### Abstracts

**
Texture clustering with asymmetric Cheeger cut**

**
Xavier Bresson**

Cheeger
cut has recently been shown to provide excellent partitioning results for two
classes. While the classical Cheeger cut favors a 50-50 partition of the graph,
we present here an asymmetric variant of the Cheeger cut which favors, for
example, a 10-90 partition. This asymmetric Cheeger cut provides a powerful tool
for unsupervised multi-class clustering. We use it in recursive bipartitioning
to detach one after the other each of the classes. This asymmetric recursive
algorithm handles equally well any number of classes, as opposed to symmetric
recursive bipartitioning which is naturally better suited for 2^{m}
classes. We apply the asymmetric Cheeger cut to perform texture classification
and show that our method outperforms related spectral methods.

**
Data-driven tight frame construction and image denoising**

**
Jian-Feng Cai**

The
regularization methods for image restoration using the ℓ_{1} norm of the
coefficients of the underlying image under some system assume that the image has
a good sparse approximation under the given system. Such a system can be a
basis, a frame or a general over-complete dictionary. One widely used system in
image restoration is wavelet tight frame. There have been enduring efforts on
seeking wavelet type of tight frames under which certain class of functions or
images can have a good sparse approximation. However, the structure of images
varies greatly in practice and a system working well for one type of images may
not work for another. This talk presents a method that derives discrete tight
frame system from the input image itself to provide a better sparse
approximation to the input image. Such an adaptive tight frame construction
scheme is applied on image denoising by constructing a tight frame tailor down
to the given noisy data. The experiments showed the proposed approach performs
better in image denoising than those wavelet tight frames designed for a class
of images. Moreover, by ensuring the system derived from our approach is always
a tight frame, our approach also runs much faster than some other adaptive
over-complete dictionary based approaches with comparable PSNR performance.

**
Decoupling noises and features based on sparse analysis**

**
Falai Chen**

In this
talk, I will discuss a new approach for decoupling noises and features on 3D
models. The approach consisits of two phases. In the first phase, a smooth base
model is generated to approximate the underlying model of the input noisy model
by a global Laplacian regularization donosing scheme. The base model is
guaranteed to asymptotically converge to the underlying model with probability
one as the sampling points go to infinity. In the second phase, a sparse
optimization scheme is proposed to recover shape features from the residual
between the input model and the smooth base model. The algorithm is based on the
observation that sharpe features can be sparsely represented in some coherent
dictionary. Experiments show that our approch can robustly recover the sharpe
features while remove noises on a 3D model.

**
Coupling Interface Method and Macromocules in Ionic Solution**

**
I-Liang Chern**

In this
talk, I will first review the coupling interface method for solving the elliptic
interface problems in arbitrary dimensions. It is a finite difference method on
an underlying regular Cartesian grid. The method takes a dimension-splitting
approach. In each dimension, it uses piecewise polynomials to approximate the
solution from both sides of the interface. The information from different
dimensions is linked through jump conditions, leading to a coupled linear
equation for the principal second order derivatives. This approach reduces the
size of stencil, and thus has mild restriction on the interfaces. It has more
potential to handle complex interface problems. A recent progress on
second-order accurate approximation for gradients on interfaces will be
reported.

Secondly,
I will show application of the coupling interface method on simulation of
macromolecules in ionic solution, in which the Poisson-Boltzmann equation is
solved. Various singularity removal techniques will be demonstrated.

**
Spline Spaces over T-meshes**

**
Jiansong Deng**

A T-mesh
is a rectangular grid that allows T-junctions. Some types of spline spaces over
T-meshes have been considered in the literature, including hierarchical
B-splines, T-splines and LR splines. In the talk, I will explain why we need
T-meshes in geometric modeling and how to define spline spaces over T-meshes. In
the talk, I will introduce a new type of spline spaces over T-meshes and given
their dimension formula and basis function construction. The applications in
computer graphics, image processing and isogeometric analysis are reviewed as
well.

