Detecting Degenerate Object Configurations with the Rational Univariate Reduction

1
Detecting Degenerate Object Configurations with
the Rational Univariate Reduction

• John Keyser
• Department of Computer Science
• Texas AM University

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Robustness in Boundary Evaluation

• Like many geometric operations, boundary
  evaluation subject to robustness problems
• Exact computation approaches, used in
  computational geometry, havent been applied as
  readily
• Exact computation with curved objects generally
  requires computer algebra methods.

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Goal(big picture)

• Support boundary evaluation for curved solids
  robustly
• Numerical error
• Degeneracies

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Need for Exact Boundary Evaluation

• Early work showed numerical problems
• Boole system, floating-point based
• Curved primitives
• Tracing approach, tolerances
• Early-mid 90s
• Krishnan et al. at UNC

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Exact Boundary Evaluation

• Solids defined by parametric surfaces
• Implicit form also stored
• Intersection curves are algebraic plane curves
• Usually no parametric representation
• Kept in patch domain
• Intersections of curves are algebraic points
• Solution of systems of polynomial equations

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ESOLID

• Exact boundary evaluation program
• Exact representations of surfaces, curves, points
• Key operations curve-curve intersection, curve
  topology provided by MAPC library
• MAPC curve-curve intersections
• Most time-consuming part of computation
• With Culver,Krishnan,Foskey,Manocha

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ESOLID (continued)

• Intersecting curves
• Resultants allow isolating x,y coordinates
  separately
• Box hits determine which points are true
• Points represented as unique root of polynomial
  system within a box.

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ESOLIDSpeedups

• Numerous speedup techniques applied
• Floating-point filters/Interval arithmetic
• Lazy evaluation
• Quick rejection
• Assumes General Position
• Does not handle any actual degeneracies
• Fails by crash, infinite loop

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ESOLIDPerformance

• Applied to real-world cases
• Performance, with speedups, within one order of
  magnitude of Boole

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Problem

• Degenerate cases cause failure
• Want way to represent points even in degenerate
  cases
• Tangential intersections
• Intersections at curve singularities
• 3 or more curves meeting at a point
• Need way to treat degeneracies

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Approach

• Define new method for representing points
• Allows certain degeneracies to be detected,
  represented cleanly
• Numerical perturbation approach to treat
  degeneracies
• Work done with Koji Ouchi, Maurice Rojas

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Rational Univariate Reduction(RUR)

• We will use it as an alternative representation
  for points
• Capable of handling degenerate situations
  smoothly
• Can be used for detecting degeneracies, or as
  part of a routine to handle them directly

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Previous Work on RUR

• Fundamental method not new known for centuries
• Prior computation approaches ineffective
• Rely on Groebner bases too slow
• Groebner-free methods are iterative dont work
  well for exact computation
• Sparse resultant Emiris (and others)

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Context for Our Implementation

• Fits into precision-driven computation model
• LEDA and EGC work
• Core library
• Extends model to handle arbitrary roots of
  polynomials
• Includes complex roots

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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RUR operation

• Given m polynomials in n variables
• Rational coefficients
• Determine all roots of system by finding a set of
  polynomials
• h(x) minimal polynomial
• h1(a), h2(a), etc. coordinate polynomials

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RUR operation (continued)

• Determine the roots (real and complex) of the
  minimal polynomial
• Evaluate those roots in coordinate equations
• Result gives coordinates of every common root of
  original system
• For positive dimensional components, gives one
  point on that component.

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Procedure for Computing RUR
Determine Matrix Labels
Minimal Polynomial
Coordinate Polynomials
Compute Perturbed System
Input Polynomials
Perturb For Each Coordinate
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Computing the Matrix Labels

• Form and compute the toric perturbation (sparse
  resultant matrix following Canny and Emiris)
• Find Newton polytopes for input equations
• Compute Minkowski sum
• Perturb randomly
• Compute mixed subdivision
• Lattice points in interior give column labels
• Rows are indexed by monomial multiplied by
  original equation
• Entries are coefficients of input polynomials

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Using the Sparse Matrix

• Determinant is the sparse resultant
• Vanishing is a necessary condition for common
  root
• Determining matrix row/column labels
• linear programming
• Do not actually put equation coefficients in
• Instead, will use perturbed system

