In this paper, we give a priori estimation of derivatives of solution of Duncan-Mortensen-Zakai (DMZ) equation up to 2nd order. We also prove the existence and uniqueness of weak solution for DMZ equation, which is a fundamental equation in nonlinear filtering.

In this paper we introduce our preliminary research results for segmentation and labeling of 3-dimensional microscopy neuron image. We segment each of stacked 2-dimensional image slices using a partial differential equation (PDE) based algorithm and project previous slice segmentation result to the next slide as an initialization condition. Then we label neurons using an efficient connected component labeling algorithm. We show sample results obtained from real neuron image data

Structured light system using a digital video projector is increasingly used for a 3-D shape measurement because of its digital nature. However, the nonlinear gamma of the projector causes the projected fringe patterns to be non-sinusoidal, which results in phase error therefore shape measurement error. Previous work showed that, by using a small look-up-table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. In this research, we prove that this type of phase error compensation method is not limited to a three-step phase-shifting algorithm. It is generic for any phase-shifting algorithm. The phase error compensation algorithm is able to theoretically eliminate the phase error caused by the gamma of the projector completely. It is based on our finding that in phase domain, the phase error due to the projector's gamma is preserved for arbitrary object's surface

Mirror symmetry conjecture identifies the complex geometry of a Calabi $ Yau manifold with the symplectic geometry of its mirror Calabi $ Yau man $ ifold. Using the SYZ mirror transform, we argue that (i) the mirror of an elliptic Calabi $ Yau manifold admits a twin Lagrangian fibration structure and (ii) the mirror of the Fourier $ Mukai transform for dual elliptic fibra $ tions is a symplectic Fourier $ Mukai transform for dual twin Lagrangian fibrations, which is essentially an identity transformation in this case.

In this sequel to [L-Y1],[LLSY], and [L-Y2](respectively arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], and arXiv: 0901.0342 [math. AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Klebanov and Witten [KW](arXiv: hep-th/9807080) and Klebanov and Strasser [KS](arXiv: hep-th/0007191). A comparison with resolutions via noncommutative desingularizations is given in the end.