Geoff Davis

Postdoctoral Training

Recent reports have called for changes to the training of postdoctoral scientists and engineers. We tested the hypothesis that the practices advocated make a measurable difference in the experiences and productivity of postdoctoral researchers using data from a large-scale survey. We found that structured oversight and professional development opportunities are associated with a broad range of positive outcomes; compensation-related measures, in contrast, have few quantifiable benefits. Postdocs who wrote research/career plans at the start of their appointments were 23% more productive than those did not. Teaching experiences, exposure to non-academic careers, and training in proposal writing and project management were also associated with multiple positive outcomes.

• Geoffrey M. Davis, "Improving the Postdoctoral Experience: An Empirical Approach." In Richard B. Freeman and Daniel L. Goroff (Eds.), Science and Engineering Careers in the United States: An Analysis of Markets and Employment, Chicago, IL: University of Chicago Press, 2009. PDF
• Geoffrey M. Davis, Doctors Without Orders. American Scientist 93 (3, supplement), 2005. PDF

Nonlinear Wavelet Transforms

We examine the central issues of invertibility, stability, artifacts, and frequency-domain characteristics in the construction of non-linear analogs of the wavelet transform. The lifting framework for wavelet construction motivates our analysis and provides new insight into the problem. We describe a new type of non-linearity for use in constructing non-linear transforms: a set of linear predictors that are chosen adaptively using a non-linear selection function. We also describe how a previously studied family of non-linear filter banks can be extended through the use of prediction functions operating on a causal neighborhood. We present preliminary results for a synthetic test image.

• Roger L. Claypoole, Jr., Geoffrey M. Davis, Wim Sweldens, and Richard G. Baraniuk "Nonlinear Wavelet Transforms for Image Coding via Lifting," IEEE Transactions on Image Processing, Vol. 12 No. 12, 1449-1459, Dec. 2003. DOI: 10.1109/TIP.2003.817237 PDF
• Roger L. Claypoole, Jr., Geoffrey M. Davis, Wim Sweldens, and Richard G. Baraniuk, "Nonlinear Wavelet Transforms for Image Coding," Asilomar Conference on Signals, Systems, and Computers, Vol. 1, 662-667, 1997. DOI: 10.1109/ACSSC.1997.680528 PDF

Wavelet-based Image Coding: An Overview

This paper presents an overview of wavelet-based image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings, and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using rate-distortion considerations as well as approximation-theoretic considerations. Finally, we give an overview of current coders in the literature.

• Geoffrey M. Davis and Aria Nosratinia, "Wavelet-Based Image Coding: An Overview," Applied and Computational Control, Signals, and Circuits, Vol. 1, No. 1, Spring 1998. PDF

Multiwavelet Constructions

Lifting provides a simple method for constructing biorthogonal wavelet bases. We generalize lifting to the case of multiwavelets. In so doing we provide useful intuition about the additional degrees of freedom available in constructing multiwavelets. We show that any compactly supported multiwavelet transform can be decomposed into a sequence of lifting steps. Finally, we compare lifting to the two-scale similarity transform construction method.

• Geoffrey M. Davis, Vasia Strela, Radka Turcajova, "Multiwavelet Construction via the Lifting Scheme," in Wavelet Analysis and Multiresolution Methods, T.-X. He (editor), Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1999. PDF

Science Policy

Young mathematicians have been facing dismal job prospects throughout the nineties. What are the effects of the current labor market problems on the mathematics community as a whole? What forces have contributed to these problems? What are effective remedies? We address each of these questions, providing partial answers when data exists, and pointing out the key gaps in our current understanding. We conclude by describing some specific steps that the mathematical societies can take to improve the current labor market situation for mathematics Ph.D.s.

