System and method for variational ball skinning for geometric modeling of ordered balls

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 Si being parameterized by positions of the ith and i+1th balls and contact angles αi, αi+1 from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment Si, each the αi affecting spline segment Si and Si−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.

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 inFIGS. 1(a)-(b). Given an ordered sequence of balls, shown inFIG. 1(a), one can produce in envelope that optimally interpolates the balls, as shown inFIG. 1(b). This envelope has two splines, an inner11and an outer12spline 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 inFIG. 2, while a two-sided envelope defines an interpolating region that has an inside and outside, as demonstrated inFIGS. 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 inFIG. 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 C1continuous 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, C1envelope 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 C1spline, 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'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 Sithat form an envelope surrounding the plurality of 2D balls, each the spline segment Sibeing parameterized by positions of the ithand i+1thballs and contact angles αi, αi+1from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment Si, each the αiaffecting spline segment Siand Si+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)Eα+kEc, where k is a constant, Eαis an arc-length term and Ecis 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 αn+1=αn−Δt∇Eαn(αn) where Δt is a time step, superscript n is a time step index for the angles, and

According to a further aspect of the invention, each spline segment is a C1cubic spline segment modeled as Si=Ait3+Bit2+Cit+Di, where tε[0,1] and where each the curve satisfies the constraints

pi=ci+[ri⁢cos⁢⁢αiri⁢sin⁢⁢αi]
where ciis the center of the ball and riis the radius, and

ti=[-ai⁢sin⁢⁢αiai⁢cos⁢⁢αi]
is the starting direction of the curve, where αiis a stiffness factor.

According to a further aspect of the invention, each αiis fixed to be half the distance between the next and previous ball centers, and where for the first and last balls, the αiis 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

Ea=∑i=1N⁢∫St′⁢ⅆt
using derivatives with respect to the angles αi, where the prime indicates a derivative with respect to t and the sum is over all the curves, and where

According to a further aspect of the invention, minimizing the curvature comprises minimizing

Ec=∑i=1N⁢∫κi2⁡(t)⁢ⅆt⁢⁢where⁢⁢κi=〈Si′,JSi′′〉〈Si′,Si′〉32⁢⁢and⁢⁢J=[01-10]
using derivatives with respect to the angles αi, where the primes indicate derivatives with respect to t and the sum is over all the curves, and where

According to a further aspect of the invention, coefficients Ai, Bi, Ci, Diare defined as
Ai=−2pi+1+2pi+ti+ti+1
Bi=3pi+1−3pi−2ti−ti+1,
Ci=ti
Di=pi
and where

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 αifor 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 Sithat interpolate an inner boundary of the plurality of 2D balls, each the spline segment Sibeing parameterized by positions of the ithand i+1thballs and contact angles αi, αi+1from the center of each respective ball to a point on the perimeter of each the ball contacting the spline segment Si, each the αiaffecting spline segment Siand Si−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)Eα+kEc, where k is a constant, Eαis an arc-length term and Ecis 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.

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 R2to 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. 2illustrates 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 Siis tangent to ball Biat point piwhere the tangent is orthogonal to a ray extended from the centerpount ciwhere the ray makes an angle αiwith 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. 2depicts a desirable envelope as a dotted line, which is a C1curve passing through the point of contact on each ball, where at each contact point, the envelope is tangent to the ball. A C1curve is a continuous curve with continuous first derivatives. The envelope S is composed of a set of envelope segments, Si, for i=1, . . . , N, where N is the total number of segments.

Segments

FIG. 3depicts a segment of an envelope. As shown inFIG. 3, one can choose to model each segment Siusing a spline that starts at point piin direction ti, and ends at point pi+1in direction ti+1. The segment can be modeled using a cubic polynomial curve
Si=Ait3+Bit2+Cit+Di,  (1)
since the four constraints require four degrees of freedom. For the ithsegment, Ai, Bi, Ci, and Diare 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:

Si⁢❘t=0=pi,⁢ⅆSiⅆt⁢❘t=0=ti,⁢Si⁢❘t=1=pi+1,⁢ⅆSiⅆt⁢❘t=1=ti+1.
With these constraints, and the derivative of the segment,

ⅆSiⅆt=3⁢Ai⁢t2+2⁢Bi⁢t+Ci(2)
one can obtain a system of four equations for the four coefficients:
Di=pi,
Ci=ti,
Ai+Bi+Ci+Di=pi+1,
3Ai+2Bi+Ci=ti+1,
which can be solved, yielding:
Ai=−2pi+1+2pi+ti+ti+1
Bi=3pi+1−3pi−2ti−ti+1
Ci=ti
Di=pi
Endpoints

Now that there is a way to model each segment of the envelope, one can determine the endpoints pi, pi+1and their respective tangents, ti, ti+1of each segment. As shown inFIG. 2, the point of contact pion the ithball can be represented as

pi=ci+[ri⁢cos⁢⁢αiri⁢sin⁢⁢αi],(4)
where riis the radius of a ball, ciis its center, and αiis an angle. In addition, the tangent can be represented as

ti=[-ai⁢sin⁢⁢αiai⁢cos⁢⁢αi],(5)
where αiis a stiffness term that controls the influence of the tangential constraint. Each αican 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 piand the tangent tiare only a function of the angle αi, since the radius of the ball is fixed.

