EUROMECH Colloquium 511 on Biomechanics of Human Motion J. Ambrosio et.al. (eds.) Ponta Delgada, Azores, Portugal, 9–12 March 2011 RELEVANCE OF ACCURATE AND VERIFIED NUMERICAL ALGORITHMS FOR VERIFICATION AND VALIDATION IN BIOMECHANICS Ekaterina Auer 1 , Andrey Chuev 1 , Roger Cuypers 1 , Stefan Kiel 1 , and Wolfram Luther 1 1 Department of Computer and Cognitive Sciences (INKO) University of Duisburg-Essen Duisburg, Germany e-mail: {auer,chuev,cuypers,kiel,luther}@inf.uni-due.de Keywords: Biomechanics, Gait, Verified methods, Distance algorithms. Abstract. In this paper, we give an overview on how accurate and verified methods can be employed for several biomechanical models. First, we consider the problem of human gait stabilization with uncertain parameters which plays an important role in many clinical appli- cations and in forward dynamical gait simulation. Next, we turn to a broader field of patient- specific preoperative surgical planning, with a focus on the superquadric (SQ) geometrical model. We show how to use it to efficiently and reliably implement important parts of the pro- cessing pipeline and describe our current endeavors to numerically verify SQ-based operations. Here, we propose two approaches: the first one is based on the Gilbert-Johnson-Keerthi algo- rithm. The calculated distance is verified a posteriori with help of the interval arithmetic. The second approach relies on hierarchical decomposition. The SQ model is decomposed into boxes using interval octrees to obtain its verified inner and outer enclosures. In this case, the problem is reduced to computing the distance between a box and a point, which is much simpler. 1
Transcript
EUROMECH Colloquium 511 on Biomechanics of Human MotionJ. Ambrosio et.al. (eds.)
Ponta Delgada, Azores, Portugal, 9–12 March 2011
RELEVANCE OF ACCURATE AND VERIFIED NUMERICALALGORITHMS FOR VERIFICATION AND VALIDATION IN
BIOMECHANICS
Ekaterina Auer1, Andrey Chuev1, Roger Cuypers1, Stefan Kiel1, and Wolfram Luther1
1Department of Computer and Cognitive Sciences (INKO)University of Duisburg-Essen
Abstract. In this paper, we give an overview on how accurate and verified methods can beemployed for several biomechanical models. First, we consider the problem of human gaitstabilization with uncertain parameters which plays an important role in many clinical appli-cations and in forward dynamical gait simulation. Next, we turn to a broader field of patient-specific preoperative surgical planning, with a focus on the superquadric (SQ) geometricalmodel. We show how to use it to efficiently and reliably implement important parts of the pro-cessing pipeline and describe our current endeavors to numerically verify SQ-based operations.Here, we propose two approaches: the first one is based on the Gilbert-Johnson-Keerthi algo-rithm. The calculated distance is verified a posteriori with help of the interval arithmetic. Thesecond approach relies on hierarchical decomposition. The SQ model is decomposed into boxesusing interval octrees to obtain its verified inner and outer enclosures. In this case, the problemis reduced to computing the distance between a box and a point, which is much simpler.
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
1 INTRODUCTION
Verified methods provide a mathematical proof for the correctness of results of computersimulations, which is especially important in the biomechanical context. Interval [34], Taylormodel [32] or affine arithmetic [10] based methods are most prominent examples of verifiedtechniques. Besides proving the correctness of the computed result, verified methods can takecare of rounding errors and propagate bounded uncertainties through systems. If such verifica-tion is not possible at some point of a process, it is nonetheless crucial to use reliable algorithmswith accurate error estimations to obtain feasible results.
In this paper, we give an overview on how accurate and verified methods can be employedfor several biomechanical models that appeared in the course of the recent project PROREOP1
[37], [3], [9]. In this project, a consortium of engineers, biomechanicians, computer and medicalscientists has developed and evaluated new tools in training, planning and assessment of totalhip replacement. A new follow-up project [33] aims at developing a highly interactive prosthesisplanning tool that would allow surgeons to assess 3D imaging data and to use geometrical,mechanical, kinematical and material/surface-specific bone features of the patient as primarysources for their decisions. This would help them to choose the best-fitting prosthesis and tofind its position and orientation in a body based on patient-specific geometric and physiognomicdata.
In this setting, there are several subtasks that need to be solved, of which we consider theproblems of human stance stabilization and bone prosthesis fitting in some detail. The firstsubproblem plays an important role in forward dynamical gait simulation, which again is ofimportance for a number of clinical evaluations. Several parameters of the task, such as posi-tion or mass of the pelvis, are influenced by uncertainty of the so-called epistemic (reducible)type. This type comprises the incertitude due to lack of knowledge, an example of which is theabsence of evidence about the probability distribution of a parameter. In our case, such param-eters are the lengths and masses of bones as well as their positions, which cannot be measuredexactly. We analyze the model using interval methods, which includes propagating the uncer-tainty through the system and computing the sensitivities of the equations of motion in the firsttime interval.
