Sampling and Reconstruction of Shapes with Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of \mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.Comment: 12 pages, 14 figure

Accurate geometry modeling of vasculatures using implicit fitting with 2D radial basis functions

Accurate vascular geometry modeling is an essential task in computer assisted vascular surgery and therapy. This paper presents a vessel cross-section based implicit vascular modeling technique, which represents a vascular surface as a set of locally fitted implicit surfaces. In the proposed method, a cross-section based technique is employed to extract from each cross-section of the vascular surface a set of points, which are then fitted with an implicit curve represented as 2D radial basis functions. All these implicitly represented cross-section curves are then being considered as 3D cylindrical objects and combined together using a certain partial shape-preserving spline to build a complete vessel branch; different vessel branches are then blended using a extended smooth maximum function to construct the complete vascular tree. Experimental results show that the proposed method can correctly represent the morphology and topology of vascular structures with high level of smoothness. Both qualitative comparison with other methods and quantitative validations to the proposed method have been performed to verify the accuracy and smoothness of the generated vascular geometric models

NEW ALGEBRAIC INVARIANTS OF IMPLICIT POLYNOMIALS FOR 3D OBJECT RECOGNITION

Abstract In this paper, we present a method for deriving the rotation invariants of 2 nd and 4 th degree implicit polynomials and we build a system for 3D object recognition using the derived invariants. Our results show that invariants derived in this paper are stable and the success of the recognition is high when the polynomial fit is successful

A wavelet based method for affine invariant 2D object recognition

Recognizing objects that have undergone certain viewing transformations is an important problem in the field of computer vision. Most current research has focused almost exclusively on single aspects of the problem, concentrating on a few geometric transformations and distortions. Probably, the most important one is the affine transformation which may be considered as an approximation to perspective transformation. Many algorithms were developed for this purpose. Most popular ones are Fourier descriptors and moment based methods. Another powerful tool to recognize affine transformed objects, is the invariants of implicit polynomials. These three methods are usually called as traditional methods. Wavelet-based affine invariant functions are recent contributions to the solution of the problem. This method is better at recognition and more robust to noise compared to other methods. These functions mostly rely on the object contour and undecimated wavelet transform. In this thesis, a technique is developed to recognize objects undergoing a general affine transformation. Affine invariant functions are used, based on on image projections and high-pass filtered images of objects at projection angles . Decimated Wavelet Transform is used instead of undecimated Wavelet Transform. We compared our method with the an another wavelet based affine invariant function, Khalil-Bayoumi and also with traditional methods