There is now a way to represent the envelope S as a set of segments Si, where each segment Siinterpolates between the points of contact pi, pi+1with balls Bi, Bi+1, subject to tangent conditions ti, ti+1respectively.

By construction of the problem, the angle αiaffects only the segment Sias well as the segment Si−1, as can be easily seen inFIG. 2. Finally, the envelope is fully parameterized by the balls and the spline angles αi. Since the balls are fixed, the objective will be to compute the angles αithat 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 αithat 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 αithat minimize
Eα=∫|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

Ea=∑i=1N⁢⁢∫Si′⁢ⅆt.
Next, the derivative of the energy is taken with respect to the angle αi. As stated above, the ithangle only affects the segments Si−1and Si. Therefore,

∂Ea∂αi=∂∂αi⁢(∫Si-1′⁢ⅆt)+∂∂αi⁢(∫Si′⁢ⅆt).(6)
Second Term

Considering the second term of EQ. (6) first, propagating the derivative with respect to αithrough the integral, it is easy to show that

∂Si′∂αi
terms can be derived using EQ. 2, yielding

∂Si′∂αi=3⁢t2⁢∂Ai∂αi+2⁢t⁢∂Bi∂αi+∂Ci∂αi.
The derivatives

∂Ai∂αi,∂Bi∂αi,and⁢⁢∂Ci∂αi
can be derived using EQ. 3, as

∂pi∂αi⁢⁢and⁢⁢∂ti∂αi
can be derived from EQS. 4 and 5 as

∂pi∂αi=[-ri⁢sin⁢⁢αiri⁢cos⁢⁢αi]∂ti∂αi=[-ai⁢cos⁢⁢αi-ai⁢sin⁢⁢αi]
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 αithrough the integral yields

∂∂αi⁢(∫Si-1′⁢ⅆt)=∫〈Si-1′,Si-1′〉-12⁢〈Si-1′,∂Si-1′∂αi〉⁢ⅆt
As before, an expression can be derived for the

∂Ai-1∂αi,∂Bi-1∂αi,and⁢⁢∂Ci-1∂αi
can be derived using EQ. 3, as

∂Ai-1∂αi=-2⁢∂pi∂αi+∂ti∂αi,⁢∂Bi-1∂αi=3⁢∂pi∂αi-∂ti∂αi,⁢∂Ci-1∂αi=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 αithat minimize
Ec=∫κ2(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

Ec=∑i=1N⁢∫κi2⁡(t)⁢ⅆt,
where κ(t) is the curvature at point t along segment Si. Next, the derivative of the energy is taken with respect to the angle αi. As stated above, the ithangle only affects the segments Si−1and Si. Therefore,

∂Ec∂αi=∂∂αi⁢(∫κi-12⁡(t)⁢ⅆt)+∂∂αi⁢(∫κi2⁡(t)⁢ⅆt).(9)
Recall that the curvature is given by

κi=Si′×Si″Si′3,
which can be re-expressed as

κi=〈Si′,JSi″〉〈Si′,Si′〉32,⁢whereJ=[01-10]
is a 90 degree rotation matrix and < > denotes an inner product. Using these equations, EQ. 9 becomes

∂Ec∂αi=∂∂αi⁢(∫[〈Si-1′,JSi-1″〉〈Si-1′,Si-1′〉32]2⁢ⅆt)+∂∂αi⁢(∫[〈Si′,JSi″〉〈Si′,Si′〉32]2⁢ⅆt).(10)
Second Term

To derive the second term of EQ. 10, the derivative with respect to αiis propagated through the integral to obtain

∂∂αi⁢(∫[〈Si′,JSi″〉〈Si′,Si′〉32]2⁢ⅆt)=∫2[〈Si′,JSi″〉〈Si′,Si′〉32]⁢(∂∂αi⁢〈Si′,JSi″〉〈Si′,Si′〉32-32⁢〈Si′,JSi″〉⁢∂∂αi⁢〈Si′,Si′〉〈Si′,Si′〉52)⁢ⅆt.
For this, one needs the derivatives

∂∂αi⁢〈Si′,JSi″〉⁢⁢and⁢⁢∂∂αi⁣〈Si′,Si′〉.
These derivatives can be shown to be

∂Si′∂αi=3⁢t2⁢∂Ai∂αi+2⁢t⁢∂Bi∂αi+∂Ci∂αi,⁢∂Si″∂αi=6⁢t⁢∂Ai∂αi+2⁢∂Bi∂αi.
The derivatives

