Abstract. It is argued that the complexity of fluid particle trajectories provides the basis for a new method, referred to as the Complexity Method (CM), for estimation of Lagrangian coherent structures in aperiodic flows that are measured over finite time intervals. The basic principles of the CM are explained and the CM is tested in a variety of examples, both idealized and realistic, and in different reference frames. Two measures of complexity are explored in detail: the correlation dimension of trajectory, and a new measure -the ergodicity defect. Both measures yield structures that strongly resemble Lagrangian coherent structures in all of the examples considered. Since the CM uses properties of individual trajectories, and not separation rates between closely spaced trajectories, it may have advantages for the analysis of ocean float and drifter data sets in which trajectories are typically widely and non-uniformly spaced.

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14A Wavelet Auditory Model (WAM) is constructed in tenns of wavelet frames and an irregular sampling algorithm for Fourier frames. Its theoretical effectiveness is demonstrated in the context of speech coding, and its original fonnulation is found in [8][9]. The presentation of WAM in this chapter emphasizes its underlying mathematical ideas, and, in particular, develops the notions from the theory of frames and irregular sampling that arise naturally in constructing WAM. IntroductionWe shall develop some of the basic theory of frames, as well as conse-, quences ofthat theory from the area of irregular sampling. Moreover, we shall see how these ideas playa natural role in modelling parts of the mammalian auditory process with the purpose of using this process to devise signal reconstruction algorithms in the field of speech coding.The theory of frames is due to Duffin and Schaeffer [22], and it was developed to address problems in non-harmonic Fourier series. Prior to [22], these problems were concerned with finding criteria on real sequences {tn} so that the closed linear span, span{e t }, of exponentials e t (f) = e'2rtjtnf would be equal to the space L2 [-0, 0] offinit~ energy signals defin~d on [-0, 0]. The origins of

The detection and measurement of the tortuosity -i.e. the bending and winding -of vessels has been shown to be potentially useful in the assessment of cancer progression and treatment response. Although several metrics for tortuosity are used, no single one measure is able to capture all types of tortuosity.This report presents a new multiscale technique for measuring vessel tortuosity. The approach is based on a methodcalled the ergodicity defect -which gives a scale-dependent measure of deviation from ergodicity. Ergodicity is a concept that captures the manner in which trajectories or signals sample the space; thus, ergodicity and vessel tortuosity both involve the notion of how a signal samples space. Here we begin to explore this connection.We first apply the ergodicity defect tortuosity measure to both 2D and 3D synthetic data in order to demonstrate the response of the method to three types of tortuosity observed in clinical patterns. We then implement the technique on segmented vessels extracted from brain tumor MRA images. Results indicate that the method can be effectively used to detect and measure several types of vessel tortuosity.