**
Total Variation, Wavelet Frames, Sparsity and Imaging**

**
Bin Dong**

I will
start my talk with a brief literature review of total variation and some more
general variational models introduced in the literature for image restoration
problems. Then I will review some basic concepts of wavelet frames and their
recent applications in image restoration. Wavelet frame based and differential
operator based variational models were generally considered as different
approaches. However, they do share the same observation that images are
generally sparse under certain. I will address the connections of wavelet frame
based models to variational models based on one of our recent theoretical
studies, which also granted geometric interpretations to the wavelet frame
transform and enabled us to extend the applications of wavelet frames. To
further utilize the property of sparse approximation by wavelet frames, I will
present a model (as well as some fast algorithms) that penalizes the 0-norm of
the frame coefficients, which has some advantages over the commonly used 1-norm
for some specific types of images.

**
Seismic Imaging**

**
Bjorn Engquist**

Seismic imaging is the mot important process for finding out properties of the
earth's interior. Seismic waves are generated and measured at the surface. An
image is then generated based on high frequency wave propagation in variable
velocity media. We will discuss imaging principles and key numerical techniques
for seismic exploration.

**
**

**
4D compressive medical imaging via Split Bregman method**

**
Hao Gao**

I will
present a few 4D compressive medical imaging examples with experimental data,
including MRI, CT and optical imaging, to illustrate how the split Bregman
method helps the real-world problems.

**
Computational Conformal Geometry Theories and Applications**

**
Xianfeng David Gu**

Conformal
geometry studies the invariants under angle preserving transformations, which
offers powerful tools for surface matching, classification and analysis.

This talk
introduces the main theorems in computational conformal geometry, including
discrete Hodge theory, discrete surface Ricci flow theory, discrete
uniformization, conformal modules, Teichmuller map and so on.

The direct
applications will be covered as well, including surface registration in computer
vision, global parameterization in graphics, geometric routing in networks,
homotopy detection in computational topology, brain mapping and virtual
colonoscopy in medical imaging and so on.

**
Blind image de-blurring for removing the spatially-varying motion-blurring from a
single image**

**
Hui Ji**

Blind
image de-blurring problem is an ill-posed inverse problem, which is about how to
remove image blurring from photographs without knowing all the information about
the corresponding blur kernel. One such often seen problem is blind motion
de-blurring for recovering the motion-blurred image due to the camera movement
during the shutter time. Blind motion de-blurring has drawn a lot of attentions
in recent years and most existing approaches consider a spatially invariant
convolution model for image blurring. However, the practical motion blurring
often is a spatially varying blurring process over the image. In this talk, I
will start with the introduction of several important results on various aspects
of blind image de-convolution. Then, based on these results, I will introduce a
two-stage approach for removing the spatially-varying motion blurring from a
single image.

**
**

**
Level Set Methods and their Applications in Biomedical Image Processing**

**
Chiu-Yen Kao**

Level set
methods are well known for their flexibility in handling the shape and
topological changes of moving interfaces. These methods have been successfully
applied to study biomedical images and provide robust analysis tools. In this
talk, we will give examples of morphology and connectome study of human brains
and the shape analysis of ciliary muscle of human eyes.

**
Solving PDEs on Point clouds and applications**

**
Ronjie Lai**

In this
talk, I will present two systematic methods to solve PDEs on manifolds
represented as meshless point clouds. Global mesh structures or
parameterizations are usually hard to construct in this case. While our methods
only rely on the local structure at each data point and scale well with the
total number of points and the intrinsic dimension of point clouds. Once the
local structure is available, we propose numerical schemes to approximate
differential operators and define integrals on point clouds, which can be used
to solve partial differential equation (PDE) and variational problem on point
clouds. The framework we proposed can be easily adapted to solve PDEs on
k-manifolds in R^n. Numerical comparisons with existing methods demonstrate the
accuracy of the proposed methods. Finally, several applications to geometric
understanding of point clouds will be discussed based on solutions of PDEs on
point clouds.