Another important part of PROREOP is modeling and analysis of a bone-prosthesis fittinginto the medullary space of the already routed femoral shaft (cf. Figure 1). The implant of endo-prostheses (EP) and artificial joints is taking a more and more important role in modern humanmedicine due to the steady increase in life expectancy of the population in most developedcountries. At present, EPs (e.g. femur prostheses) are classified according to a limited set ofgeometrical, mechanical and material-specific parameters (so-called features). However, thesecharacteristics are not very reliable if a real patient is concerned because bones are usually de-scribed via scaled standard templates. The parameter values in question are often derived eitherfrom a computer supported 2D surgery planning based on manually reconstructed contour linesand landmarks extracted from X-Ray imagery [20] or from 3D reconstructed pelvis and femurmodels [27],[1] (cf. Figure 2). This conceptual gap between two different descriptions obstructsthe appropriate selection of patient-specific EPs. In this paper, we describe an accurate and ef-ficient geometric fitting and reconstruction approach using the same composite superquadric(SQ) and voxel or surface models for bones, prostheses and receiving canals. Filling and con-tact constraints for cementless and cemented components are handled using tolerance modeling
1Development of a new prognosis system to optimize patient-specific preoperative surgical planning for thehuman skeletal system (2007-2008)
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
Figure 1: Truncated medullary space and hip joint prosthesis [8].
and epsilon-geometry.One of the important subtasks of the fitting process is minimizing the Euclidian distance
between the SQ based prosthesis model and the femoral shaft [8]. As the shaft is modeled witha set of interconnected triangles, it is essential to calculate verified distance between a convexSQ and a triangle. Due to convexity of SQ models, we can use a version of the Gilbert-Johnson-Keerthi (GJK) algorithm, the result of which is verified a posteriori with the help of intervalarithmetic. Note that our interval version of GJK algorithm from [6] cannot be applied to SQsdirectly. Another approach we describe in this paper relies on hierarchical decomposition. TheSQ model is decomposed into boxes using interval octrees to obtain its verified inner and outerenclosures. In this case, the problem can be reduced to computing the distance between a boxand a point, which is much simpler.
This paper is structured as follows. In Section 2, we give a short overview of theory, methodsand techniques needed to tackle the tasks mentioned above. Then we focus on handling ofuncertainty in the problem of stance stabilization with interval methods in Section 3. In thenext section, we describe the process of prosthesis insertion and computing the final pose usingreliable and accurate methods. Finally, we concentrate on our current endeavors to numericallyverify the implementation of all SQ-based operations beginning with the especially importantsubtask of distance computation. We present the GJK based technique and the one employingthe hierarchical object decomposition. We conclude by recapitulating the main results and byproviding a perspective on our future work.
2 METHODOLOGIES AND TECHNIQUES
In this section, we touch upon the theoretical background for our research. First we givea brief overview of verified arithmetics and libraries, then concentrate on superquadrics andtechniques for distance computation.
2.1 Verified methods and libraries
We chose to use interval methods [34] to model and simulate stabilization of human stanceand to verify distance computations. An interval [x,x], where x is the lower, x the upper bound,is defined as
x = [x,x] = {x ∈ R|x ≤ x ≤ x}.
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Figure 2: Modeling part of a new prognosis system (left) and measurement of bone features (right) [37],[8].
Elementary operations and functions can be defined on intervals in such a way as to resultin intervals. To be able to work with this definition on a computer using a finite precisionarithmetic, the concept of a machine interval is necessary. Machine intervals are represented byfloating point numbers for the lower and upper bounds. To obtain the corresponding machineinterval for the real interval [x,x], the lower bound is rounded down to the largest representablemachine number equal or less than x, and the upper bound is rounded up to the smallest machinenumber equal or greater than x. These notions can be extended to define interval vectors andmatrices.
There exist interval analogs to higher-level numerical algorithms such as those for solvinglinear, nonlinear or differential systems of equations. The usual algorithms are reformulatedin such a way as to guarantee the correctness of the computed outcome. That means that theenclosure they produce is proved to contain the exact result, for example, via a fix point theorem.Almost all algorithms need at least one derivative of the right side of system model equations tobe able to work. That is, it is necessary to obtain derivatives of code automatically [21]. Thereare several libraries [39] implementing this concept which employ either overloading or codetransformation.
There also exist further verified arithmetics, for example, affine [10] or Taylor model [32]based ones. They try to overcome one methodical difficulty always present in interval compu-tations, namely, the dependency problem. According to the principles of interval computations,the two variables x, for example, in the expression x − x, are not considered to be the same(and therefore dependent) but rather treated independently. That is, we actually work with theexpression x − y. This is a source of considerable overestimation (too pessimistic bounds forthe result, as in x− x 6= [0, 0]) in interval arithmetic.
There are a number of software libraries implementing this theory in different programminglanguages such as C++ or FORTRAN and computer algebra packages such as MAPLE or MAT-LAB. We use PROFIL/BIAS [26], FILIB++ [28] and C-XCS [22] for basic interval operationsas well as FADBAD++ [39] for algorithmic differentiation in this paper.
