Peter J. Kostelec

Multiresolution elastic image registration

We have developed a multiscale algorithm for elastic (or molded) alignment of images. There is a wide array of medical applications of elastic (as opposed to strictly rigid) alignment: Subtraction of previous images from current ones to identify changes is perhaps the most obvious. We present preliminary results of this molding technique on a variety of images, and conclude with some closing remarks about this and future directions and goals of this work.

Differential affine motion estimation for medical image registration

We have applied techniques from differential motion estimation in the context of automatic registration of medical images. This method uses optical-flow and Fourier technique for local/global registration. A six parameter affine model is used to estimate shear, rotation, scale and translation. We show the efficacy of this method with images of similar and different contrasts.

The use of Hamilton's principle to derive time-advance algorithms for ordinary differential equations

Hamiltons principle is applied to derive a class of numerical algorithms for systems of ordinary differential equations when the equations are derivable from a Lagrangian. This is an important extension into the time domain of an earlier use of Hamiltons principle to derive algorithms for the spatial operators in Maxwells equations. In that work, given a set of expansion functions for spatial dependences, the Vlasov-Maxwell equations were replaced by a system of ordinary differential equations in time, but the question of solving the ordinary differential equations was not addressed. Advantageous properties of the new time-advance algorithms have been identified analytically and by numerical comparison with other methods, such as Runge-Kutta and symplectic algorithms. This approach to time advance can be extended to include partial differential equations and the Vlasov-Maxwell equations. An interesting issue that could be studied is whether a collisionless plasma simulation completely based on Hamiltons principle can be used to obtain a convergent computation of average properties, such as the electric energy, even when the underlying particle motion is characterized by sensitive dependence on initial conditions.

Automated multiscale elastic image registration using correlation

We present an automatic, multi-resolution correlation based approach for elastic image registration. The technique presented assumes no a priori information (such as landmarks or segmentation), which makes it suitable for a wide class of image registration tasks. We also present preliminary results of the technique on a variety of images.

Weak Harmonic Maaß Forms and the Principal Series for SL(2, \mathbb{R})

We use the representation theory of \(SL(2, \mathbb{R})\) to construct examples of functions with transformation properties associated to classical modular forms and Maas wave forms. We show that for special eigenvalues of the Laplacian, a Maas wave form may be associated naturally with both a weak harmonic Maas form and a classical modular form, leading to examples of weak harmonic Maas forms for all even negative integer weights.

Interpolating Uniquely with Only a Finite Class of Polynomials

This paper draws from theorems in transcendental number theory to answer questions about interpolation with a finite class of multivariable polynomials. In particular we describe sets of data points at which a unique polynomial from a particular class gives us a good approximation of the output.