Topologically faithful fitting of simple closed curves

Keren, Daniel

doi:10.1109/tpami.2004.1261095

Cited by 17 publications

(8 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a better approximation of the contour, we use a second degree polynomial fitting [3,15,16,28] applied to the contour points vector V [P(x i , y i )], using the least square method, (Fig. 16b).…”

Section: Methods 2: Vertebra Contours Detectionmentioning

confidence: 99%

X-ray image segmentation for vertebral mobility analysis

Benjelloun

Mahmoudi

2008

Int J CARS

View full text Add to dashboard Cite

Objective The goal of this work is to extract the parameters determining vertebral motion and its variation during flexion-extension movements using a computer vision tool for estimating and analyzing vertebral mobility. Materials and MethodsTo compute vertebral body motion parameters we propose a comparative study between two segmentation methods proposed and applied to lateral X-ray images of the cervical spine. The two vertebra contour detection methods include (1) a discrete dynamic contour model (DDCM) and (2) a template matching process associated with a polar signature system. These two methods not only enable vertebra segmentation but also extract parameters that can be used to evaluate vertebral mobility. Lateral cervical spine views including 100 views in flexion, extension and neutral orientations were available for evaluation. Vertebral body motion was evaluated by human observers and using automatic methods. Results The results provided by the automated approaches were consistent with manual measures obtained by 15 human observers. Conclusion The automated techniques provide acceptable results for the assessment of vertebral body mobility in flexion and extension on lateral views of the cervical spine. Electronic supplementary materialThe online version of this article (

show abstract

Section: Methods 2: Vertebra Contours Detectionmentioning

confidence: 99%

X-ray image segmentation for vertebral mobility analysis

Benjelloun

Mahmoudi

2008

Int J CARS

View full text Add to dashboard Cite

show abstract

“…, 125, along a nested quintic curve, cf. [12], degraded by noise. For the noise we use the generalised p-order model [10] discussed in Example 1 with p = 2, α = 1.0 and β = 10.0, truncated at ten Fourier coefficients.…”

Section: Simulated Examplementioning

confidence: 99%

A spectral mean for random closed curves

Lieshout

2016

Spatial Statistics

View full text Add to dashboard Cite

We propose a spectral mean for closed curves described by sample points on its boundary subject to mis-alignment and noise. First, we ignore mis-alignment and derive maximum likelihood estimators of the model and noise parameters in the Fourier domain. We estimate the unknown curve by back-transformation and derive the distribution of the integrated squared error. Then, we model mis-alignment by means of a shifted parametric diffeomorphism and minimise a suitable objective function simultaneously over the unknown curve and the mis-alignment parameters. Finally, the method is illustrated on simulated data as well as on photographs of Lake Tana taken by astronauts during a Shuttle mission.In memory of J. Harrison.Throughout this paper we model the boundary of the random object of interest by a smooth (simple) closed curve.Consider the class of functions Γ : I → R 2 from some interval I to the plane. Define an equivalence relation ∼ on the function class as follows. Two functions Γ and Γ ′ are equivalent, Γ ∼ Γ ′ , if there exists a strictly increasing function ϕ from I onto another interval I ′ such that Γ = Γ ′ • ϕ. Note that ϕ is a homeomorphism. The relation defines a family of equivalence classes, each of which is called a curve. Its member functions are called parametrisations.Since the images of two parametrisations of the same curve are identical, we shall, with slight abuse of notation, use the symbol Γ for a specific parametrisation, for a curve and for its image.A curve is said to be continuous if it has a continuous parametrisation, in which case all parametrisations are continuous; it is simple if it has a parametrisation that is injective. A Jordan curve has the additional property of being closed, in other words, it is the image of a continuous function Γ from [p, q] to R 2 that is injective on [p, q) and for which Γ(p) = Γ(q). By the Jordan-Schőnflies theorem, the complement of any Jordan curve in the plane consists of exactly two connected components: a bounded one and an unbounded one separated by Γ. The bounded component is called the interior of Γ and can be thought of as the object. Since closed curves have neither a 'beginning' nor an 'end', a rooted parametrisation is provided by a point on the curve together with a cyclic parametrisation from that point in a given direction (say with the interior to the left). For convenience, we shall often rescale the definition interval to [−π, π],In the statistical inference to be discussed in the next section, we need derivatives. In this context, it is natural to assume a curve to be parametrised by some function Γ that is C 1 and the same degree of smoothness to hold for the functions ϕ that define the equivalence relation between parametrisations. In effect, ϕ should be a diffeomorphism. See [17, Chapter 1] for further details. Fourier representationLet Γ = (Γ 1 , Γ 2 ) : [−π, π] → R 2 be a C 1 function with Γ i (−π) = Γ i (π), i = 1, 2. Recall that the family of functions {cos(jx), sin(jx) : j ∈ N 0 } forms an orthogonal basis for L 2 ([−π, π]),

show abstract

“…In order to get a closed contour, we apply an edge closing method to the contours obtained, a polynomial fitting to each face for each vertebra. Indeed, for a better approximation of vertebra contours, we use a second degree polynomial fitting [9,10]. We achieve this 2D polynomial fitting by the least square method, Figure 6.…”

Section: Polar Signaturementioning

confidence: 99%