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2.2 Superquadrics and distance computation
A superquadric F is an implicit surface in 3D space that consists of all the points (x, y, z)with F (x, y, z) = 1. It generalizes quadrics like ellipsoids with parameters a1, a2 and a3
defining the scale of the shape in x, y and z direction and exponents ε1 and ε2 specifying itsroundness:
F (x, y, z) =
((x
a1
) 2ε2
+
(y
a2
) 2ε2
) ε2ε1
+
(z
a3
) 2ε1
= 1 . (1)
A SQ described with Eq. 1 is convex when 0 ≤ ε1, ε2 ≤ 2 holds.To describe more complex shapes with the SQ model, we can use bent SQs as defined in [23],
page 45. Bending is described there by the curvature parameter k and the angle α specifyingthe direction of the bending plane. Using the rule given by Eq. (2), the original coordinates(X, Y, Z) are transformed into bent coordinates (x, y, z). They can be evaluated using theimplicit formula for SQs (1).
x = X − (R− r) cosαy = Y − (R− r) sinαz = 1
kγ
, (2)
with
γ = arctanZ
1k−R
, r =1
k−
√Z2 +
(1
k+R
)2
, R =√X2 + Y 2 cos
(α− arctan
Y
X
).
There are quite a lot of distance calculation methods for convex polyhedra [18], [7], [40],[19]. Two of them are known to be general enough to be extended to arbitrary convex objects:the GJK [19] and the interior point optimization algorithm (IP) [7]. A general method fordistance calculation using IP is proposed in [7]. We applied this method to find the distancebetween a point and a convex SQ using IPOPT optimization library [41]. The produced resultsare comparable to [7]. The main drawback of the method is its slowness. The authors reportGJK to be up to 80 times faster than IP on equal tasks. Similar benchmarks were observed inour experiments (cf. Section 5, Table 6).
However, GJK has not been previously applied to general convex SQs because it is difficultto calculate corresponding support mapping functions in this case. A support mapping for theobject A is defined as follows [40]:
Theorem 1. If for some convex objectA the normal at the surface point ~p has the same directionas ~v than sA(~v) = ~p is a support mapping function for A.
Proof. The point ~p lies on the surface of A hence ~p ∈ A. Let ~n be the normal at ~p. It followsthat ~n · (~x− ~p) = 0 is tangent plane at ~p. Therefore it holds ~n · (~x− ~p) ≤ 0 for all points ~x thatbelong to A. As ~v and ~n have same direction it follows ~n = s~v with s ≥ 0. We conclude thats~v · (~x− ~p) ≤ 0 which implies ~v · ~x ≤ ~v · ~p and hence ~v · ~p = max {~v · ~x : ~x ∈ A}.
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Table 1: Some uncertain parameters in stance stabilization.
Force related parametersω [0.5, 6.28]s−1 frequencyFx [0, 200] N force along the x axisFy [0, 50] N force along the y axis
Mass related parametersmp [35, 65] kg pelvis masspx [0.05, 0.1] m x-position of pelvispy [0.1, 0.5] m y-positionpz [-0.05, 0.05] m z-position
Contact related parametersrff [0.04, 0.2] m radius of forefootrhf [0.02, 0.15] m radius of hindfooteN [0.01, 0.2] normal restitutioneT [0.01, 0.2] tangential restitutionµst [0.5, 2.0] static friction coef.µd [0.08, 2.3] dynamic friction coef.
Methods available for computing support mappings rely on finding roots of polynomialswith arbitrary fractional exponents which is numerically inefficient [7]. Therefore, we calculatethe support mapping function for convex superquadrics analytically by means of trigonometricequations (cf. Section 5).
In this context, the works [36], [30] should be mentioned. The authors of the first paperexamine the task of contact detection between convex SQ surfaces by formulating it a nonlin-ear constrained optimization problem. The straight line between two points belonging to thesurfaces with minimum distance is normal to both surfaces and is described by using cartesianor spherical coordinates. The optimization problem is solved by using the already mentionedIP method and a sequential quadratic programming approach. In the second article, the authorspropose a methodology to compute the minimum distance between SQ surfaces together with acouple of points on the surfaces with common normal vectors by solving a non-linear equationsystem with the Newton-Rapson method. The task of contact detection is again in the focus ofresearch here. However, numerical efficiency and result verification issues are not addressed inthese works.
3 VERIFIED UNCERTAINTY ANALYSIS OF A MODEL FOR HUMAN STANCESTABILIZATION
In this section, we describe how uncertainties in several measured parameters influence themodel for the human stance stabilization. The process can be divided into three stages [29],[2].First, human skeleton has to modeled. The model consists of nine segments: the pelvis repre-senting the whole upper body, then right and left femur, right and left tibia as well as right andleft foot composed of a forefoot and hindfoot each. These segments are connected by appropri-ate joints. The second, most important stage is the modeling of the foot contact. It is achievedby choosing two cylinders as contact surfaces for the foot and using a Hunt-Crossley contactscheme. Finally, a PID controller is applied to stabilize the stance.
At each stage of the process, there are characteristics known with some large or small incer-titude (cf. Table 1). For example, the pelvis mass of [35,65] kg or its position on the x axis([0.05,0.1] m) constitute the group of mass parameters. and belong roughly to the first stage ofthe human stance modeling. Such parameters as the radii of forefoot or hindfoot or static anddynamic friction coefficients influence mainly the second stage. Forces along the x and y axescan be counted to the last stage. This problem has 26 degrees of freedom.
