A bound for the range of a narrow light beam in the near field.

We investigate the possibility of light beams that are both narrow and long range with respect to the wavelength. On the basis of spectral electromagnetic field representations, we have studied the decay of the evanescent waves, and we have obtained some bounds for the width and range of a light beam in the near-field region. The range determines the spatial bound of the near field in the direction of propagation. For a number of representative examples we found that narrow beams have a short range. Our analysis is based on the uncertainty relations between spatial position and spatial frequency.

Ultrasonic Synthetic-Aperture Interface Imaging.

Synthetic-aperture (SA) imaging is a popular method to visualize the reflectivity of an object from ultrasonic reflections. The method yields an image of the (volume) contrast in acoustic impedance with respect to the embedding. Typically, constant mass density is assumed in the underlying derivation. Due to the band-limited nature of the recorded data, the image is blurred in space, which is quantified by the associated point spread function. SA volume imaging is valid under the Born approximation, where it is assumed that the contrast is weak. When objects are large with respect to the wavelength, it is questionable whether SA volume imaging should be the method-of-choice. Herein, we propose an alternative solution that we refer to as SA interface imaging. This approach yields a vector image of the discontinuities of acoustic impedance at the tissue interfaces. Constant wave speed is assumed in the underlying derivation. The image is blurred in space by a tensor, which we refer to as the interface spread function. SA interface imaging is valid under the Kirchhoff approximation, where it is assumed that the wavelength is small compared to the spatial dimensions of the interfaces. We compare the performance of volume and interface imaging on synthetic data and on experimental data of a gelatin cylinder with a radius of 75 wavelengths, submerged in water. As expected, the interface image peaks at the gelatin-water interface, while the volume image exposes a peak and trough on opposing sides of the interface.

A multiplicative regularization approach for deblurring problems.

In this work, an iterative inversion algorithm for deblurring and deconvolution is considered. The algorithm is based on the conjugate gradient scheme and uses the so-called weighted L2-norm regularizer to obtain a reliable solution. The regularizer is included as a multiplicative constraint. In this way, the appropriate regularization parameter will be controlled by the optimization process itself. In fact, the misfit in the error in the space of the blurring operator is the regularization parameter. Then, no a priori knowledge on the blurred data or image is needed. If noise is present, the misfit in the error consisting of the blurring operator will remain at a large value during the optimization process; therefore, the weight of the regularization factor will be more significant. Hence, the noise will, at all times, be suppressed in the reconstruction process. Although one may argue that, by including the regularization factor as a multiplicative constraint, the linearity of the problem has been lost, careful analysis shows that, under certain restrictions, no new local minima are introduced. Numerical testing shows that the proposed algorithm works effectively and efficiently in various practical applications.

Contrast source inversion methods in elastodynamics.

In this paper a nonlinear inversion method is presented for determining the mass density of an elastic inclusion from the knowledge of how the inclusion scatters known incident elastic waves. The algorithm employed is an extension of the multiplicative regularized contrast source inversion method (MR-CSI) to elasticity. This method involves alternate determination of the mass density contrast and the contrast sources (the product of the contrast and the fields) in each iterative step. The simple updating schemes of the method allow the introduction of an extra regularization term to the cost functional as a multiplicative constraint. This so-called MR-CSI method (MR-CSI) has been proven to be very effective for the acoustic and electromagnetic inverse scattering problems. Numerical examples demonstrate that the MR-CSI method shows excellent edge preserving properties by robustly handling noisy data very well, even for more complicated elastodynamic problems.