A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Non-uniqueness in the inverse source problem in acoustics and electromagnetics

A unified theoretical framework based on the scalar wave equation is presented for transmission tomography of weakly inhomogeneous objects embedded in a known uniform background medium The developed formulation differs from the usual treatments of diffraction tomography in that it is not limited to applications employing plane-wave illumination and planar measurement boundaries and can be extended to weakly inhomogeneous objects embedded in non-uniform backgrounds and to the class of strongly scattering objects The new formulation is based on the mathematical operations of propagation and backpropagation which are generalisations of the conventional tomographic operations of projection and backprojection to the case of diffracting wavefields The projection and backprojection operations are then obtained from the generalised operations in the zero wavelength limit A class of reconstruction algorithms are developed within the new formulation that are generalisations, to the case of diffraction tomography, of the class of iterative ART (algebraic reconstruction technique) algorithms of conventional (X-ray) tomography The reconstruction algorithms developed (called generalised ART algorithms) are shown to be applicable to situations where the tomographic data are sparse (limited-view problem) and to the incorporation of a priori information into the reconstruction process

Reconstructive tomography with diffracting wavefields

The synthetic aperture focusing technique (SAFT) is briefly reviewed and addressed as a heuristic digital ultrasonic imaging scheme which exploits the idea of back-propagating a set of measured and digitally stored A-scans. It is shown that for a far-field experimental set-up, ie for small, isolated defects remote to the transducer, SAFT reduces to the filtered back-projection imaging scheme which is well known within the framework of conventional X-ray computer tomography. Therefore, alternative data processing via Fourier transforms only, similar to the Fourier slice theorem of tomography, is possible, which sheds considerable light upon the heuristic SAFT pixel-space envelope-detection scheme. The resulting imaging identity has been termed POFFIS (physical optics far-field inverse scattering). The far-field assumption is then omitted yielding a Fourier-transform-SAFT algorithm (FT-SAFT) whose results are identical to back-propagation imaging with the definite advantage of fast processing capabilities based upon standard hardware and allowing immediate implementation of high resolution procedures as well as inclusion of mode-conversion effects; the theoretical background is pulse-echo diffraction tomography. The above results are supported and illustrated by application of all three algorithms — SAFT, POFFIS, FT-SAFT — to experimental data obtained from scanning a line aperture for several test specimens.

Synthetic aperture focusing technique signal processing

The use of ultrasonic waves in nondestructive evaluation is reviewed. Perspective is provided by a summary of the role played by nondestructive evaluation in the manufacture and utilization of structural components. Included are discussions of the methodologies for predicting part serviceability and of the constraints placed on the nondestructive measurement process by the mechanical designs and functions of structural components. The principles of active (ultrasonic) and passive (acoustic-emission) testing techniques are then reviewed. Specific examples of applications from several industries illustrate the capabilities and limitations of state-of-the-art techniques based on these principles. Detailed discussion is then given of present research and development activities designed to overcome these limitations, with particular emphasis on those areas best addressed by the tools of electrical engineering. Included are discussions of instrumentation, inspection performance modeling, flaw sizing techniques (including imaging systems), material property measurement techniques, and acousticemission techniques. The paper concludes with a consideration of important directions of future work.