We implemented the model described above in SMARTMOBILE [4]. SMARTMOBILE(based on MOBILE [24]) is one of the first integrated environments providing result verifica-
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Table 2: The abridged wrench vector [w1 w2 w4 w6] for different sets of uncertain parameters (directed roundingto the second digit after the decimal point).
all parameters from Tab. 1 uncertain mp, px and Fx uncertain nominalw1 [0,200] N·m [0,200]N·m [99.99,100.00]N·mw2 [-940.00,-595.69]N·m [-915.00,-620.69]N·m [-767.85,-767.84]N·mw4 [-31.89,31.89]N [0,0]N [0,0]Nw6 [-50.17,45.49]N [-50.17,45.49]N [1.33,1.34]N
tion for kinematic and dynamic simulations of mechanical systems. Models in both tools areexecutable C++ programs built of the supplied classes for transmission elements and solvers.The advantage of SMARTMOBILE is its flexibility due to the template structure: the user canchoose the kind of (non)verified arithmetics according to his task. Advanced users are notlimited to the already defined classes for these arithmetics and are free to plug in their ownimplementations.
The transmission elements necessary to model the stance along with their parameters areimported into a C++ executable model from an XML file using the XERCES-C++ XML parser.To be able to read interval-related data from the XML description directly, we extended theoriginal XML tags with additional attributes for defining the deviation. We need the classpwFunc [2] to represent piecewise smooth functions such as |x| or sign(x).
The parameters of interest are the pelvis mass mp, the position of the pelvis center of masson the x axis px and the applied force along the x axis Fx. We consider the first, second, fourthand sixth coordinates w1, w2, w4 and w6 of the wrench vector from the equations of motion(moments about the x and y axes, forces along the x and z axes, respectively). In Table 2,we show interval evaluations for these characteristics under influence of two sets of uncertainparameters and for nominal parameters. The term nominal parameters means that midpoints ofrespective parameter ranges, represented as point intervals, were considered in computations.The sensitivity (that is, the first partial derivative of the characteristic wrt. a parameter) of w1,w2, w4 and w6 to mp, px and Fx under uncertainty in mp, px and Fx is shown in Table 3. Asa comparison, we computed the sensitivities for nominal parameters along with the resultingreference uncertainty (cf. Table 4). We define the reference uncertainty as r =
∑i
|∂x/∂pi| ·pi,
where x is the characteristic of interest, pi the parameters and pi intervals comprising theiruncertainty.
The tables show that the force-induced part of equations of motion depends most substan-tially on the position and the mass of the pelvis. This holds especially for w6 (force in z direc-tion), which is most sensitive to px.
The results show that contact mechanics can lead to large variations of dependent data whenparticular parameters are varied. This behavior might be damped by numerical simulation,leading to slightly smaller variations in the integrated dynamical behavior. However, the majorchallenge while simulating dynamics of the stance stabilization in a verified way is the foot
Table 3: Interval sensitivity (directed rounding to the second digit after the decimal point).
contact stage. The main reason is that the equations of motion for it change their right side aswell as in some cases their left side in dependence on the zeros of a certain switching function.For this reason, one needs to handle a hybrid system in which one switches between differentmodes of operation. Verified treatment of such situations is infrequent, but have some advan-tages, for example, for contact area modeling. In general, the contact between a cylinder and aplain is not a point but a small area for small angles between the corresponding normals. Theusual method is to project the center of this area into a point. Verified methods could offer apossibility to work with the original contact area as an interval.
4 INSERTION PROCESS AND FINAL POSE COMPUTATION
Worldwide, there exist many tools for total hip replacement (THR) planning which are uti-lized in real life surgeries. These tools employ 3D surface models to assist surgeons duringoperations or to allow them to perform operations virtually [11]. However, the traditional meth-ods have several drawbacks. For example, surgeons often have to use 2D imagery for 3Dreconstruction or employ standardized bone and muscle models scaled only roughly for the in-dividual patient. That is why we propose methods using patient-specific MRI-, CT- and X-Raydata of the human pelvis and lower limbs in combination with analytically described modelswith robust geometrical parameters. The whole process is outlined in Figure 3.
Typical parameters for THR are leg length, femoral offset and anteversion (angle between theaxes of the femoral neck and shaft), so-called extraosseous aspects of the reconstruction [20],prosthetic acetabular center, femur length, fit-and-fill limitations as well as contact constraints
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
(intraosseous considerations concerning, e.g., the femoral canal). The center of the prostheticfemoral head should coincide with the center of the acetabular cup. Rotational components aredescribed within a standardized coordinate system and accessed via gait analysis. The situationis made more complicated by the fact that parameters collected initially for primary THR differfrom those for a revision surgery where a failing orthopaedic implant is replaced with a newone.
The femoral bone is decomposed in a way that the individual components can be accuratelyapproximated by translated and rotated SQ shapes. The inserted stem of the hip joint prosthe-sis has to fit the medullary space. Thus, the femur cavity volume must be modeled. This canbe done by constructing a triangulated surface fitting the point cloud obtained by intelligentsegmentation algorithms. To construct the mesh surface, a tetrahedral mesh generation algo-rithm of the average complexity O(n log n) together with alpha shapes is used. For a non-zeroalpha value, only edges, faces, or tetrahedra contained within the sphere (of radius alpha) areproduced. Then, a correction of the triangle normal vectors is realized in such a way that allnormals point inward the surface. Finally, the mesh is smoothed by averaging the normals andby filling gaps (cf. Figure 4).
