Summer 2009 REU Projects

Problem: What is the best dimension reduction method for hyperspectral imagery?

While a standard color image contains 3 bands for Red Green Blue (RGB) light, a hyperspectral image typically contains about 200 bands. Each band is a grayscale image representing the sensor response to a particular wavelength of light. Often the relevant information in the image can be described in a much smaller number of image bands. That is, a 200 band image can often be compressed to a 4 band image that still describes the important image classes (tree, grass, bulding, etc.). Many different dimension reduction (DR) methods have been developed. These methods include classical linear methods such as PCA, ICA, and MDA. Nonlinear methods such as ISOMAP, Diffusion Maps, and LLE have proven to be more effective in reducing the dimension while still preserving the image information, but these methods are much slower and generally have large memory requirements. In 2008, researchers developed hybrid linear methods such as GPCA that represent a compromise between the speed of the linear methods and the effectiveness of the nonlinear methods. To our knowledge, hybrid linear methods have not been tested on hyperspectral data.

Despite the wide variety of DR methods and the availability of code, there has been little work done in comparing these methods and deciding which DR method is best in a given situation. A student team will test the effect of DR methods on the three most important tasks in hyperspectral image processing: classification, anomaly/target detection, and linear demixing. Students will be expected to gather data sets, in some cases preparing synthetic images for testing purposes. A major part of this project will involve developing metrics and numerical experiments for comparing the performance of the DR methods. In other words, the students will be asked to determine the best DR method for hyperspectral images, but the hard part is defining what is meant by the word "best".

Prerequisites: Students should have a background in linear algebra. Some background in statistics and numerical analysis would be helpful, but is not required.

Problem: Can we develop a mathematical model of criminal behavior that reflects real crime data?

In a differential equations course, one of the first partial differential equations that students encounter is the 1D heat equation:

Researchers at UCLA have recently developed a differential equation model to explain the movement of criminals. This model generates crime patterns which qualitatively seem reasonable, but are based on purely theoretical assumptions and not real situations. Using crime data provided by the Los Angeles and Long Beach police departments as well as GIS data for California, a student team will fit the model to real data. The hope is to produce a model of real crime in California that can be used for tracking and prediction. An accurate model of criminal activity might help the police develop law enforcement strategies to deter that activity. The model fitting process could also reveal the nature of criminal behavior by giving a better understanding of the forces which drive the model, such as criminal mobility and neighborhood attractiveness.

Prerequisites: Students should have taken a basic course in differential equations and have some familiarity with partial differential equations. A background in numerical analysis and optimization would also be helpful.

Problem: Can we improve the LAPD's crime analysis software by incorporating GIS data?

The Los Angeles Police Department (LAPD) uses sophisticated software for tracking criminal behavior. This software gives the police a better understanding of criminal activity and helps them develop strategies to deter crime and apprehend criminals. For example, a computer program might examine patterns arising in incidents of auto theft to extrapolate likely locations of chop shops. A student team will gain familiarity with the LAPD software and evaluate its strengths and weaknesses. The students will then try to improve and extend the existing software. In particular, there is a desire to merge the software with GIS data to give more realistic results. For example, when extrapolating the location of an auto chop shop we could rule out many areas such as bodies of water, public parks, cemeteries, etc.

Prerequisites: Students should have experience with a high-level programming language such as C++. A background in statistics and graph theory would be helpful, but is not required.

Problem: How do we register two images from different modalities?

There are many applications when two images need to be lined up. For example, suppose we want to evaluate the effect of global warming by comparing the ice shelf over the Arctic today against a picture of the Arctic ten years ago. Before we can compute the difference, we first need to accurately align the images, a process called "image registration". Image registration is a necessary first step for many tasks such as change detection, image fusion, and pan-sharpening. In general, image registration is a very difficult problem. In our example of Arctic images, we want to register the images so we can detect the differences, but it is difficult to align the images because they are different!

Many image registration methods have been developed, but the choice of registration method appears to be application dependent. A method developed for registering MRI brain images may not be appropriate for registering satellite images of the earth. Focusing on data gathered from earth-observing satellites (like pictures on Google Maps), students will test a variety of image registration methods. The students will investigate both rigid and non-rigid registration techniques as well as both area-based and feature-based approaches. Possible methods include normalized cross correlation, variational methods, wavelet-based methods, and feature matching. Students will examine existing software for image registration such as the Matlab central library, ITK, and AIR. A particular area of interest is to register images from different modalities, such as a color image and a hyperspectral image. This topic is of particular interest to faculty at UCLA working on image fusion.

Prerequisites: Students should have some experience with computer programming. Some background in image processing would be helpful, but is not required.

Problem: How do we detect important changes in aerial images?

Suppose we have two pictures taken from an airplane over Chicago, one taken during the summer and one during the winter. We want to detect important differences, like new buildings sprouting up. But pixel by pixel, the images will be very different. The leaves have fallen off the trees, the shadows from the tall buildings are in different positions, and there is snow everywhere. Locating the non-trivial changes cannot be done by simply looking for changes in color between two corresponding pixels. Given two images of the same scene, the process of identifying and locating important differences is called change detection. Of course, this requires a definition of what constitutes a real "change," an ambiguity which makes the problem even harder. Change detection has important applications to surveillance, environmental monitoring, weather tracking, and population studies.

A student team will examine different change detection techniques and evaluate their performance on aerial imagery. The students will learn about hypothesis testing and shading and background models. The students will gather aerial datasets and test different change detection algorithms. In particular, students will investigate if sparsity constraints can improve change detection in situations when the changes are assumed to be on a small scale.

Prerequisites: Students should have some background in statistics. A background in image processing would be helpful, but is not required.

Problem: Can we guide the path of a cantilever on an atomic force microscope to reduce the imaging time?

Atomic Force Microscopy (AFM) is a revolutionary technology which allows researchers to view materials at the molecular level. Scientists use AFM to study the structure of materials and to observe processes such as the binding of proteins. But AFM doesn't work like a camera that takes snapshot of a scene in one flash. AFM more closely resembles a record player, where a tiny needle called the cantilever gently rides the material surface, recording the precise height as it moves. Typically, the cantilever moves back and forth across the material row by row until an entire image is scratched out. Gathering one image typically takes 10 seconds. This is acceptable when the material being studied is static, but in some applications the material is in motion. For example, biologists would like to observe the microscale binding of proteins, a process which lasts less than 1 second. So by the time the cantilever has arrived at the protein location, the process we were trying to observe has already finished! Rather than gathering an image of the entire material surface, an alternative would be to localize the cantilever over the protein, in effect "cropping" the image to a specific area.

A student team will investigate techniques for guiding the cantilever to a region of interest, with the ultimate goal of reducing the time taken to gather the image. Using ideas from boundary tracking and pattern recognition, students will develop a simulator in Matlab that shows how the cantilever path can be restricted to a region. Note that the region of interest depends on the application, so the students will examine data from different sources such as the crystalline structure of semiconductors and the intertaction of proteins.

Prerequisites: Students should have some experience with computer programming. Some background in image processing would be helpful, but is not required.