Research in Image Deblurring and Processing

Research in Functional Analysis

My research in image deblurring involves a theoretical analysis for using the Poisson negative log likelihood data fidelity functional for astronomical image deblurring.

My primary research in functional analysis involves analysis of operators between algebras of continuous functions and between algebras of bounded linear operators on Banach spaces.

1) Total-variation Penalized Poisson Likelihood Estimation for Ill-posed Problems, Johnathan Bardsley and Aaron Luttman, Adv. Comput. Math., Special Volume on Mathematical Methods for Image Processing, 31 (2009), no. 1, 35-59 (PDF).

1) Uniform Algebra Isomorphisms and Peripheral Multiplicativity, Aaron Luttman and Thomas Tonev, Proc. Amer. Math. Soc., 135 (2007), no. 11, 3589-3598. (PDF).

2) L^1 Regularized Poisson Likelihood Estimation for Inverse Problems in Astronomical Imaging, Aaron Luttman, preprint.

2) Weakly Peripherally Multiplicative Operators Between Uniform Algebras, Scott Lambert, Aaron Luttman, and Thomas Tonev, Contemp. Math., 435 (2007), 265-281. (PDF).

3) Norm Conditions and Uniform Algebra Isomorphisms, Aaron Luttman and Scott Lambert, Cent. Eur. J. Math., 6 (2008), no. 2, 272-280. (PDF).

Research in Image Segmentation

4) Weakly Peripherally Multiplicative Surjections of Pointed Lipschitz Algebras, A. Jiménez-Vargas, Aaron Luttman, and Moisés Villegas-Vallecillos, Rocky Mountain J. Math., to appear. (PDF).

5) Algebra Isomorphisms Between Standard Operator Algebras, Thomas Tonev and Aaron Luttman, Studia Math., 191 (2009), 163-170. (PDF).

Ph. D. Dissertation

We use variational methods (based on the Chan-Vese technique) for segmenting videos of leaves fluorescing.  An example segmentation can be found here.  The original data was supplied by the Complexity and Stomatal Behavior research lab at Utah State University.

1) A Variational Approach to Video Segmentation for Botanical Data, Aaron Luttman and Johnathan Bardsley, SIAM J. Sci. Comput., 29 (2007), no. 4, 1550-1566. (PDF).

My doctoral dissertation focused on my research in image segmentation as described to the left.

2) Analysis of Pattern Formation on the Surface of Respiring Leaves, Aaron Luttman, Emily Stone, and Johnathan Bardsley, Physica D, 232 (2007), no. 2, 142-155. (PDF).

1) A Three-dimensional Variational Approach to Video Segmentation, Aaron Luttman, University of Montana, (2006). (PDF).

3) A Fixed-Point Formulation of the k-Means Algorithm and a Connection to Mumford-Shah, Johnathan Bardsley and Aaron Luttman, Applied Math. E-Notes, to appear.