The general aim of a THR surgery is to align the implant so that the fitness with the femoralstem is maximized. Among the criteria that define the fitness value are:
• Minimization of the distance between the mechanically relevant areas of the implant andthe medullary space, for example, smaller than 2 mm in the cementless completion; agiven distance should be respected in the cemented completion (cf. Figure 5).
• Maximization of implant component size without exceeding the space given by the un-covered medullary space opening.
• Preservation of the individual bone features leg length, femoral offset and anteversion.
To find an optimal fit, it is necessary to solve a minimization problem for the distance be-tween the implant and the femoral stem. Depending on the geometric model used for the implantand the femoral bone, this involves the repeated calculation of the distance between multi com-ponent SQ models or a multi component SQ model and the meshed surface. If the task consistsin inserting an SQ-based implant into a mesh-based medullary space, the latter provides a pointset C ⊆ R3 with fixed element number n, that means a discretization of the surface. Then theobjective function of the fit can be formulated as follows:
maximize : F (T ) =n∑i=1
f(dist(S, T, ci)) , ci ∈ C
constraints : Il ≤ I ≤ Ir
. (4)
I is a surface-type characterization vector of parameters which can be bounded by intervals[Il, Ir]. S represents the multicomponent superquadric surface, T the transform matrix for thecurrent implant pose parameters and ci ∈ C surface points of the bone. The distance functiondist is signed. It becomes negative, if ci lies inside the implant. The function f is a fitnessquality function which returns comparatively large values for implant poses conforming withthe previously mentioned quality criteria. An example of such a function can be found in [35].
Since the implant has to be navigated through the opened spongiosa and medullary space,effectively more bone substance has to be removed than is necessary for a tight fit of the implant,especially in the areas of neck and most proximal shaft. Therefore, our most important goal is
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
Figure 4: a) Point cloud b) Delaunay triangulation with backspace culling c) Corrected direction of normal vectorsd) Mesh smoothing [8].
to achieve a close proximity in areas which are critical for a stable fit, like the distal part ofthe shaft. This requires the decomposition of the implant into different areas with respectivepriorities. The mean difference between the positioned implant and the bone was measured inseveral tests to be between 1 mm (critical areas) and 5 mm (less critical areas). An extensiveclinical evaluation of the proposed methods and a comparison to manual methods has yet to bedone.
5 VERIFYING THE INSERTION PROCESS
In Fig. 5, the distance between a SQ modeled EP and a meshed cavity is shown. It is neces-sary to compute this distance as reliably as possible because the red sector in this figure needsa tight fitting with the small orange colored clearing. In this section, we propose interval basedmethods allowing us to compute distance in a verified way. Generally, this process should beaccompanied by an additional model validation. Note that verification means a mathematicalproof of the correctness of the used methods whereas validation addresses model fidelity bydefining a metric and comparing the outcomes of the simulation and experiments. Our workin progress is to evaluate CT and MRI data of 25 patients, which will allow us to compare thenew modeling and simulation approach with traditional measurement and surgical preparationtechniques. First results are published in [8].
5.1 Verified distance between a convex SQ and a primitive
First, we derive a method to verify the distance between a point and a convex SQ. The sameSQ as in Eq. (1) can be parametrically described as a set of points ~p with
~p =
a1cosε1η cosε2ωa2cosε1η sinε2ω
a3sinε1η
,−π/2 ≤ η ≤ π/2−π ≤ ω ≤ π
. (5)
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
Figure 5: Distance between SQ modeled EP and meshed cavity.
As the exponential function xy is not defined for x < 0 we use here and everywhere else in thispaper the extended exponential function
xy = sign(x)|x|y . (6)
To calculate ~p from Theorem 1 on the surface of a SQ with the normal at ~p having the same di-rection as the given vector ~v, we propose the following. The set of SQ normals can be describedparametrically [5], [23] by
~n =
1a1
cos2−ε1η cos2−ε2ω1a2
cos2−ε1η sin2−ε2ω1a3
sin2−ε1η
,−π/2 ≤ η ≤ π/2−π ≤ ω ≤ π
(7)
If we set~n = s~v (8)
for the given ~v, we obtain a system of three nonlinear equations that can be solved for η, ω ands ∈ R. We divide the second component of Eq. (8) by the first one and obtain
sin2−ε2 ω
cos2−ε2 ω= tan2−ε2 ω =
a2v2
a1v1
. (9)
Then we raise the first and the second component of Eq. (8) to a power of 2/(2 − ε2) and addtheir corresponding left and right sides
cos2
2−ε12−ε2 η(cos2 ω + sin2 ω) = (sa1v1)
22−ε2 + (sa2v2)
22−ε2 . (10)
By using the main trigonometric identity cos2 ω + sin2 ω = 1 we get
cos2−ε12−ε2 η =
√(sa1v1)
22−ε2 + (sa2v2)
22−ε2 . (11)
We raise the left and right side of Eq. (11) to a power of (2− ε2) and factor out s
cos2−ε1 η = s
(√(a1v1)
22−ε2 + (a2v2)
22−ε2
)2−ε2(12)
so that we can divide the third component of Eq. (8) by Eq. (12) obtaining
sin2−ε1 η
cos2−ε1 η=
a3v3(√(a1v1)
22−ε2 + (a2v2)
22−ε2
)2−ε2 . (13)
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From Eqs. (9) and (13), we can derive the final expressions for η and ω:
tanω =
(v2a2
v1a1
) 12−ε2
, (14)
tan η =
v3a3(√(v1a1)
22−ε2 + (v2a2)
22−ε2
)2−ε2
1
2−ε1
(15)
with ~v = (v1v2v3)T . We solve Eqs. (14) and (15) for η and ω taking into account that ω and
η must lie in the same quadrants as the points (v1, v2) and (v3,√v2
1 + v22), respectively. This
can be solved by using the two argument inverse tangent function atan2 that is available inthe most programming languages. The resulting ω and η are substituted in Eq. (5) to calculatethe value of the support mapping function for a convex SQ according to Theorem 1. Note thats = n3/v3.