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Procedure for Computing RUR
Determine Matrix Labels
Minimal Polynomial
Coordinate Polynomials
Compute Perturbed System
Input Polynomials
Perturb For Each Coordinate
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Computing the Minimal Polynomial

• If input system is f1f2fn0 in variables
  x1,,xn
• Compute perturbed system
• f0u0u1x1u2x2unxn
• fifi-sfi for i1 to n
• ui are chosen randomly
• fi are polynomials with same support as fi, but
  with random coefficients

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Computing the Minimal Polynomial (continued)

• Find sparse resultant of perturbed system
• The minimal polynomial is found from this
  (perturbed) resultant, substiuting a variable T
  for u0.

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Procedure for Computing RUR
Determine Matrix Labels
Minimal Polynomial
Coordinate Polynomials
Compute Perturbed System
Input Polynomials
Perturb For Each Coordinate
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Computing Coordinate Polynomials

• To compute the coordinate polynomial hi, we
  substitute ui/-1 for ui, and compute the
  determinant again (twice).
• We find the square-free parts, and using the two
  determinants, we again introduce the variable T,
  with a gcd calculation.
• Finally, the coefficient polynomial is found
  through division.

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Computing Polynomials(Summary)

• Use linear programming to determine sparse matrix
• Evaluate that matrix several times for various
  entries
• Perform some polynomial operations (gcd,
  division) to determine RUR.

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ExtensionsAffine Roots

• Can compute affine roots (coordinate is zero) by
  appending origin to the support of each input
  polynomial
• May lead to spurious additional roots
• Will need to test each root found to see whether
  it should be kept (i.e. solves the actual system).

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ExtensionsNon-square Systems

• Can deal with non-square systems.
• If too few polynomials, just repeat one
  polynomial
• If too many polynomials, form new set of
  polynomials from generic linear combinations of
  input polynomials
• Will need to test final results, as spurious
  roots may be found.

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Computing with the RUR

• Given the RUR, can find roots of minimal
  polynomial
• We need all roots, real and complex
• Various techniques Aberths method is one for
  iteratively converging to roots.
• Substitute these into coordinate polynomials to
  determine (complex) coordinates of roots.

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Procedure for Computing RUR
Find Roots (Aberths Method)
Minimal Polynomial

a1
a2
an
z-coordinate Polynomial
y-coordinate Polynomial
x-coordinate Polynomial
a1,x
a1,y

a1,z
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Dealing with Complex Numbers

• Do not fit into existing root-bound approaches
• Can determine real/imaginary parts separately
• Root bounds can be determined for real and
  imaginary parts independently

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Finding Real Roots

• Often only care about real roots
• To find which roots are real, use root bounds to
  show that imaginary part is 0.
• Remember complex roots come in conjugate pairs
• In many cases, no complex roots near 0 this
  test is easy.

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Representing Roots

• The evaluation process can determine bounding
  intervals (boxes) around each root
• Can be used in geometric computations just like
  previous methods e.g. for quick rejection tests

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Functionality

• Each root is found by one point
• Degeneracies handled cleanly
• Tangential intersections
• Singularities in curves/surfaces
• Points lying on curves/surfaces
• Coincident points/curves/surfaces

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Functionality

• At least one point found on each positive
  dimensional component
• Two determinations with different generic values
  will output two different sets of points
• Differences indicate presence of positive
  dimensional components

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Performance

• Implemented in C, using Gnu MP for
  multiprecision arithmetic
• Exact implementation of sparse resultant
• Based off of Emiris implementation
• Uses exact computation throughout

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Determining Polynomials

• Use interpolation of polynomials to avoid
  symbolic computation.
• Substitute (random) values and determine
• Vandermonde interpolation
• Used when evaluating sparse resultant matrix
  determinant.
• Have degree bounds for all polynomials

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Timing Results

• Quadric curve intersections arising from
  real-world boundary evaluation cases
• MAPC .017 -.024 seconds
• RUR .317-1.772 seconds
• Approximately 20-100 times slower!
• Cases with degenerate intersections, positive
  dimensional components, higher dimension, all
  successful

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Timing Breakdown

• Slows rapidly with higher degrees/dimensions
• For lower dimension/degree, the size of the
  matrix tends to determine running time
• Increases quickly with degree/dimension
• At higher dimension/degree, coefficient size of
  coordinate polynomials grows very quickly and
  tends to dominate time

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Checking Root Bounds

• Surprisingly, gt98 of time was spent in
  computation of the RUR itself, not in checking
  root bounds.
• Root bounds could conceivably take longer, with
  repeated construction in precision-driven system.

