′     ,     S   i   ′       〉       =   2       ⁣       〈       S   i   ′     ,       ∂     S   i   ′         ∂     α   i           〉     .           
Next, derive an expression for the
 
               ∂     S   i   ′         ∂     α   i                 and               ∂     S   i   ″         ∂     α   i             
terms using EQ. 2, yielding
 
                   ∂     S   i   ′         ∂     α   i         =       3   ⁢     t   2     ⁢       ∂     A   i         ∂     α   i           +     2   ⁢   t   ⁢       ∂     B   i         ∂     α   i           +       ∂     C   i         ∂     α   i             ,     
     ⁢         ∂     S   i   ″         ∂     α   i         =       6   ⁢   t   ⁢       ∂     A   i         ∂     α   i           +     2   ⁢         ∂     B   i         ∂     α   i         .                 
The derivatives
 
                 ∂     A   i         ∂     α   i         ,       ∂     B   i         ∂     α   i         ,     and   ⁢           ⁢       ∂     C   i         ∂     α   i                 
are given in EQ. 7. All the derivatives needed to compute the second term in EQ. 10 are now present.
 
First Term
 
     The first term of EQ. 10 is very similar the second term derived above. Propagating the derivative with respect to α i  through the integral, one obtains 
                 ∂     ∂     α   i         ⁢     (     ∫         [       〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉         〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       3   2         ]     2     ⁢     ⅆ   t         )       =     ∫       2   [       〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉         〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       3   2         ]     ⁢     (           ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉           〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       3   2         -       3   2     ⁢         〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉     ⁢     ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉           〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       5   2             )     ⁢       ⅆ   t     .               
For this, the derivatives
 
               ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉     ⁢           ⁢   and   ⁢           ⁢     ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉           
are needed. It can be shown that these derivatives are
 
                   ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     JS     i   -   1     ″       〉       =       〈         ∂     S     i   -   1     ′         ∂     α   i         ,     JS     i   -   1     ″       〉     +     〈       S     i   -   1     ′     ,     J   ⁢       ∂     S     i   -   1     ″         ∂     α   i             〉         ,     
     ⁢         ∂     ∂     α   i         ⁢     〈       S     i   -   1     ′     ,     S     i   -   1     ′       〉       =     2   ⁢       〈       S     i   -   1     ′     ,       ∂     S     i   -   1     ′         ∂     α   i           〉     .               
The expressions for
 
                 ∂     S     i   -   1     ′         ∂     α   i         ⁢           ⁢   and   ⁢           ⁢       ∂     S     i   -   1     ″         ∂     α   i               
are
 
                   ∂     S     i   -   1     ′         ∂     α   i         =       3   ⁢     t   2     ⁢       ∂     A     i   -   1           ∂     α   i           +     2   ⁢   t   ⁢       ∂     B     i   -   1           ∂     α   i           +       ∂     C     i   -   1           ∂     α   i             ,     
     ⁢         ∂     S     i   -   1     ″         ∂     α   i         =       6   ⁢   t   ⁢       ∂     A     i   -   1           ∂     α   i           +     2   ⁢         ∂     B     i   -   1           ∂     α   i         .                 
The derivatives
 
                 ∂     A     i   -   1           ∂     α   i         ,       ∂     B     i   -   1           ∂     α   i         ,     and   ⁢           ⁢       ∂     C     i   -   1           ∂     α   i                 
are given in EQ. 8.
 
     Thus, all the derivatives needed to compute the first term of EQ. 10 have been 
     Boundary Conditions 
     The integrals in EQ. 10 are evaluated for each angle α i . However, for the first ball, i=1, there is no segment S i−1 , so the first integral is skipped in EQ. 10. Likewise, for the last ball, i=N, there is no segment S i , so the second integral in the equation is skipped. 
     Thus, the gradient of energy functionals E α  and E c  has been derived with respect to angles, α i . The derivation consisted implicitly of several steps via the chain rule, as the energy is the squared curvature, which in turn is a function of the envelope, which in turn is a function of the segment constants A i , B i , C i , D i  and A i−1 , B i−1 , C i−1 , D i−1 , which in turn are functions of the angles α i . 
     Energy Minimization 
     The energies E α  and E c  can be combined together, as
 