We use interval arithmetic [34] to calculate the guaranteed distance enclosure. Let ~q be thegiven point and ~p an approximation of the nearest SQ point to ~q calculated with GJK. The point~p may be inaccurate due to numerical errors. To accurately calculate ~p we first find the smallestpossible interval inclusion of ~p that is guaranteed to contain at least one point of the SQ. Inorder to obtain this enclosure, we take the line ~l(t) = ~p + (~q − ~p)t and calculate the quantitiest1 ≤ 0 and t2 ≥ 0 with the smallest absolute value |t| so that F (~l(t1)) ≤ 1 and F (~l(t2)) ≥ 1(see Eq. (1)). This can be achieved using the bisection method. The desired interval inclusionis therefore ~P = [~l(t1),~l(t2)]. As the point ~q is supposed to be accurate its interval inclusion issimply ~Q = [~q, ~q].
At the next step we obtain the normal ~N at the point ~P as ~N = ∇F (~P ) using the implicitSQ equation (1). After that we calculate the distance between ~Q and ~P as
U = ‖~P − ~Q‖2 (16)
and the distance between ~Q and the interval plane ~N · ( ~X − ~P ) = 0 as
L =( ~Q− ~P ) · ~N‖ ~N‖2
. (17)
The plane ∇F (~y) · (~x − ~y) = 0 is a separating plane for every ~y that lies at the surface of aconvex object. This implies that the distance between ~q and the plane ∇F (~p) · (~x − ~p) = 0
is the lower bound for the distance between ~q and the SQ. We know that ~p ∈ ~P . From theinclusion properties of interval extensions we conclude that ∇F (~p) ∈ ∇F (~P ) and thereforethe true distance between ~q and the separating plane at ~p is included in L (see Eq. (17)). Theaccurate distance D between the point ~q and the SQ is the interval union
D = [inf L, supU ] . (18)
We can construct a method for verified distance calculation between a SQ and a triangleusing the approach described above. We assume that the triangle is defined by its three vertices.A triangle normal can be calculated from these vertices and used to find the closest point onthe SQ to the plane containing the triangle (see Eqs.(14), (15) and (5)). The point on the SQ is
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
projected onto the triangle plane along its normal to find the point closest to the SQ. If the pointis contained inside the triangle we have the solution. If that is not the case then the point onthe plane is projected again onto the triangle feature (edge or vertex) closest to it. One can usegeneralized Voronoi regions to solve this task [18].
If we want to obtain a verified distance we have to perform the above described steps usinginterval arithmetic. However, if the closest point lies on one of the triangle edges then we haveno exact solution but rather an axis aligned bounding box (AABB) that contains the closestpoint. In order to calculate verified distance between the AABB and the SQ we enclose theAABB in a sphere. We calculate verified distance between the sphere center and the SQ andinflate both interval bounds by the sphere’s radius. If the closest point on the plane lies onseveral triangle features the distance is calculated separately for each feature and the union ofresulting intervals is taken as a result.
The method to calculate verified distance between a point and a convex SQ can be extendedto handle a pair of convex SQs. Let A and B be convex SQs with the corresponding implicitdescriptions FA(~x) ≤ 1 and FB(~y) ≤ 1. Let ~p ∈ A and ~q ∈ B be the unverified closest points.First we calculate cubes ~P and ~Q containing these unverified solutions ~p ∈ ~P , ~q ∈ ~Q with~P ∩A 6= ∅ and ~Q∩B 6= ∅. Then the upper bound of the distance between A and B is obtainedas
sup ‖~P − ~Q‖2 (19)
by interval evaluation. We compute then the interval normal vector to the separating plane at~P as ~N = ∇FA(~P ). Using the interval extensions of Eqs. (14), (15) and (5) it is possible tocalculate the verified inclusion ~R of the point ~r ∈ B with ∇FB(~r) = −s∇FA(~p) for somepositive s. The affine transformation theorem for support mapping functions can be used tohandle translations and rotations of A and B [40]. Thus we obtain verified enclosures for theparallel supporting planes ∇FA(~p) · (~x − ~p) = 0 and ∇FA(~p) · (~y − ~r) = 0 with the distancebetween them d ≤ ‖~p− ~r‖2. The lower bound of the distance between A and B is obtained as
inf~N · (~R− ~P )
‖ ~N‖2. (20)
We implemented and tested the distance calculation method based on the GJK algorithmusing the SOLID library [40] to obtain an approximation. The C++ FILIB++ [28] was used forinterval computations. As test objects, we chose three SQs with the following parameters:
Table 5: Description of the superquadrics used in our test cases.
and 105 random points for each SQ. SOLID was running every time with the accuracy parameterof 10−6. The average computation times and the absolute (width of the interval) and relativeerrors (interval diameter to midpoint relations) are summarized in Table 6. They show that thenumerical modeling error can be neglected in comparison to the input data uncertainty and thefitting error.
