Thomas J. Asaki

A Variational Approach to Reconstructing Images Corrupted by Poisson Noise

We propose a new variational model to denoise an image corrupted by Poisson noise. Like the ROF model described in [1] and [2], the new model uses total-variation regularization, which preserves edges. Unlike the ROF model, our model uses a data-fidelity term that is suitable for Poisson noise. The result is that the strength of the regularization is signal dependent, precisely like Poisson noise. Noise of varying scales will be removed by our model, while preserving low-contrast features in regions of low intensity.

ACOUSTIC RADIATION FORCE ON A BUBBLE DRIVEN ABOVE RESONANCE

The positions and sizes of stably levitated bubbles on the Earth in an acoustic standing wave have been measured for bubbles larger than resonance size. The equilibrium positions are sensitive to the radiation force on the bubble. The results are in good agreement with the radiation force approximation of Eller [J. Acoust. Soc. Am. 43, 170–171 (1968)] as well as that of the more general results of Crum and Prosperetti [J. Acoust. Soc. Am. 73, 121–127 (1983)] and Lee and Wang [J. Acoust. Soc. Am. 93, 1637–1640 (1993)]. Previous experimental studies have measured forces on bubbles of smaller than resonance size or have been concerned with forces on larger bubbles in nonequilibrium situations.

Shape oscillations of bubbles in water driven by modulated ultrasonic radiation pressure: Observations and detection with scattered laser light

Steady‐state quadrupole shape oscillations of air bubbles trapped in water were excited by amplitude modulation of the acoustic radiation pressure used for levitation. This method of exciting controlled shape oscillations may make possible noncontact dynamical measurements of the rheological properties of bubbles. Bubble sizes ranged from 1.6‐ to 12‐mm diameter corresponding to observed quadrupole mode frequenices of 190 to 17 Hz. Small‐amplitude oscillations were detected by interference of scattered laser light. Some larger amplitude oscillations were detected by the unaided eye or with a television camera. The structure of the acoustic field in the levitator needed for the levitation of large bubbles is discussed. In the absence of modulation the levitated bubbles had an oblate shape.

Free decay of shape oscillations of bubbles acoustically trapped in water and sea water

Asymptotic results for the free decay of shape oscillations of viscous liquid spheres have been extended to include higher-order terms in the ratios of the inner and outer viscous penetration lengths to the radius. The new expressions are shown to be important for studies in which the two fluids have dissimilar densities and viscosities such as air/liquid systems. The analysis also includes an expansion for the frequency of maximum response of driven oscillations. The calculations are supported by measurements of the small-amplitude quadrupole mode free decay of nearly spherical bubbles acoustically levitated in clean water. The bubble radii ranged from 400 μm to 1400 μm. The results are interpreted in light of the initial-value problem. The lack of excess damping suggests that the interface behaves ideally for times up to two hours after bubble injection. Measurements were also carried out on bubbles in 0.5 m NaCl solution and in sea water. Larger bubbles (radius > 800 μm) in clean water exhibit damping two to four times larger than predicted by theory. The transition from this anomalous damping to theoretical damping is a very abrupt function of radius. All observations were carried out with similar acoustic fields for counteracting buoyancy. The excess damping appears to be associated with some nonlinear response of the bubble.

Abel inversion using total-variation regularization

In the case of radiography of a cylindrically symmetric object, the Abel transform is useful for describing the tomographic measurement operator. The inverse of this operator is unbounded, so regularization is required for the computation of satisfactory inversions. We introduce the use of the total variation seminorm for this purpose, and prove the existence and uniqueness of solutions of the corresponding variational problem. We illustrate the effectiveness of the total-variation regularization with an example and comparison with the unregularized inverse and the H1 regularized inverse.

