Dates:  Jan 17-18, 2008.

Recently several power methods have been discovered for minimization problems related to level set methods and total variation minimization.

We shall use this two-day workshop to get people interested in these applications together in CIPR Bergen. We shall try to give several lectures on methods related to graph-cut and mean cut. Then we shall present our own research results. In the end, we shall set time to have discussions to see the possibility to use these fast numerical techniques for our problems.

If you are interested to present something related to level set methods and fast algorithms, please send me a message. People just interested to participate, please drop me a message. List of participants and the final program will be available in the webpage.

Sincerely Xuecheng Tai

tai@math.uib.no

Final schedule:

Time: Jan 17-18, 2008.

Location: CIPR, Allegate 41, Bergen.

Jan 17:

12:00 -- 13:00, Lunch, CIPR, room 4060, Bergen.

13:15 –14:00, Geir Dahl, CMA, "Minimum cuts: basic theory and algorithms".

14:00-14:45, Truls Flatberg, CMA and SINTEF, "Minimum cuts: applications in image

analysis".

14:45-15:15, coffee break

15:15-16:00, Xue-Cheng Tai, CIPR and CMA, “ Piecewise constant level set

methods and integer programming”.

16:15-16:45, Ørjan Knudsen, Mathematics Institute, UIB, “Piecewise Constant Level

Set Methods for Multiple-Surface Segmentation and Reconstruction Problems”.

Jan 18:

09:00-09:45, Eigil Samset, CMA and the interventional Center, UIO, “Image

processing for medical applications”.

09:45—10:15, Egil Bae, Mathematics Institute, UIB, "PDEs for interpolation and

compression of surfaces".

10:15-10:45, coffee break

10:45—11:15, S. Bruvoll* and M. S. Floater, CMA, UIO, “Transfinite mean value

interpolation in general dimemsion”.

11:15-11:45, S. Bruvoll, CMA, UIO, “Network flow theory based disparity maps".

12:00-13:00 Lunch, Lunch, CIPR, room 4060, Bergen.

13:00 -13:30, David Moreno, CIPR, UIB, “Coupling level set methods and the EnKF

for permeability identification and reconstruction”.

13:30-14:00, Hongwei Li, CIPR, UIB, “Permeability estimation with the PCLSM”.

14:00-15:00, discussions.

Abstracts

Speaker: Geir Dahl, CMA.

Title: "Minimum cuts: basic theory and algorithms"

Abstract: The purpose of this talk is to present the minimum cut problem in a graph and its relationship to

another problem, the maximum flow problem. These problems belong a set of core problems in

combinatorial optimization/network flows that have a great number of applications, in various fields, and for

which highly efficient algorithms exist. After presenting the problems we discuss some basic theoretical

results and how these results lead to efficient algorithms for solving the problems. The talk is intended for

those who are new to this area of applied mathematics.

Speaker: Truls Flatberg, CMA and SINTEF.

Title: "Minimum cuts: applications in image analysis"

Abstract: Minimum cuts and corresponding maximum flow algorithms have recently become popular in

image processing, often profiled as "graph cuts". In this talk we will give an introduction to the use of

minimum cuts, and cuts in general, in image analysis with a focus on image segmentation and restoration.

Speaker: Xue-Cheng Tai

Title: “ Piecewise constant level set methods and integer programming”.

Abstract: Level set method has proven to be a versatile tool for many interface problems. It has application to

be useful for image processing, computer vision, shape design, inverse problems. In this talk, we show that

we can use piecewise constant level set methods to replace the traditional level set methods. By doing so, we

have transferred many interface problems into integer programming problems. Thus, fast method related to

graph-cut, mean-cut can potentially be efficient methods for solving this kind of problems.

Speaker: Ørjan Knudsen.

Title: Piecewise Constant Level Set Methods for Multiple-Surface Segmentation/Reconstruction Problems.

Abstract: The Piecewise Constant Level Set Method (PCLSM) is an interesting formulation for many

problems in image processing. Like other level set methods, it simplifies the process of tracking moving

boundaries in 2-D and 3-D. What sets it apart from other level set methods such as the signed distance level

set method (SDLSM) is the effective representation of multiple phases or regions of interest (ROIs), as well

as avoiding the need for costly re-initialization of the level set function. We follow a probabilistic approach

to surface reconstruction after Kolev, Brox and Cremers(2006), and modify it to support the PCLSM. Results

from the application of this modified algorithm to various problems will be shown.

Speaker: Eigil Samset, CMA and the interventional Center, UIO

Title: Image processing for medical applications

Abstract: "Image Processing plays an important role in computer aided diagnosis and image-guided surgery.

Many of the current methods fall short in addressing the requirements for real-time and autonomous

performance that are present in cases where intra-operative imaging is being used. Some examples will be

presented showing how advanced image progressing can be used intra-operatively as well as examples of

unsolved challenges in this area"

Speaker: Egil Bae, Mathematics Institute, UIB

Title: PDEs for interpolation and compression of surfaces

Abstract: "Recently, there has been a growing attention to partial differential equations (PDEs) for

interpolation and compression of images. In this talk we will present similar PDE based methods for

interpolation of surfaces from point clouds, and show its application to compression of surface data.

Triangulated surfaces will be the main attention when presenting experiments and our choice of

discretization technique. Since the underlying continuous equations are most important, this talk is also

intended for those who are interested in the level set method or other discretization techniques."

Speaker: S. Bruvoll, University of Oslo, Norway. solveio@ifi.uio.no.

Title: Transfinite mean value interpolation in general dimemsion

Abstract: In this talk we study mean value interpolation over domains in general dimension. We derive

conditions on the boundary of the domain to guarantee interpolation when the data is continuous. By deriving

the normal derivative of the interpolant and of a mean value weight function we construct a transfinite

Hermite interpolant.

Speaker: Solveig Bruvoll

Title: Network flow theory based disparity maps.

Abstract: Given a pair of stereo images produced by aligned cameras, we explain how network flow theory

can be applied to determine a disparity map, i.e. a table containing the point correspondences of

the two images. We construct an energy function concerning intensity difference, occlusion and smoothness.

By the construction of a graphcorresponding to the energy function, we are able to minimize

the energy functional through application of the Max-flow min-cut theorem from network flow theory.

Speaker: David Meron.

Title: Coupling level set methods and the EnKF for permeability identification and reconstruction.

Abstract: A methodology for continuous model updating based on the ensemble Kalman filter(EnKF) and the

level set method is presented. We show how by using the level

set method in combination with the EnKF we have managed to identify and reconstruct the underlaying

permeability of an unknown reservoir distribution. The method allows the incorporation of prior information

about the reservoir and does not require time consuming adjoint or gradient calculations and it is well

suitable for continuous data incorporation and assimilation. Applications to a synthetic and an outcrop case

are presented.

Speaker: Hongwei Li

Title: Permeability estimation with the PCLSM

Abstract: We consider the case that permeability field can be represented by some piecewise constant

function. To condition the reservoir by adjusting this permeability field, we combine the Piecewise

constantlevel set method (PCLSM) with the adjoint method. Numerical experiments will be presented to

showthe effectiveness of the proposed method.