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
Object time (s) verified, time (s) absolute error relative errorGJK, point to SQ
Table 6: Sample run times, in seconds, for both unverified and verified distance computation with the averagemaximal relative errors. All data were obtained on a 1.6 GHz Pentium Core-2-Duo with 1 GB of RAM.
5.2 Verified closest points algorithm
The methods described in the previous sections provide a verified enclosure only for thedistance d between two strictly convex SQs A and B. To obtain the unique closest points ~p ∈ Aand ~q ∈ B on the corresponding SQs with D = ‖~p − ~q‖2, we have to determine a verifiedenclosure of the nonlinear equation system below. It describes formal relations between ~q and~p.
Let A be a convex SQ with the parameters a1, a2, a3, ε1 and ε2 that describe its shapeaccording to Eqs. (1) and (5). Similarly letB be a convex SQ with the corresponding parametersb1, b2, b3, δ1 and δ2. The SQ B can be parametrically described as a set of points ~q with
where the vector ~u and angles τ0 and θ0 describe the translation and rotation of B relative to A.According to [23], the normal ~n to A at the point ~p can be expressed in terms of ~p as
~n =
cos2η cos2ω
p0
cos2η sin2ω
p1
sin2η
p2
,−π/2 ≤ η ≤ π/2−π ≤ ω ≤ π
. (22)
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
The normal ~m to B at the point (~q − ~u) is similarly described as
~m =
cos2(τ − τ0) cos2(θ − θ0)
q0 − u0
cos2(τ − τ0) sin2(θ − θ0)
q1 − u1
sin2(τ − τ0)q2 − u2
,−π/2 ≤ τ ≤ π/2−π ≤ θ ≤ π
. (23)
In order to calculate the verified enclosure of the unique closest points ~p ∈ A and ~q ∈ B, weneed to solve the following system of equations for (η, ω, τ, θ, s, d)
ni = smi
pini + dn2i
‖~n‖2= s(qi − ui)mi + suimi
(24)
for i = 1, 2, 3. As we can approximately compute ~p and ~q we already have good startingvalues for (η, ω, τ, θ, s, d). The system of equations (24) can be solved with the interval Newtonmethod [34].
5.3 Hierarchical decomposition
Another approach for deriving the verified distance between implicit surfaces is hierarchicaldecomposition. The implicit objects of interest are decomposed into much simpler objects, forexample axis-aligned boxes. The distance is computed between these boxes, which reduces theproblem object-object to box-box.
An octree [38] is a data structure for constructing and storing a hierarchical decompositionof an object that uses labeled nodes. The construction process usually starts with the unit cube.If the cube is completely filled by the object the corresponding node is labeled black. If they aredisjoint the node’s color is set to white. Otherwise, we assign the gray color to the node. Afterthat, we split the gray node into eight equally sized disjoint subcubes and repeat the process forthem. We stop if there are no gray leaf nodes anymore or if some predefined subdivision depthis reached.
To obtain the verified enclosure of distance, we need to verify the object decomposition first.We can adapt the octree structure to an interval octree in two simple steps.
1. Represent the octree nodes with boxes
2. Use an interval extension of the implicit function describing the object to determine thenodes’ colors
The first step is straightforward, as the octree cubes are already axis-aligned boxes so that wecan represent them directly using interval vectors. If an object o is described by an implicitfunction f and F is an interval extension of f we can determine the color of a box x as follows:
COLOR(x) =
black if F (x) ≤ 0white if F (x) > 0
gray else
Interval octrees deliver a verified object enclosure because it can be proved computationallywith their help that every black box lies completely inside the object and white boxes are disjoint
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
with it. However, gray nodes constitute the area of uncertainty, where it is unknown if a partof the object lies there. Note that this approach works even if the considered objects are notconvex, as is the case for bent superquadrics described by 2.
The femur model does not consist of a single SQ but is rather modeled with several SQswhich are combined in a CSG-tree. Duff [12] proposed rules for simplifying CSG-trees overa box. Although the simplified tree is valid for this box only, it can be used efficiently inhierarchical structures because it becomes valid for a node and all its child nodes in this case.The simplified tree can be used instead of the original one for determining the color of a node.Therefore, we proposed a combined CSG-octree in [15]. This is a new hybrid octree structuredesigned for objects described by CSG trees with quadric primitives. Simplified CSG trees arestored inside its nodes and used for determining the node color. An improved version of the datastructure is presented and compared to usual octrees without simplification in [17]. Using thisnew model, it is possible to stop the subdivision process if a node contains only one primitive.The node is then labeled as terminal-gray.