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Optimizations

• Other optimizations are not implemented yet.
• Prior experience shows filtering and similar
  approaches can yield significant speedups
• Difficult to filter the sparse resultant matrix
  calculations
• May be able to filter over coefficients of the
  coefficient equations
• Possibly make higher degree/dimension more
  practical, but unlikely to ever beat MAPC

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Direct Implementation

• RUR approach could be used always
• Represent all points using RUR
• All degeneracies handled smoothly
• Checks between points are easy
• Evaluating point on curve is easy
• However, much slower for cases where there is no
  degeneracy

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Incorporating for Degenerate Situations

• Idea hybrid representation of points
• Use standard MAPC approach first
• When it does not seem to be working (e.g.
  repeated iterations), determine RUR
• Thereafter RUR can be used instead
• Would allow a uniform treatment of all points,
  with lower overhead than all-RUR
• Bounding boxes for both approaches
• Might not need RUR itself after first evaluation

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Detecting Degeneracies

• Several degenerate situations in solid modeling
  (boundary evaluation)
• Some are dealt with more easily by other means
• Overlapping surfaces dealt with through
  factorization or vanishing of intersection curve
• Usually want to detect degeneracies
• Apply special case code or perturbation

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Degeneracies
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Specific Degeneracy Detection

• Assume points represented as RUR
• Coincident Points
• Check sign (need root bounds) for difference of
  two points
• Four or more surfaces meeting at a point
• Appears as a point lying on a curve in a patch
  domain
• Substitute coordinate values of point into curve
• Can generate root bounds to guarantee answer

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Specific Degeneracy Detection

• Curve lying on surface, or overlapping curves
• Test for positive dimensional components
• Surfaces tangent at a point
• Point found normally, curve topology checks
• Tangent curves
• Intersection found normally, examine curve
  topology to determine if tangent

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Numerical Perturbation

• Predicated on a test to detect degeneracies
• Idea Since we have exact computation, we can
  eliminate degeneracies by perturbing the data
  numerically (not symbolically)
• Symbolic must relate predicates directly to input
• Filtering should make perturbed data not much
  slower in non-degenerate cases.

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Numerical Perturbation

• Key is capturing the designers intent
• Random perturbation of input surfaces in boundary
  evaluation can easily lead to undesirable results
• True for numeric or symbolic perturbation
• New approach assumes only one degenerate
  situation, at known point in CSG-style tree.

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Generating Perturbation

• Idea need to perturb in or out at level
  where degeneracy is occurring.
• Capture designers intent
• Union perturb both out
• Intersection perturb both in
• Difference (A-B) A in, B out
• Perturbation will be less than some tolerance
  value

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Propagating Information

• Propagate to children, all the way to leaves
• For difference A-B, must propagate opposite
  perturbation direction
• Apply scale in/out at leaves
• Results in perturbed primitives
• Must repeat calculations for entire tree
• When bad node is reached, no degeneracy

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Outline

• Motivation
• The Rational Univariate Reduction
• Background
• Computation Process
• Implementation
• Incorporating the RUR
• Numerical Perturbation
• Conclusion and Future Work

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Summary

• RUR method can find solutions to degenerate
  systems easily
• Application to several degenerate cases
• Less efficient than other general-position
  methods
• Numerical perturbation technique to eliminate
  degeneracies

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Future Work

• Near term
• Add RUR check to ESOLID system for degeneracy
  detection
• Add numerical perturbation approach for
  degeneracies
• Optimizations of RUR implementation
• Long term
• Boundary evaluation for higher degree surfaces,
  handling all degeneracies

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Questions?

• Thank you for listening (and staying)!
• Contact
• John Keyser keyser_at_cs.tamu.edu
• J. Maurice Rojas rojas_at_math.tamu.edu
• Koji Ouchi kouchi_at_cs.tamu.edu
• Funding
• NSF/DARPA CARGO
• NSF ITR (small)