 E =(1− k ) E   α   +kE   c ,
 
where k is a constant used to weight the arc length minimization relative to the curvature minimization. Setting k=0 results in the arc length minimization, while setting k−1 gives the curvature minimization. Convex combinations of the two can be selected using kε[0,1]. Therefore, the combined energy minimization is given by
 
                   ∂   E       ∂     α   i         =         (     1   -   k     )     ⁢       ∂     E   a         ∂     α   i           +     k   ⁢       ∂     E   c         ∂     α   i               ,         
where
 
               ∂     E   a         ∂     α   i             
is given in EQ. 6 and
 
               ∂     E   c         ∂     α   i             
is provided in EQ. 9. In all of the experiments herein disclosed, k=0.9, to encourage smoother solutions. This value of k is exemplary and non-limiting, and can take on other values in other embodiments of the invention.
 
     These equations are a set of differential equations that can be used to optimize the envelope by manipulating the angles α=[α 1 , . . . , α N ] T . One exemplary, non-limiting method of optimizing the envelope is through a gradient descent procedure. A flowchart of such a method for computing an interpolating envelope of an ordered set of 2D balls is depicted in  FIG. 9 . A method according to an embodiment of the invention starts by providing at step  91  a digitized image of a blood vessel that has been identified using a 2D variant of a ball packing algorithm, which places numerous balls of different radii that fit snugly inside an imaged blood vessel. To start the optimization procedure, since the envelope is fully specified by the angles α i , one only needs to initialize an angle for each ball at step  92 . According to an embodiment of the invention, one way to do this is to choose the angle α for each ball that matches the ray orthogonal to the centerline, as shown in  FIG. 8 , connecting adjacent ball center points.  FIG. 8  shows ball  80  with center point  83  and rays  81  and  82  orthogonal to centerline  84 . However, there are two possible angles in the figure, with respect to the rays  81  and  82 . For two-sided envelopes, one would use both, while for one-sided envelopes, one would choose the angle always on the left or right side of the centerline. Then, at step  93 , letting α i   n  be the i th  angle at iteration n, one can then update the angles by moving them in the negative gradient direction, i.e.,
 
α n+1 =α n   −Δt∇E   α     n   (α n ),  (11)
 
where Δt is a time step and
 
               ∇       E     α   n       ⁡     (     α   n     )         =         [         ∂   E       ∂     α   1   n         ,   …   ⁢           ,       ∂   E       ∂     α   N   n           ]     T     .           
The updating of step  93  is repeated until the energy reaches a suitable minimum, at step  94 . It is to be understood, however, that the disclosure of a gradient descent procedure is for expository purposes only, and is a special case of the more general formulation
 
α n+1 =α n   +Δt ·ƒ( E|α   n |).
 
Other techniques can be used to minimize the energy functional in other embodiments of the invention.
 