A verified lower bound on the distance between CSG octrees can be derived with algorithmsfrom [14], [16]. The one from [16] uses a geometric approach for calculating the distancebetween two quadric primitives directly as soon as a pair of two terminal-gray nodes occur. Thisapproach cannot be easily extended to more complex surfaces like superquadrics. Therefore weonly consider the approach from [14] in this paper.
The general idea of this algorithm is as follows. A list L is defined to contain pairs of nodes(x, y) that might minimize the distance together with the current lower distance bound dinf . Thefirst candidate pair is taken from L in each step. The larger node (e.g. x) is split, the proximitybetween each of the two new nodes and the y is computed and, if necessary, dinf is updated.As the proximity between boxes is computed according to [6], the algorithm does not have tobe limited to quadric surfaces. However, the implementation and tests from [14] consider onlyquadric primitives.
Based on our framework UniVerMeC (Unified Framework for Verified GeoMetric Compu-tations) [3] and the ideas from [13], we propose an improved distance algorithm described indetail in [25]. This algorithm can also work on superquadrics because the framework allows usto use arbitrary implicit surfaces given by their closed-form expressions. In contrast to the orig-inal version, we are now able to calculate a verified distance enclosure between two arbitraryimplicit surfaces, that is, a lower and, additionally, an upper bound on the minimum distance.Our algorithm uses an adaptive hierarchical object decomposition constructed during its run-time, ensuring that the resulting enclosure is at least as tight as defined by the user beforehand.However, it can not handle the CSG-trees used for modeling the complete bones in its currentimplementation, which remains a topic for our future work.
This approach is slower than the GJK based one presented in Section 5.1. Its advantage lies,as mentioned before, in its ability to handle the non-convex case (e.g. bent superquadrics). InTab. 7, the results for computing the distance between the SQ given in Table 5.1 are shown forthe convex cases as defined in Section 5.1 (the first three scenarios) as well as for the non-convexcases with the bent superquadric models given below (the second three):
1 k = 0.00001 α = 0, 3 k = 0.001 α = 70 .
Time is given in seconds, the error describes the width of the derived interval enclosure andε1 = 0.1, ε2 = 0.01 are the goal interval tightnesses defined by the user.
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
Scenario Time for ε1 Err 1 Time for ε2 Err 21 - 2 85 0.06 574 0.0091 - 3 33 0.09 557 0.0012 - 3 38 0.09 344 0.0091 - 2 bent 224 0.091 - 3 bent 1206 0.172 - 3 bent 1284 0.12
Table 7: CPU times for calculating the distances using the hierarchical decomposition approach.
6 CONCLUSIONS
We presented an overview on how accurate and verified methods can be employed in abiomechanical context. In particular, we showed a first verified sensitivity analysis of the stancestabilization model. We used SMARTMOBILE for this purpose, a tool providing verified kine-matics, dynamics and sensitivity analysis options for several classes of (bio)mechanical sys-tems. The equations of motion for the stance stabilization were particularly sensitive to theposition of the pelvis and the pelvis mass. Next, we showed how the overall fitting processcould be enhanced by using SQ models and accurate algorithms. Finally, we introduced anaccurate and efficient method to calculate verified distance between a point and a superquadric.With only minor extensions, it can be used to calculate the distance between either a triangleand a SQ or a pair of convex SQs. On the other hand, we can use a hierarchical decompositionof an SQ modeled object and reduce computations to calculating the distance between a boxand a point.
Our future work will include the development of a verified solver for systems with nons-mooth equations of motion and modeling of the contact area between a cylinder and a plainwith the help of intervals. Further, we would like to investigate spatial and temporal coherenceof the GJK method in order to speed up the calculations. There could be a significant perfor-mance increase as the parameters change slowly during a fitting process. Another interestingquestion is handling of deformed SQs. For example, the cylindrical SQs in [9] can be slightlybent. This leads to the loss of convexity in general. However, such SQs can usually be subdi-vided in convex and nonconvex parts using a single longitudinal cut. The convex part can thenbe processed as presented in Section 5. The nonconvex part can be approximated with convexpolytopes. After that we are free to use general verified distance algorithms between a pointand polyhedron or a pair of convex polyhedra, for example, those described in [31].
On the other hand, we plan to add handling for the CSG-models using the CSG octrees ap-proach as described in Section 5.3. It also seems promising to combine both approaches forderiving the distance in the following way. Assuming we have the two CSG models with bentSQ as primitives describing bone and medullary space as the input, we can use the new distancealgorithm from [25] in combination with the CSG octrees for constructing an adaptive subdivi-sion. As soon as the subdivision process encounters terminal gray nodes where the superquadricis strictly convex, we can switch to the much faster method described in Section 5.1.
If the global and local properties of geometric models described in this paper are used, itis possible to determine functional parameters and valid points of bones, mark canals and EPsautomatically and accurately by using surface data that is simultaneously segmented and markedup with respect to object components. These data should be completed by mechanical andmaterial-specific parameters in order to construct distance measures between bones and EPs.In the end, the personalized model data and methods for its analysis may serve for diagnostic
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Ekaterina Auer, Andrey Chuev, Roger Cuypers, Stefan Kiel and Wolfram Luther
use and are to be incorporated into a biomechanical simulation and visualization environment,so that they can be easily applied in the context of 3D surgical planning or specific trainingscenarios.
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