**
A Grid Based Particle Method for Solving Variational Problems on Manifolds**

**
Shingyu Leung**

We propose
a numerical approach to solve variational problems on manifolds represented by
the Grid Based Particle Method (GBPM). In particular, we propose a splitting
algorithm for image segmentation on manifolds represented by unconnected
sampling particles. To develop a fast minimization algorithm, we propose a new
splitting method by generalizing the augmented Lagrangian method (ALM). To
efficiently implement the resulting method, we incorporate with the local
polynomial approximations of the manifold in the GBPM. The resulting method is
flexible for segmentation on various manifolds including closed or open or even
surfaces which are not orientable. This is a joint work with Jun Liu.

**
Teichmuller Extremal Mapping and its Applications**

**
Ronald Lok Ming Lui**

Registration, which aims to find an optimal 1-1 correspondence between shapes,
is an important process in different research areas. Conformal mappings have
been widely used to obtain a diffeomorphism between shapes that minimizes
angular distortion. Conformal registrations are beneficial since it preserves
the local geometry well. However, when landmark constraints are enforced,
conformal mappings generally do not exist. This motivates us to look for a
unique landmark matching quasiconformal registration, which minimizes the
conformality distortion. Under suitable condition on the landmark constraints, a
unique diffeomporphism, called the Teichmuller extremal mapping between two
surfaces can be obtained, which minimizes the maximal conformality distortion.
In this talk, an efficient iterative algorithm, called the Quasi-conformal (QC)
iterations, to compute the Teichmuller mapping will be presented. The basic idea
is to represent the set of diffeomorphisms using Beltrami coefficients (BCs),
and look for an optimal BC associated to the desired Teichmuller mapping. The
associated diffeomorphism can be efficiently reconstructed from the optimal BC
using the Linear Beltrami Solver(LBS). Using BCs to represent diffeomorphisms
guarantees the diffeomorphic property of the registration. Using the proposed
method, the Teichmuller mapping can be accurately and efficiently computed
within 10 seconds. The obtained registration is guaranteed to be bijective. This
proposed algorithm can also be practically applied to real applications. In the
second part of my talk, I will present how Teichmuller extremal mapping can be
used for brain landmark matching registration, constrained texture mapping and
face recognition.

**
On reinitializing level-set functions name: Chohong Min and Frederic Gibou
affilation**

**
Chohong Min**

The
level-set method has been successfully applied to image processing problems such
as active contouring and computational fluid problems with interfaces. One of
the hardest parts in its applications is the reinitialization of the level-set
function, avoiding slopes near interface too steep or too flat. The
reinitialization is implemented by a Hamilton-Jacobi equation with discontinuous
source term that requires special cares. We present a stable and accurate
spatial and temporal discretizations of the equation. With adaptive mesh and
parallel implementation added, it becomes a state-of-the-art solution to
reinitialization.

**
Variational Methods for Image Stitching**

**
Michael Ng**

We discuss
image stitching algorithms to compute (i) weighting mask functions automatically
on input images and stitch them together; and (ii) to handle color inconsistency
across input images.

Experimental results will be shown to demonstrate the performance of the
proposed algorithm.

**
High-Order Factorization Based High-Order Hybrid Fast Sweeping Methods for
Point-Source Eikonal Equations**

**
Jianliang Qian**

The
solution for the eikonal equation with a point-source condition has an upwind
singularity at the source point as the eikonal solution behaves like a distance
function at and near the source. As such, the eikonal function is not
differentiable at the source so that all formally high-order numerical schemes
for the eikonal equation yield first-order convergence and relatively large
errors. Therefore, it is a long standing challenge in computational geometrical
optics how to compute a uniformly high-order accuratesolution for the
point-source eikonal equation in a global domain. In this paper, we propose
high-order factorization based high-order hybrid fast sweeping methods for
point-source
eikonal equations to compute
just such solutions. Observing that the
squared eikonal is differentiable at the source, we propose to factorize the
eikonal into two multiplicative or additive factors, one of which is specified
to approximate the eikonal up to arbitrary order of accuracy near the source,
and the other of which serves as a higher-order correction term. This
decomposition is achieved by using the eikonal equation and applying power
series expansions to both the squared eikonal and the squared slowness function.
We develop recursive formulas to compute the approximate eikonal up to arbitrary
order of accuracy near the source. Furthermore, these approximations enable us
to decompose the eikonal into two factors either multiplicatively or additively
so that we can design two new types of hybrid, high-order fast sweeping schemes
for the point-source eikonal equation. We also show that the hybrid first-order
fast sweeping methods are monotone and consistent so that they are conver-gent
in computing viscosity solutions. Two- and three-dimensional numerical examples
demonstrate that a hybrid p-th order fast sweeping method yields desired,
uniform, clean p-th order convergence in a global domain by using a p-th order
factorization. This is a joint work with Songting Luo and Robert Burridge.