     The computational complexity of an algorithm according to an embodiment of the invention depends on the number of balls N+1 and the number of points L on a segment where the points and derivatives are evaluated. For each iteration of the gradient descent procedure, the computational complexity is O(NL). The number of iterations required depends on the time step Δt as well as how close the initial envelope is to the final solution. Note that the gradient descent approach only guarantees a locally optimal solution; however, given the constraints of the problem formulation, the energy functional is rather convex. 
     Results 
       FIGS. 4(   a )-( c ) show a simple example ball skinning. The initial spline  41  is shown in  FIG. 4(   a ), the result  42  after 50 iterations is shown in  FIG. 4(   b ), and the converged result  43  after 100 iterations is shown in  FIG. 4(   c ). Here, four balls of radius 50, 75, 50, and 25 pixels, respectively were set along the x-axis. The initial angles for this experiment were 0.57, 1.07, 1.57 and 2.07 radians, respectively; the initial envelope is shown in  FIG. 4(   a ) of the figure. The angles were iteratively updated using EQ. 11. An intermediate solution after 50 iterations in shown in  FIG. 4(   b ), at this stage, the envelope is considerably smoother while still satisfying the constraints of the problem. The result after 100 iterations are shown in  FIG. 4(   c ), at which point the energy has reached a minimum and the angles have converged. The solution (all 100 iterations) is computed in 47 milliseconds using C++ code compiled on a machine with a 3.0 GHz single-core processor. 
       FIGS. 5(   a )-( c ) shows a slightly more complicated example for which some balls overlap and others do not. The initial envelope  51  is shown in  FIG. 5(   a ), an intermediate result  52  after 70 iterations in  FIG. 5(   b ), and the final result  53  upon convergence after 140 iterations in  FIG. 5(   c ). The solution (all 140 iterations) is computed in 143 milliseconds. 
       FIGS. 6(   a )-( c ) show an example of generating an interpolating region for a collection of balls. In this case, there are two envelopes, one defining the interior boundary of the region, and another defining the exterior boundary. For each ball, there are two points of contact: one from the interior envelope and one for the exterior envelope; however, these points of contact are constrained to be separated by 180 degrees. Therefore, for each ball there is only one angle α i  to be determined as in the examples above. The angle is solved for all the balls, with each envelope contributing a term in EQ. 11.  FIG. 6(   a ) shows the initializations  61 ,  66 ,  FIG. 6(   b ) shows intermediate results  62 ,  67  after 50 iterations, and  FIG. 6(   c ) shows the final converged results  63 ,  68  after 100 iterations. Convergence for this example occurs in 190 milliseconds. 
     More examples are provided in  FIGS. 7(   a )-( c ) and  FIGS. 1(   a )-( b ).  FIGS. 7(   a )-( c ) illustrate ball skinning for balls arranged on a sine wave and having a variable radius. The initial splines  71 ,  72  are shown in  FIG. 7(   a ), the intermediate results  73 ,  74  are shown in  FIG. 7(   b ), and the final converged results  75 ,  76  are shown in  FIG. 7(   c ). Convergence of the skinning algorithm, starting from a set of angles far from the optimal result, takes 775 milliseconds. In  FIGS. 1(   a )-( b ), the variable radius balls are arranged in a spiral, and  FIG. 1(   b ) shows the converged results  11 ,  12 . The envelopes are generated in 2.5 seconds. 
     System Implementation 
     It is to be understood that embodiments of the present invention can be implemented in various forms of hardware, software, firmware, special purpose processes, or a combination thereof. In one embodiment, the present invention can be implemented in software as an application program tangible embodied on a computer readable program storage device. The application program can be uploaded to, and executed by, a machine comprising any suitable architecture. 
       FIG. 10  is a block diagram of an exemplary computer system for implementing a method for computing an interpolating envelope of an ordered set of 2D balls, according to an embodiment of the invention. Referring now to  FIG. 10 , a computer system  101  for implementing the present invention can comprise, inter alia, a central processing unit (CPU)  102 , a memory  103  and an input/output (I/O) interface  104 . The computer system  101  is generally coupled through the I/O interface  104  to a display  105  and various input devices  106  such as a mouse and a keyboard. The support circuits can include circuits such as cache, power supplies, clock circuits, and a communication bus. The memory  103  can include random access memory (RAM), read only memory (ROM), disk drive, tape drive, etc., or a combinations thereof. The present invention can be implemented as a routine  107  that is stored in memory  103  and executed by the CPU  102  to process the signal from the signal source  108 . As such, the computer system  101  is a general purpose computer system that becomes a specific purpose computer system when executing the routine  107  of the present invention. 
     The computer system  101  also includes an operating system and micro instruction code. The various processes and functions described herein can either be part of the micro instruction code or part of the application program (or combination thereof) which is executed via the operating system. In addition, various other peripheral devices can be connected to the computer platform such as an additional data storage device and a printing device. 
     It is to be further understood that, because some of the constituent system components and method steps depicted in the accompanying figures can be implemented in software, the actual connections between the systems components (or the process steps) may differ depending upon the manner in which the present invention is programmed. Given the teachings of the present invention provided herein, one of ordinary skill in the related art will be able to contemplate these and similar implementations or configurations of the present invention. 
     While the present invention has been described in detail with reference to a preferred embodiment, those skilled in the art will appreciate that various modifications and substitutions can be made thereto without departing from the spirit and scope of the invention as set forth in the appended claims.