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Math 290J (section 2) current literature in applied mathematics
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Mathematical models in image processing and analysis

**Organizer:** Luminita Vese. E-mail: lvese@math.ucla.edu

**Presentations, Time and Location:**

Please note that some meetings will be at a different time and location. All meetings will
be announced by email and posted on the class webpage.

** Format:**
Graduate students give presentations from current literature in the field. Sometimes, we
may also have a guest lecturer.

Monday Jan 10, 4-5pm, MS 5117

Guest Lecturer: Rachel Ward (Courant Institute)
* Fast Dimensionality Reduction: Improved Bounds and Implications for Compressed Sensing *

**Abstract** Embedding high-dimensional data sets into subspaces of much lower dimension is important for reducing storage cost and speeding up computation in several applications, including numerical linear algebra, manifold learning, and computer science. The relatively new field of compressed sensing is based on the observation that if the high-dimensional data are sparse in a known basis, they can be embedded into a lower-dimensional space in a manner that permits their efficient recovery through l1-minimization. We first give a brief overview of compressed sensing, and discuss how certain statistical procedures like cross validation can be naturally incorporated into this set-up. The latter part of the talk will focus on a "near equivalence" of two fundamental concepts in compressed sensing: the Restricted Isometry Property and the Johnson-Lindenstrauss Lemma; as a consequence of this result, we can improve on the best-known bounds for dimensionality reduction using structured, or "fast" linear embeddings. Finally, we discuss the Restricted Isometry Property for structured measurement matrices formed by subsampling orthonormal polynomial systems, and high-dimensional function approximation from a few samples.

Wednesday, January 13, 4-5pm, MS 6229

Guest Lecturer: Corey Toler-Franklin (Princeton University , Computer Science)
*Matching, Visualizing and Archiving Cultural Heritage Artifacts Using Multi-Channel
Images *

** Abstract** Recent advancements in low-cost acquisition technologies have made it more practical to acquire real-world datasets on a large scale. This has lead to a number of computer-based solutions for reassembling, archiving and visualizing cultural heritage artifacts. In this talk, I will show how we can combine aspects of these technologies in novel ways and introduce algorithms to improve upon their overall efficiency and robustness. First, I will introduce a 2D acquisition system to address the challenge of acquiring higher resolution color and normal maps than those available with 3D scanning devices. Next, I will incorporate these normal maps into a novel multi-cue matching system that uses machine learning to reassemble small fragments of artifacts. I will show examples of how this system is used by archaeologists at the Akrotiri Excavation Laboratory of Wall Paintings in Santorini Greece for reconstructing the Theran Frescoes. I will then present a non-photorealistic render ing pipeline for illustrating geometrically complex objects using images with multiple channels of information (including RGBN images). Finally, I will talk about an approach for visualizing historic artifacts from digital museum collections.
**Bio** Corey Toler-Franklin is a Ph.D. student in computer science at Princeton University. Her research area is Computer Graphics, focusing on algorithms for acquiring, reassembling, and visualizing complex real-world datasets of cultural heritage significance. Corey?s most recent projects include the development of a multi-feature matching system for reassembling the Theran frescoes at the Akrotiri excavation site in Santorini Greece, and a non-photorealistic rendering pipeline for visualizing museum collections. Corey obtained a MS degree in Computer Graphics and a Bachelor's degree in Architecture from Cornell University. Before joining the Princeton Graphics Group, she worked as a Software Engineer at Autodesk where she implemented platform enhancements to the 3D Graphics System of AutoCAD. She also led a pilot project between Autodesk and two international architecture firms, HOK and Gensler, to encourage the adoption of new computer technologies by the design industry.

Wednesday, January 19, 4-5pm, MS 6229

Guest Lecturer: Neus Sabater (Caltech)
*Reliability and Accuracy in Stereovision (Joint work with J.-M. Morel and A. Almansa)*

**Abstract** This work is a contribution to stereovision written in the framework of the MISS (Mathematics for Stereoscopic Space Imaging) project launched by CNES (French Space Agency). This project has the ambitious goal to model a stereo satellite, using two almost simultaneous views of the Earth with small baseline in urban areas. Its main goal is to get an automatic chain of urban reconstruction at high resolution from such pairs of views. The project faces fundamental problems that our work aims at solving. The first problem is the rejection of matches that could occur just by chance, particularly in shadows or occlusions, and the rejection of moving objects (vehicles, pedestrians, etc.). In this work we have proposed a method for rejecting false matches based on the "a contrario" methodology. The mathematical consistency of this rejection method will be shown and it will be validated. The reliable accepted matches reach a 40% to 90% density in the tested pairs.
The second issue is the accuracy. Indeed, the type of considered stereoscopy requires a very low baseline between the two views, which are visually almost identical. To get a proper relief, an extremely accurate shift must be estimated, and the noise level that allows this accuracy must be calibrated. In this work a subpixel disparity estimation method is proposed, which will be proved optimal by experimental and mathematical arguments.

Wednesday, January 26, 4-5pm, MS 6229

Guest Lecturer: Jerome Gilles, UCLA
* Turbulence Image Restoration*

** Abstract** In this talk, we address the problem of image restoration in the case of turbulence deformations. Indeed, in long range imaging, the optical path is subject to the atmosphere behavior. One of the most limiting effect is the deformation induced by the turbulence. In order to correct this effect, we propose to model it as a deformation field which under a weak assumption can be written as a linear convolution operator. Then, we introduce this operator in a classical or non-local total variation framework. We minimize it by an iterating scheme where we estimate both the deformation field and the restored image. Numerical experiments shows that this algorithm needs relatively less iterations and converges fast.