**
One-Bit Compressive Sampling**

**
Lixin Shen**

In this
talk, we address the problem of 1-bit compressive sampling. We introduce an
optimization model for reconstruction of sparse signals from 1-bit measurements
and propose an algorithm for obtaining an approximation reconstruction of the
sparse signals from it. Our model is to minimize the ℓ_{0} norm of the
signal of interest subject to the signal satisfying convex consistency
constraints. Our approach is to obtain a sequence of optimization problems by
successively approximating the ℓ_{0} norm and to solve resulting
problems by exploiting the proximity operator. Our approach does not require
prior knowledge on the sparsity of the signal. Convergence analysis of our
algorithm is presented. We examine the performance of our proposed algorithm and
compare it with the binary iterative hard thresholding (BIHT) a state-of-the-art
algorithm for 1-bit compressive sampling reconstruction.

**
Inverse Lax-Wendroff Procedure for Numerical Boundary Conditions of Hyperbolic
Equations**

**
Chi-Wang Shu**

We develop
a high order finite difference numerical boundary condition for solving
hyperbolic Hamilton-Jacobi equations and conservation laws on a Cartesian mesh.
The challenge results from the wide stencil of the interior high order scheme
and the fact that the boundary may not be aligned with the mesh and can
intersect the grids in an arbitrary fashion. Our method is based on an inverse
Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the
partial differential equation to write the normal derivatives to the inflow
boundary in terms of the tangential derivatives and the time derivatives (for
time dependent equations). With these normal derivatives, we can then impose
accurate values of ghost points near the boundary by a Taylor expansion. At the
outflow boundaries, we use Lagrange extrapolation or least squares extrapolation
if the solution is smooth, or a weighted essentially nonoscillatory (WENO) type
extrapolation if a shock is close to the boundary. Extensive numerical examples
are provided to illustrate that our method is high order accurate and has good
performance when applied to one and two dimensional scalar or system cases with
the physical boundary not aligned with the grids and with various boundary
conditions including the solid wall boundary condition. This is a joint work
with Ling Huang and Mengping Zhang (for the Hamilton-Jacobi equations) and with
Sirui Tan (for the time dependent conservation laws).

**
A Fast Global Optimization-Based Approach to Evolving Contours with Generic Shape
Prior**

**
Xue-Cheng Tai**

In this
talk, we present a new global optimizationbased approach to contour evolution,
with or without the novel variational shape constraint that imposes a generic
star shape using a continuous max-flow framework. In theory, the proposed
continuous max-flow model provides a dual perspective to the reduced continuous
min-cut formulation of the contour evolution at each discrete time frame, which
proves the global optimality of the discrete time contour propagation. The
variational analysis of the flow conservation condition of the continuous
max-flow model shows that the proposed approach does provide a fully time
implicit solver to the contour convection PDE, which allows a large time-step
size to significantly speed up the contour evolution. For the contour evolution
with a star shape prior, a novel variational representation of the star shape is
integrated to the continuous max-flow-based scheme by introducing an additional
spatial flow. In numerics, the pro-posed continuous max-flow formulations lead
to efficient duality-based algorithms using modern convex optimization theories.
Our approach is implemented in a GPU, which significantly improves computing
efficiency. We show the high performance of our approach in terms of speed and
reliability to both poor initialization and large evolution step-size, using
numerous experiments on synthetic, real-world and 2D/3D medical images.