• Geoffrey M. Davis, "Mathematicians and the Market," Notices of the American Mathematical Society, Vol. 44, No. 10, pp. 1307-1315, 1997. PDF HTML

Significance Tree Quantization

A number of recent embedded transform coders, including Shapiro's EZW scheme, Said and Pearlman's SPIHT scheme, and Xiong et al.'s EZDCT scheme employ a common algorithm called significance tree quantization (STQ). Each of these coders have been selected from a large family of significance tree quantizers based on empirical work and a priori knowledge about transform coefficient behavior. We describe an algorithm for selecting a particular form of STQ that is optimized for a given class of images. We apply our optimization procedure to the task of quantizing 8x8 DCT blocks. Our algorithm yields a fully embedded, low-complexity coder with performances from 0.6 to 1.9 dB better than baseline JPEG for standard test images.

• Geoffrey M. Davis and Sumit Chawla, "Image Coding Using Optimized Significance Tree Quantization," Proc. Data Compression Conference, 1997, 387-396, 1997. DOI: 10.1109/DCC.1997.582064 PDF

Joint Source and Channel Coding

We describe a joint source/channel allocation scheme for transmitting images lossily over a block erasure channel such as the Internet. The goal is to reduce image transmission latency. Our subband-level and bitplane-level optimization procedures give rise to an embedded channel coding strategy. Source and channel coding bits are allocated in order to minize an expected distortion measure. More perceptually important low frequency subbands of images are shielded heavily using channel codes, and higher frequencies are shielded lightly. The result is a more efficient use of channel codes that can reduce channel coding overhead. The reduction is most pronounced on bursty channels for which the uniform application of channel codes is particularly expensive. We derive optimal source/channel coding tradeoffs for our block erasure channel. We describe an implementation of this algorithm and compare its performance on the Internet to lossless TCP/IP transmission of the same images. In our experiments our lossy image transmission scheme delivered images significantly faster than TCP/IP during periods of heavy traffic.

• Geoffrey M. Davis and John Danskin, "Joint Source and Channel Coding for Image Transmission Over Lossy Packet Networks," SPIE Conference on Wavelet Applications of Digital Image Processing XIX, Denver, August 1996. PDF
• John Danskin, Geoffrey M. Davis, and Xiyong Song, "Fast Lossy Internet Image Transmission," ACM Multimedia, San Francisco, November 1995. PDF

A Wavelet-Based Analysis of Fractal Image Compression

Recipient of the 2000 IEEE Leon K. Kirchmayer Prize Paper Award

Why does fractal image compression work? What is the implicit image model underlying fractal block coding? How can we characterize the types of images for which fractal block coders will work well? These are the central issues we address in the papers below. We introduce a new wavelet-based analytical framework for block-based fractal compression schemes. Within this framework we are able to draw upon insights from the well-established transform coder paradigm in order to address the issue of why fractal block coders work. We show that fractal block coders are a form of Haar wavelet subtree quantization scheme. We generalize fractal block coders to smooth wavelet subtree coders and in so doing obtain a simpler fractal-based coding scheme that gives performance comparable to the best fractal schemes in the literature. Our wavelet framework gives new insight into the convergence properties of fractal block coders, and leads us to develop an unconditionally convergent scheme with a fast decoding algorithm. Our experiments with this new algorithm indicate that fractal coders derive much of their effectiveness from their ability to efficiently represent wavelet zerotrees. Finally, our framework enables us to examine some of the fundamental limitations of current fractal compression schemes.

• Geoffrey M. Davis, "A Wavelet-based Analysis of Fractal Image Compression," IEEE Transactions on Image Processing, Vol. 7, No. 2, 141-154, Feb. 1998. Recipient of the 2000 IEEE Leon K. Kirchmayer Prize Paper Award. DOI: 10.1109/83.660992 PDF
• Geoffrey M. Davis, "Implicit Image Models in Fractal Image Compression," Invited paper, SPIE Conference on Wavelet Applications in Signal and Image Processing IV, Denver, August 1996. PDF
• Geoffrey M. Davis, "Adaptive Self-Quantization of Wavelet Subtrees: A Wavelet-Based Theory of Fractal Image Compression," SPIE Conference Wavelet Applications in Signal and Image Processing III, San Diego, July 1995. Earlier versions of this work appeared at the 1995 IEEE Data Compression Conference in Snowbird, Utah and at SPIE Wavelet Applications for Dual Use 1995, Orlando. PDF