EQUILIBRIUM SHAPE OF AN ACOUSTICALLY LEVITATED BUBBLE DRIVEN ABOVE RESONANCE

The equilibrium shapes of stably levitated bubbles on the Earth in an ultrasonic standing wave have been measured and calculated for bubbles larger than resonance size. The shape and equilibrium position are sensitive to the acoustic field used for levitation. Measured static oblate shapes are in good agreement with predictions based upon the quadrupole projection of the radiation pressure on a compressible sphere taking into account the monopole response of the bubble. For the size range of bubbles considered, the monopole response is limited by inertia and radiation damping. A simplified theory (based on the deformation resulting from the Bernoulli pressure of incompressible potential flow around a sphere) is generally consistent with the measurements for the smaller of the bubbles observed.

The effects of a soluble surfactant on quadrupole shape oscillations and dissolution of air bubbles in water

Single air bubbles were ultrasonically trapped in aqueous solutions of the soluble surfactant Triton X-100. Quadrupole shape oscillations were induced by modulation of the radiation pressure and the free decay was recorded using an optical extinction technique. The frequency and damping were found to be maximal at a bulk surfactant concentration which only weakly affects the surface tension. At higher concentrations the frequency is reduced due to surface tension depression and the damping approaches a value several times that expected for a clean interface. These results are in qualitative agreement with theoretical predictions based on interfacial viscoelastic considerations. Bubble size and dissolution curves were obtained through the use of digitized bubble images. Dissolution rates are shown to be significantly enhanced by the presence of the surfactant.

Abel inversion using total variation regularization: applications

We apply total-variation (TV) regularization methods to Abel inversion tomography. Inversions are performed using the fixed-point iteration method and the regularization parameter is chosen such that the resulting data fidelity approximates the known or estimated statistical character of the noisy data. Five one-dimensional (1D) examples illustrate the favorable characteristics of TV-regularized solutions: noise suppression and density discontinuity preservation. Experimental and simulated examples from X-ray radiography also illustrate limitations due to a linear projection approximation. TV-regularized inversions are shown to be superior to squared gradient (Tikhonov) regularized inversions for objects with density discontinuities. We also introduce an adaptive TV method that utilizes a modified discrete gradient operator acting only apart from data-determined density discontinuities. This method provides improved density level preservation relative to the basic TV method.

Response of bubbles to ultrasonic radiation pressure: Dynamics in low gravity and shape oscillations

A dual-frequency acoustic levitator containing water was developed for studying bubble and drop dynamics in low gravity. It was flown on the Space Shuttle on USML-1 where it was used in NASA’s Glovebox facility. High frequency (21 or 63 kHz) ultrasonic waves were modulated by low frequencies to excite shape oscillations on bubbles and oil drops ultrasonically trapped in the water. Bubble diameters were typically close to 1 cm or larger. When such large bubbles are acoustically trapped on the earth, the acoustic radiation pressure needed to overcome buoyancy tends to shift the natural frequency for quadrupole (n = 2) oscillations above the prediction of Lamb’s equation. In low gravity, a much weaker trapping force was used and measurements of n = 2 and 3 mode frequencies were closer to the ideal case. Other video observations in low gravity include: (i) the transient reappearance of a bulge where a small bubble has coalesced with a large one, (ii) observations of the dynamics of bubbles coated by oil indicating that shape oscillations can shift a coated bubble away from the oil-water interface of the coating giving a centering of the core, and (iii) the agglomeration of bubbles induced by the sound field.

Quantitative Object Reconstruction Using Abel Transform X-Ray Tomography and Mixed Variable Optimization

This paper introduces a new approach to the problem of quantitative reconstruction of an object from few radiographic views. A mixed variable programming problem is formulated in which the variables of interest are the number and types of materials and geometric parameters. To demonstrate the technique, we considered the problem of reconstructing cylindrically symmetric objects of multiple layers from a single radiograph. The mixed variable pattern search algorithm for linearly constrained problems was applied by means of the NOMADm MATLAB software package. Numerical results are presented for several test configurations and show that, while there are difficulties yet to be overcome, the method is promising for solving this class of problems.