This talk
is based in a joint work by: J. Yuan, E. Ukwatta1, X.C. Tai, A. Fenster1, C.
Schnorr.

**
An Implicit Interface Boundary Integral Method for Poisson’s and Helmholtz
Equations on Arbitrary Domains**

**
Richard Tsai**

We propose
a simple formulation for constructing boundary integral methods to solve
Poisson’s and Helmholtz equations on domains with piecewise smooth boundaries
defined through their signed distance function. Our formulation is based on
averaging a family of parameterizations of an integral equation defined on the
boundary of the domain, where the integrations are carried out in the level set
framework using an appropriate Jacobian. By the coarea formula, the algorithm
operates in the Euclidean space and does not require any explicit
parameterization of the boundaries. We present numerical results in two and
three dimensions.

**
Data Regularization Using Beams Decomposition and Sparse Optimization**

**
Yanfei Wang**

In seismic
exploration, the process of acquisition records the continuous wavefield which
is generated by thesource. In order to restore the seismic data correctly, the
acquisition should satisfy the *Nyquist/Shannon* sampling theorem, i.e.,
the sampling frequency should be at least twice of the maximum frequency of
original signal. In seismic acquisition, because of the influence of obstacles
at land surface, rivers, bad receivers, noise, acquisition aperture, restriction
of topography and investment, the obtained data usually does not satisfy the
sampling theorem. A direct effect of the limitations of acquisition is the
sub-sampled data will generate aliasing in the frequency domain; therefore, it
may affect the subsequent processing such as filtering, de-noising, AVO
(amplitude versus offset) analysis, multiple eliminating and migration imaging
[1-5].

In order
to remove the influence of sub-sampled data, the seismic data regularization
technique is often used. Let us denote by *m *the original seismic
wavefield, *d *the sampled data, and *L *the sampling operator, the
data regularization can be written as

*
Lm *= *d.
*(1)

Our
purpose is to restore *m *from the sampled data *d*. Since *d *is
usually incomplete and *L *is an underdetermined operator, this indicates
that there are infinite solutions satisfying the seismic imaging equation (1).
Hence, seismic data regularization is an ill-posed inverse problem.

In this
research work, we develop some sparse optimization methods for the wavefield
reconstruction problem. We consider Gaussian beams decomposition methods and
sampling techniques and solve the problem by constructing different kinds of
regularization models and study sparse optimization methods for solving the
regularization model. The *l**p*-*l**q
*model with
*p *= 2 and *q *= 0*, *1 is fully studied. Solving methods for
the optimization problem are addressed. Numerical experiments are performed for
solving the ill-posed data regularization problem. The results revealed that the
proposed method can greatly improve the quality of wavefield recovery.

References

1. Wang
Y.F., Liu P., Li Z. H., Sun T., Yang C. C. and Zheng Q. S. Data regularization
using Gaussian beams decomposition and sparse norms. *Journal of Inverse and
Ill-posed Problems*, DOI: 10.1515/jip-2012-0030, 2012.

2. Wang
Y.F. Sparse optimization methods for seismic wavefields recovery. *Proceedings
of the Institute of Mathematics and Mechanics *(Yekaterinburg) (2012) 18, No.
1, 42-55.

3. Wang
Y.F., Yang C.C. and Cao J.J. On Tikhonov regularization and compressive sensing
for seismic signal processing. *Mathematical Models and Methods in Applied
Sciences *(2012) 22, No. 2, 1150008-1-1150008-24.

4. Wang
Y.F., Cao J.J. and Yang C.C. Recovery of seismic wavefields based on compressive
sensing by an *l*1-norm
constrained trust region method and the piecewise random sub-sampling. *
Geophysical Journal* *International *(2011) 187, 199-213.

5. Wang
Y.F., Stepanova I.E., Titarenko V.N. and Yagola A.G. *Inverse Problems in
Geophysics and Solution Methods*. Higher Education Press, Beijing, 2011.

