The average singular value of a complex random matrix decreases with dimension

We obtain a recurrence relation in $d$ for the average singular value $\alpha (d)$ of a complex valued $d\times d$\ matrix $\frac{1}{\sqrt{d}}X$ with random i.i.d., N( 0,1) entries, and use it to show that $\alpha (d)$ decreases monotonically with $d$ to the limit given by the Marchenko-Pastur distribution.\ The monotonicity of $\alpha (d)$ has been recently conjectured by Bandeira, Kennedy and Singer in their study of the Little Grothendieck problem over the unitary group $\mathcal{U}_{d}$ \cite{BKS}, a combinatorial optimization problem. The result implies sharp global estimates for $\alpha (d)$, new bounds for the expected minimum and maximum singular values, and a lower bound for the ratio of the expected maximum and the expected minimum singular value. The proof is based on a connection with the theory of Tur\'{a}n determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.Comment: 11 page

Simultaneous control of volumetric light distribution through turbid media using real-time three-dimensional optoacoustic feedback

Focusing light through turbid media presents a highly fascinating challenge in modern biophotonics. The unique capability of optoacoustics for high resolution imaging of light absorption contrast in deep tissues can provide a natural and efficient feedback to control light delivery in scattering medium. While basic feasibility of using optoacoustic readings as a feedback mechanism for wavefront shaping has been recently reported, the suggested approaches may require long acquisition times making them challenging to be translated into realistic tissue environments. In an attempt to significantly accelerate dynamic wavefront shaping capabilities, we present here a feedback-based approach using real-time three-dimensional optoacoustic imaging assisted with genetic-algorithm-based optimization. The new technique offers robust performance in the presence of noisy measurements and can simultaneously control the scattered wave field in an entire volumetric region.Comment: 4 pages, 3 figure

A qq-linear analogue of the plane wave expansion

We obtain a $q$-linear analogue of Gegenbauer's expansion of the plane wave. It is expanded in terms of the little $q$-Gegenbauer polynomials and the \textit{third} Jackson $q$-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.Comment: 12 pages, to appear in Adv. in Appl. Math. arXiv admin note: text overlap with arXiv:0909.006

Visual Quality Enhancement in Optoacoustic Tomography using Active Contour Segmentation Priors

Segmentation of biomedical images is essential for studying and characterizing anatomical structures, detection and evaluation of pathological tissues. Segmentation has been further shown to enhance the reconstruction performance in many tomographic imaging modalities by accounting for heterogeneities of the excitation field and tissue properties in the imaged region. This is particularly relevant in optoacoustic tomography, where discontinuities in the optical and acoustic tissue properties, if not properly accounted for, may result in deterioration of the imaging performance. Efficient segmentation of optoacoustic images is often hampered by the relatively low intrinsic contrast of large anatomical structures, which is further impaired by the limited angular coverage of some commonly employed tomographic imaging configurations. Herein, we analyze the performance of active contour models for boundary segmentation in cross-sectional optoacoustic tomography. The segmented mask is employed to construct a two compartment model for the acoustic and optical parameters of the imaged tissues, which is subsequently used to improve accuracy of the image reconstruction routines. The performance of the suggested segmentation and modeling approach are showcased in tissue-mimicking phantoms and small animal imaging experiments.Comment: Accepted for publication in IEEE Transactions on Medical Imagin