∂Ai∂αi,∂Bi∂αi,and⁢⁢∂Ci∂αi
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 αithrough the integral, one obtains

∂∂αi⁢(∫[〈Si-1′,JSi-1″〉〈Si-1′,Si-1′〉32]2⁢ⅆt)=∫2[〈Si-1′,JSi-1″〉〈Si-1′,Si-1′〉32]⁢(∂∂αi⁢〈Si-1′,JSi-1″〉〈Si-1′,Si-1′〉32-32⁢〈Si-1′,JSi-1″〉⁢∂∂αi⁢〈Si-1′,Si-1′〉〈Si-1′,Si-1′〉52)⁢ⅆt.
For this, the derivatives

∂∂αi⁢〈Si-1′,JSi-1″〉⁢⁢and⁢⁢∂∂αi⁢〈Si-1′,Si-1′〉
are needed. It can be shown that these derivatives are

∂Si-1′∂αi⁢⁢and⁢⁢∂Si-1″∂αi
are

∂Si-1′∂αi=3⁢t2⁢∂Ai-1∂αi+2⁢t⁢∂Bi-1∂αi+∂Ci-1∂αi,⁢∂Si-1″∂αi=6⁢t⁢∂Ai-1∂αi+2⁢∂Bi-1∂αi.
The derivatives

∂Ai-1∂αi,∂Bi-1∂αi,and⁢⁢∂Ci-1∂αi
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 αi. However, for the first ball, i=1, there is no segment Si−1, so the first integral is skipped in EQ. 10. Likewise, for the last ball, i=N, there is no segment Si, so the second integral in the equation is skipped.

Thus, the gradient of energy functionals Eαand Echas been derived with respect to angles, αi. 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 Ai, Bi, Ci, Diand Ai−1, Bi−1, Ci−1, Di−1, which in turn are functions of the angles αi.

Energy Minimization

The energies Eαand Eccan be combined together, as
E=(1−k)Eα+kEc,
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

∂Ea∂αi
is given in EQ. 6 and

∂Ec∂αi
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=[α1, . . . , αN]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 inFIG. 9. A method according to an embodiment of the invention starts by providing at step91a 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 αi, one only needs to initialize an angle for each ball at step92. 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 inFIG. 8, connecting adjacent ball center points.FIG. 8shows ball80with center point83and rays81and82orthogonal to centerline84. However, there are two possible angles in the figure, with respect to the rays81and82. 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 step93, letting αinbe the ithangle at iteration n, one can then update the angles by moving them in the negative gradient direction, i.e.,
αn+1=αn−Δt∇Eαn(αn),  (11)
where Δt is a time step and

∇Eαn⁡(αn)=[∂E∂α1n,…⁢,∂E∂αNn]T.
The updating of step93is repeated until the energy reaches a suitable minimum, at step94. 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
αn+1=αn+Δt·ƒ(E|αn|).
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 spline41is shown inFIG. 4(a), the result42after 50 iterations is shown inFIG. 4(b), and the converged result43after 100 iterations is shown inFIG. 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 inFIG. 4(a) of the figure. The angles were iteratively updated using EQ. 11. An intermediate solution after 50 iterations in shown inFIG. 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 inFIG. 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 envelope51is shown inFIG. 5(a), an intermediate result52after 70 iterations inFIG. 5(b), and the final result53upon convergence after 140 iterations inFIG. 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 αito 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 initializations61,66,FIG. 6(b) shows intermediate results62,67after 50 iterations, andFIG. 6(c) shows the final converged results63,68after 100 iterations. Convergence for this example occurs in 190 milliseconds.

More examples are provided inFIGS. 7(a)-(c) andFIGS. 1(a)-(b).FIGS. 7(a)-(c) illustrate ball skinning for balls arranged on a sine wave and having a variable radius. The initial splines71,72are shown inFIG. 7(a), the intermediate results73,74are shown inFIG. 7(b), and the final converged results75,76are shown inFIG. 7(c). Convergence of the skinning algorithm, starting from a set of angles far from the optimal result, takes 775 milliseconds. InFIGS. 1(a)-(b), the variable radius balls are arranged in a spiral, andFIG. 1(b) shows the converged results11,12. The envelopes are generated in 2.5 seconds.

System Implementation

FIG. 10is 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 toFIG. 10, a computer system101for implementing the present invention can comprise, inter alia, a central processing unit (CPU)102, a memory103and an input/output (I/O) interface104. The computer system101is generally coupled through the I/O interface104to a display105and various input devices106such as a mouse and a keyboard. The support circuits can include circuits such as cache, power supplies, clock circuits, and a communication bus. The memory103can 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 routine107that is stored in memory103and executed by the CPU102to process the signal from the signal source108. As such, the computer system101is a general purpose computer system that becomes a specific purpose computer system when executing the routine107of 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.