**
Proximity Algorithms for Solving Convex
Optimization Problems Arising from Image Processing**

**
Yuesheng Xu**

We consider a class of convex non-smooth optimization problems in the context of
image denoising/deblurring. Solutions of the optimization problems are
characterized as fixed-point equations in terms of the proximity operator of the
functions appearing in their objective functions. Proximity algorithms are
developed using the characterizations of the solutions. Convergence of the
algorithms are shown by introducing the notion of the weakly firmly
non-expansive operators. Existing algorithms are identified as special cases of
the proposed algorithms. Numerical results will be presented to demonstrate the
efficiency and accuracy of the proposed algorithms.

**
The Alternating Direction Method of Multipliers**

**
Wotao Yin**

Through
examples, we argue that the alternating direction method of multipliers (ADMM)
is suitable for problems arising in image processing, conic programming, machine
learning, compressive sensing, as well as distributed data processing, and the
method is able to solve very large
scale problem instances. The development of this method dates back to the 1950s
and has close relationships with the Douglas-Rachford splitting, the augmented
Lagrangian method, Bregman iterative algorithms, proximal-point algorithms, etc.
This "old" method has recently become popular among researchers in image/signal
processing, machine learning, and distributed/decentralized computation. After a
brief overview, we explain its convergence behavior and then demonstrate how to
solve very large scale conic programming problems and machine learning problems
such as LASSO by ADMM.

**
A Fast Modified Newton's Method for Curvature Based Denoising of 1D Signals**

**
Andy
Yip**

In this
talk, we present a fast numerical method for denoising of 1D signals based on
curvature minimization. Motivated by the primal-dual formulation for total
variation minimization introduced by Chan, Golub, and Mulet, the proposed method
makes use of some auxiliary variables to reformulate the stiff terms presented
in the Euler-Lagrange equation which is a fourth-order differential equation. A
direct application of Newton’s
method to the resulting system of equations often fails to converge. We propose
a modified Newton’s
iteration which exhibits local superlinear convergence and global convergence in
practical settings. The method is much faster than other existing methods for
the model. Unlike all other existing methods, it also does not require tuning
any additional parameter besides the model parameter. Numerical experiments are
presented to demonstrate the effectiveness of the proposed method.

**
A primal-dual fixed point algorithm for conver separable
minimization with applications to image restoration**

**
Xiaoqun Zhang**

Recently,
minimization of a sum of two convex functions has received considerable
interests in

variational image restoration model. In this paper, we propose a general
algorithmic framework for solving separable convex minimization problem from the
point of view of fixed point algorithms based on proximity operators. Motivated
from proximal forward-backward splitting (PFBS) and fixed point algorithms based
on the proximity operator (FP2O) for image denoising,

we design a primal-dual fixed point algorithm based on proximity operator
(PDFP2O for ∈ [0, 1))

and obtain a scheme with close form for each iteration. Using the firmly
nonexpansive properties of the proximity operator and with the help of a special
norm over a product space, we achieve the convergence of the proposed
PDFP2Oalgorithm. Moreover, under some stronger assumptions, we can prove the
global linear convergence of the proposed algorithm. We also give the connection
of the proposed algorithm with other existing first order methods. Finally, we
illustrate the efficiency of PDFP2O through some numerical examples on image
supperresolution, computerized tomographic reconstruction and parallel magnetic
resonance imaging. Generally speaking, our method PDFP2O is comparable
with other state-of-the-art methods in numerical performance, while it has some
advantages on parameters selection in real applications.

**
Quantitative Photoacoustic Imaging**

**
Hongkai Zhao**

I will
present two recent works for quantitative photoacoustic imaging. Photoacoustic
is a hybrid imaging modality that can achieve ultrasound resolution for optical
contrast. Quantitative photoacoustic imaging includes two key steps. In the
first step, one has to solve an inverse source problem for the acoustic wave to
reconstruct initial acoustic source distribution from boundary measurements. We
present a Neuman series based iterative algorithm that can recover the initial
wave field efficiently and accurately. The second step is to reconstruct optical
properties of the medium using internal measurements, namely, using the
reconstructed initial acoustic source distribution from the first step. We
propose a hybrid reconstruction procedure that uses both interior measurement
and boundary current data, which is usually available in diffuse optical
tomography.