Frequency domain of weakly translation invariant frame MRAs

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

Body Region-Specific 3D Age-Scaling Functions for Scaling Whole-Body Computed Tomography Anatomy for Pediatric Late Effects Studies

Purpose: Radiation epidemiology studies of childhood cancer survivors treated in the pre-computed tomography (CT) era reconstruct the patients’ treatment fields on computational phantoms. For such studies, the phantoms are commonly scaled to age at the time of radiotherapy treatment because age is the generally available anthropometric parameter. Several reference size phantoms are used in such studies, but reference size phantoms are only available at discrete ages (e.g.: newborn, 1, 5, 10, 15, and Adult). When such phantoms are used for RT dose reconstructions, the nearest discrete-aged phantom is selected to represent a survivor of a specific age. In this work, we (1) conducted a feasibility study to scale reference size phantoms at discrete ages to various other ages, and (2) evaluated the dosimetric impact of using exact age-scaled phantoms as opposed to nearest age-matched phantoms at discrete ages. Methods: We have adopted the University of Florida/National Cancer Institute (UF/NCI) computational phantom library for our studies. For the feasibility study, eight male and female reference size UF/NCI phantoms (5, 10, 15, and 35 years) were downscaled to fourteen different ages which included next nearest available lower discrete ages (1, 5, 10 and 15 years) and the median ages at the time of RT for Wilms’ tumor (3.9 years), craniospinal (8.0 years), and all survivors (9.1 years old) in the Childhood Cancer Survivor Study (CCSS) expansion cohort treated with RT. The downscaling was performed using our in-house age scaling functions (ASFs). To geometrically validate the scaling, Dice similarity coefficient (DSC), mean distance to agreement (MDA), and Euclidean distance (ED) were calculated between the scaled and ground-truth discrete-aged phantom (unscaled UF/NCI) for whole-body, brain, heart, liver, pancreas, and kidneys. Additionally, heights of the scaled phantoms were compared with ground-truth phantoms’ height, and the Centers for Disease Control and Prevention (CDC) reported 50th percentile height. Scaled organ masses were compared with ground-truth organ masses. For the dosimetric assessment, one reference size phantom and seventeen body-size dependent 5-year-old phantoms (9 male and 8 female) of varying body mass indices (BMI) were downscaled to 3.9-year-old dimensions for two different radiation dose studies. For the first study, we simulated a 6 MV photon right-sided flank field RT plan on a reference size 5-year-old and 3.9-year-old (both of healthy BMI), keeping the field size the same in both cases. Percent of volume receiving dose ≥ 15 Gy (V15) and the mean dose were calculated for the pancreas, liver, and stomach. For the second study, the same treatment plan, but with patient anatomy-dependent field sizes, was simulated on seventeen body-size dependent 5- and 3.9-year-old phantoms with varying BMIs. V15, mean dose, and minimum dose received by 1% of the volume (D1), and by 95% of the volume (D95) were calculated for pancreas, liver, stomach, left kidney (contralateral), right kidney, right and left colons, gallbladder, thoracic vertebrae, and lumbar vertebrae. A non-parametric Wilcoxon rank-sum test was performed to determine if the dose to organs of exact age-scaled and nearest age-matched phantoms were significantly different (p<0.05). Results: In the feasibility study, the best DSCs were obtained for the brain (median: 0.86) and whole-body (median: 0.91) while kidneys (median: 0.58) and pancreas (median: 0.32) showed poorer agreement. In the case of MDA and ED, whole-body, brain, and kidneys showed tighter distribution and lower median values as compared to other organs. For height comparison, the overall agreement was within 2.8% (3.9cm) and 3.0% (3.2cm) of ground-truth UF/NCI and CDC reported 50th percentile heights, respectively. For mass comparison, the maximum percent and absolute differences between the scaled and ground-truth organ masses were within 31.3% (29.8g) and 211.8g (16.4%), respectively (across all ages). In the first dosimetric study, absolute difference up to 6% and 1.3 Gy was found for V15 and mean dose, respectively. In the second dosimetric study, V15 and mean dose were significantly different (p<0.05) for all studied organs except the fully in-beam organs. D1 and D95 were not significantly different for most organs (p>0.05). Conclusion: We have successfully evaluated our ASFs by scaling UF/NCI computational phantoms from one age to another age, which demonstrates the feasibility of scaling any CT-based anatomy. We have found that dose to organs of exact age-scaled and nearest aged-matched phantoms are significantly different (p<0.05) which indicates that using the exact age-scaled phantoms for retrospective dosimetric studies is a bette

Crossover scaling functions in the asymmetric avalanche process

We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. The exact expressions for the first two scaled cumulants of the particle current are obtained in the large time limit t ! ∞ via the Bethe ansatz and a perturbative solution of the TQ-equation. The results are presented in an integral form suitable for the asymptotic analysis in the large system size limit N ! ∞. In this limit the first cumulant, the average current per site or the average velocity of the associated interface, is asymptotically finite below the critical density and grows linearly and exponentially times power law prefactor at the critical density and above, respectively. The scaled second cumulant per site, i.e. the diffusion coefficient or the scaled variance of the associated interface height, shows the O(N-1⁄2) decay expected for models in the Kardar-Parisi-Zhang universality class below the critical density, while it is growing as O(N3⁄2) and exponentially times power law prefactor at the critical point and above. Also, we identify the crossover regime and obtain the scaling functions for the uniform asymptotics unifying the three regimes. These functions are compared to the scaling functions describing crossover of the cumulants of the avalanche size, obtained as statistics of the first return area under the time space trajectory of the Vasicek random process.

-NONUNIFORM MULTIRESOLUTION ANALYSIS

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

Optimization in the construction of cardinal and symmetric wavelets on the line

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

In this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.