Parameterized families of polynomials for bounded algebraic curve and surface fitting

Taubin, Gabriel; Cukierman, Fernando; Sullivan, Steve; Ponce, Jean; Kriegman, David

doi:10.1109/34.276128

Cited by 135 publications

(66 citation statements)

References 24 publications

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“…In 1994, Keren et al (1994) and Taubin et al (1994) independently developed algorithms for a bounded (closed) algebraic curve with approximate squares distance. We will now introduce the definition and properties of a bounded algebraic curve.…”

Section: Bounded and Stably Bounded Algebraic Curve And Surfacementioning

confidence: 99%

Dimension Reduction Methods

Mizuta

2011

Handbook of Computational Statistics

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.

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Section: Bounded and Stably Bounded Algebraic Curve And Surfacementioning

confidence: 99%

Dimension Reduction Methods

Mizuta

2011

Handbook of Computational Statistics

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“…However, the cost function does not necessarily express the Euclidean distances of the data points to the zerosurface; therefore, the fit might be somewhat unintuitive, especially in regions of high surface curvature, as has been shown in previous works [16], [8], [15]. A cost function that approximates the sum of the squares of the Euclidean distances is: …”

Section: Introductionmentioning

confidence: 99%

Fitting curves and surfaces with constrained implicit polynomials

Keren

Gotsman

1999

IEEE Trans. Pattern Anal. Machine Intell.

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Abstract-A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: Although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. This paper suggests a different approach-to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful recent results from real algebraic geometry.

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“…Better results are achieved by using cost functions which provide a more accurate approximation to the geometric distance between the zero-set and the points [9], [11], [20], [1], [17], [13], [15], [3], [8], [19].…”

Section: Introductionmentioning

confidence: 99%

Topologically faithful fitting of simple closed curves

Keren

2004

IEEE Trans. Pattern Anal. Machine Intell.

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Abstract-Implicit representations of curves have certain advantages over explicit representation, one of them being the ability to determine with ease whether a point is inside or outside the curve (inside-outside functions). However, save for some special cases, it is not known how to construct implicit representations which are guaranteed to preserve the curve's topology. As a result, points may be erroneously classified with respect to the curve. The paper offers to overcome this problem by using a representation which is guaranteed to yield the correct topology of a simple closed curve by using homeomorphic mappings of the plane to itself. If such a map carries the curve onto the unit circle, then a point is inside the curve if and only if its image is inside the unit circle.

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Parameterized families of polynomials for bounded algebraic curve and surface fitting

Cited by 135 publications

References 24 publications

Dimension Reduction Methods

Dimension Reduction Methods

Fitting curves and surfaces with constrained implicit polynomials

Topologically faithful fitting of simple closed curves

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