My research mainly focuses on computational quasi-conformal geometry
(CQC) and its applications to medical imaging, computer vision and
computer graphics.
Many important problems in the real world don't have unique
solutions. It is thus important for machine learning models to be
capable of proposing different plausible solutions with meaningful
probability measures.
In this work we introduce such a deep learning framework that learns
the one-to-many relationships between the inputs and outputs. The key
of our approach is the use of a discrete latent space, where each item
represents a latent mode hypothesis for a particular type of
input-output relationship.
The discrete latent representations and the uncertainty associated to
any input are learned jointly during training.
We thus call our framework modal uncertainty estimation.
We extensively validate our framework on both real and synthetic
datasets.
We develop a framework for shape analysis of partial surfaces
using inconsistent surface mapping technqiue. Traditional
landmark-based geometric morphometrics methods suffer from the limited
degrees of freedom, while most of the more advanced non-rigid surface
mapping methods rely on a strong assumption of the global consistency
of two surfaces. From a practical point of view, given two anatomical
surfaces with prominent feature landmarks, it is more desirable to have
a method that automatically detects the most relevant parts of the two
surfaces and finds the optimal landmark-matching alignment between
those parts, without assuming any global 1-1 correspondence between the
two partial surfaces. Our method is capable of solving this problem
using an inconsistent surface mapping technique based on
quasi-conformal theory. It further enables us to quantify the
dissimilarity of two shapes using quasi-conformal distortion and
differences in mean and Gaussian curvatures, thereby providing a
natural way for shape classification. Experiments on Platyrrhine molars
demonstrate the effectiveness of our method and shed light on the
interplay between function and shape in nature. Project
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We address the problem of registering two surfaces, of which a
natural bijection between them does not exist. More precisely, only a
partial subset of the source domain is assumed to be in correspondence
with a subset of the target domain. We call such a problem an
inconsistent surface registration problem. This problem is challenging
as the corresponding regions on each surfaces and a meaningful
bijection between them have to be simultaneously determined. In this
paper, we propose a variational model to solve the inconsistent surface
registration problem by minimizing mapping distortions. Mapping
distortions are described by the Beltrami coefficient as well as the
differential of the mapping. Registration is then guided by feature
landmarks and/or intensities, such as curvatures, defined on each
surfaces. The key idea of the approach is to control angle and scale
distortions via quasiconformal theory as well as minimizing landmark
and/or intensity mismatch. A splitting method is proposed to
iteratively search for the optimal corresponding regions as well as the
optimal bijection between them. Bijectivity of the mapping is easily
enforced by a thresholding of the Beltrami coefficient. We test the
proposed method on both synthetic and real examples. Experimental
results demonstrate the efficacy of our proposed model.
Detection of abnormal deformations from normal motions is
crucial in image analysis, especially for medical image analysis. For
instance, accurate extraction of abnormal cardiac motions is a
necessary procedure for cardiac disease analysis. The combination of
nomral motion and abnormal deformations bring challenges to extract
abnormalities. In this work, we propose a novel method to extract
abnormal deformation from normal (periodic) motions by using the
Beltrami coefficients (BC). BCs are used to represent a sequence of
deformations over a sequence of images capturing an object of interest
in motion. RPCA is then performed on the BCs to decompose the overall
motion into nomral motion and abnormal deformation. Experiments have
been carried out on both synthetic and real medical image sequence,
which demonstrate the efficacy of our proposed method.