Noiselets are functions which are noise-like in the sense that they are totally uncompressible by orthogonal wavelet packet methods. We describe a library of such functions and demonstrate a few of their noise-like properties. © 2001 Academic Press INTRODUCTIONAs the reader undoubtedly knows, various effective algorithms exist for using wavelets and wavelet packets to process data, for example, for compression or noise removal. In these algorithms, analysis of data is achieved because one is able to find rapid decay in the distribution of values of the data, when it is transformed into wavelet or wavelet packet bases.In practice one finds that the few large values in the transformed data describe the interesting part of the data, and the vast majority of values, which are small, represent a noise term. See, for example, [6].The performance of these algorithms is impressive and might lull one into the belief that analysis of any "interesting" structure can be carried out via wavelet packet analysis. Of course this cannot be so, and this paper gives constructions of large families of functions which give worst case behavior for orthogonal wavelet packet compression schemes.Noiselets are functions which give worst case behavior for the aforementioned type of orthogonal wavelet packet analysis. In particular, this paper gives explicit examples of (complex-valued) noiselets for which all Haar-Walsh wavelet packet coefficients have exactly the same absolute value. So, in some sense, noiselets are "noise-like," and in particular, noiselets are totally uncompressible by orthogonal wavelet packet methods.Although noiselets are noise-like in the sense of being spread in time and frequency, there are patterns lurking in them. Certain families of noiselets arise as bases for the spaces of the Haar multiresolution analysis. These bases are computationally good in the same way that wavelet packets are; they come with fast algorithms for forward and inverse transforms and there are trees of bases with the structure needed to support the best-basis algorithm. These good properties of noiselets are no coincidence. Noiselets are constructed via a multiscale iteration in exactly the same way as wavelet packets, but with a twist. So in some sense noiselets have the structure of wavelet packets. Because of this fast computational structure, the possibility exists that noiselets will be valuable tools for certain applications, rather than simply representing counterexamples.Another source of pattern within noiselets is that one finds within their construction certain classical fractal generating mechanisms. In fact, a whole class of noiselets are nothing but the distributional derivatives of the classical paper folding curves (see [4] for an introduction to paper folding). Hence noiselets provide a counterexample to the philosophical view of analysis with which this note began. Indeed, one sees that certain interesting multiscale mechanisms can produce well-organized data which are nonetheless invisible to our standard analysis tools...

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We apply a unique micro-optoelectromechanical tuned light source & new algorithms to the hyper-spectral microscopic analysis of human colon biopsies. The tuned light prototype (Plain Sight Systems Inc.) transmits any combination of light frequencies, range 440nm 700nm, trans-illuminating H & E stained tissue sections of normal (N), benign adenoma (B) and malignant carcinoma (M) colon biopsies, through a Nikon Biophot microscope. Hyper-spectral photomicrographs, randomly collected 400X magnication, are obtained with a CCD camera (Sensovation) from 59 different patient biopsies (20 N, 19 B, 20 M) mounted as a microarray on a single glass slide. The spectra of each pixel are normalized & analyzed to discriminate among tissue features: gland nuclei, gland cytoplasm & lamina propria/lumens. Spectral features permit the automatic extraction of 3298 nuclei with classification as N, B or M. When nuclei are extracted from each of the 59 biopsies the average classification among N, B and M nuclei is 97.1%; classification of the biopsies, based on the average nuclei classification, is 100%. However, when the nuclei are extracted from a subset of biopsies, and the prediction is made on nuclei in the remaining biopsies, there is a marked decrement in performance to 60% across the 3 classes. Similarly the biopsy classification drops to 54%. In spite of these classification differences, which we believe are due to instrument & biopsy normalization issues, hyper-spectral analysis has the potential to achieve diagnostic efficiency needed for objective microscopic diagnosis.

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