Ultrasonics in nondestructive evaluation

I Localization, Identification, and Tracking of Signal Sources.- The Problem of Orientation in Photogrammetry from the Phototheodolite to Earth Resources Satellites.- Digital Array Signal Processing Methods Applied for Advanced Direction Finding Operations.- Remote Sensing and Imaging with Microwave Sensors.- The Application of Spread Spectrum Techniques in Satellite Orbit Determination and Navigation Systems.- Passive Bearing Estimation of a Moving Sound Source Using an Adaptive Filter.- Second-Order Representation of Signals.- The Phonochrome - A Coherent Spectro-Temporal Representation of Sound.- Directional Hearing and the Acoustical Near-Field of a Human and an Artificial Head.- Determination of the Transfer Function of a Time-Varying Transfer Channel by a Correlation Method. Illustrated on Sound Propagation Through a Turbulent Medium.- Redundance of Acoustic Localization.- Sound Localization in the Horizontal Plane by the House Mouse (Mus musculus).- Localization of Water Surface Waves with the Lateral Line System in the Clawed Toad (Xenopus laevis Daudin).- Determination of Source-Distance by the Surface-Feeding Fishes Apbcheilus lineatus (Cyprinodontidae) and Pantodon buchholzi (Pantodontidae).- Determination of Stimulus Direction by the Topminnow Apbcheilus lineatus.- Mathematical Description of the Stimulus for the Lateral Line Organ.- Image Sequence Analysis for Target Tracking.- Detection Performance in Visual Search Tasks.- Prediction of Human Pattern Discrimination.- On the Application of Associative Neural Network Models to Technical Control Problems.- Control of Hand-Movements Towards a Target.- How the Primate Brain Performs Detection and Pursuit of Moving Visual Objects.- Non-Linear Interaction of Vestibulo-Ocular Reflex and Pursuit Eye Movements.- How Saccadic Eye Movements Find their Target.- Visual Orientation of Flies in Flight.- II Orientation and Path Control.- Roll-Stabilization During Flight of the Blowfly's Head and Body by Mechanical and Visual Cues.- Activation of Flight Control Muscles by Neck Reflexes in the Domestic Pigeon (Columba livia var. domestica).- Dynamics of Groundspeed Control in Drosophila.- Control of Flight Speed by Minimization of the Apparent Ground Pattern Movement.- Short-Term Integration: A Possible Mechanism in the Optomotor Reaction of Walking Crickets?.- Computation of Motion and Position in the Visual System of the Fly (Musca). Experiments with Uniform Stimulation.- On the Alignment of Movement Detectors Mediating the Landing Response in the Blowfly, Calliphora erythrocephala.- The Stimulus Efficiency of Intensity Contrast, Spectral Contrast, and Polarization Contrast in the Optomotorics of Pachnoda marginata (Coleoptera: Cetoniinae).- A Possible Neuronal Mechanism of Velocity and Direction Detection.- Optical Monitoring of Ego-Motion for Movement with Respect to a Plane Surface.- Perception of Differences Between Visually and Mechanically Presented Motion Information.- Paired Tread Wheels as a Tool for the Analysis of Orientation Behaviour in Arthropods.- Wind Orientation in Walking Carrion Beetles (Necrophorus humator F.) Under Closed and Open Loop Conditions.- Correcting Behaviour - a Mechanism to Support Course Stability?.- Acoustical Pattern Recognition in Crickets.- Neural Enhancement of Directionality in Sensory Pathways of Two Arthropods, the Cricket and the Crayfish.- Homing by Systematic Search.- III Localization and Identification of Targets by Active Systems.- The Superlaminated Torus in the Electric Fish Eigenmannia, a Multimodal Space Integrator.- The Performance of Bat Sonar Systems.- Short-Time Spectral Analysis of the Ultrasound-Echoes of Flying Insects.- Control of Echolocation Pulses in the CF-FM-Bat Rhinolophus rouxi: Neurophysiological Investigations of the Function of the Brain Stem Motor Nucleus Innervating the Larynx: the Nucleus Ambiguus.- Echolocation as Seen from the Viewpoint of Radar/Sonar Theory.- Remarks on Recent Developments in Inverse Scattering Theory.- Signal Extraction and Target Detection for Localization by Radar.- Environment and Factors Influencing Optimum Multi-Radar Processing in Air Traffic Control.- Collision Avoidance in Air-Traffic - Conflict Detection and Resolution.- A Ship Encounter - Collision Model.- IV Navigation, Bird Migration and Homing.- Pacific Systems of Trade and Navigation.- Interplanetary Navigation: An Overview.- Image-Supported Navigation.- Range Measuring Techniques in Aviation.- The Geometrical Characteristics of the Navigation Satellite Systems.- Elements of Bird Navigation.- The Development of the Navigational System in Young Homing Pigeons.- The Homing Behavior of Pigeons Raised in a Reversed Magnetic Field.- The Magnetic Compass as an Independent Alternative to the Sun Compass in the Homing Performance of Young Pigeons.- Typical Migratory Birds: Endogenously Programmed Animals.- The Influence of the Earth Magnetic Field to the Migratory Behaviour of Pied Flycatchers (Ficedula hypoleuca PALLAS).- Migratory Birds: Path-Finding Using Madar? (Magnetic Detection and Ranging).

Localization and Orientation in Biology and Engineering

For the purpose of ultrasonic nondestructive testing of materials, holography in connection with digital reconstruction algorithms has been proposed as a modern tool to extract crack sizes from ultrasonic scattering data. Defining the typical holographic reconstruction algorithm as the application of the scalar Kirchhoff diffraction theory to backward wave propagation, we demonstrate its general incapability of reconstructing equivalent sources, and hence, geometries of scattering bodies. Only the special case of a planar measurement recording surface, that is to say, a hologram plane, and a planar crack with perfectly rigid boundary conditions parallel to the hologram plane and perpendicular to the incident field yields a nearly perfect correlation between crack size and reconstructed image; the reconstruction algorithm is then referred to as the Rayleigh-Sommerfeld formula; it therefore represents the optimal case matched to that special geometrical situation and, hence, may be interpreted as a quasi-matched spatial filter. Using integral equation theory and physical optics, we compute synthetic holographic data for a linear cracklike scatterer for both plane and spherical wave incidence, the latter case simulating a synthetic aperture impulse echo situation, thus illustrating how the Rayleigh-Sommerfeld algorithm or its Fresnel approximation increasingly fail for cracks inclined to the hologram plane and excited nonperpendicularly. Furthermore, we point out how the physical data recording process may additionally influence the reconstruction accuracy, and, finally, guidelines for a careful and serious application of these holographic reconstruction algorithms are given. The theoretical results are supported by measurements.

Potential and limits of holographic reconstruction algorithms

Generalized holography is understood as a heuristic solution of an inverse-scattering problem based on the time-harmonic formulation of Huygens’s principle. It is checked against a scalar two-dimensional numerical-scattering experiment: Simulated data for a cylindrical object with arbitrary elliptical cross section are inserted into the algorithm, and the results are critically discussed. We find that generalized holography yields only sparse improvement over conventional holography—the images obtained are only poor solutions to the inverse-scattering problem—but, on the other hand, its mathematical consequences, such as the Porter–Bojarski integral equation, may serve as an intuitive means to predict the performance of holographic schemes for certain specialized canonical inverse-scattering problems; once the limitations are known, considerable advantage can be drawn for practical applications, and improved extensions are readily proposed yielding very general diffraction tomography algorithms.

Imaging performance of generalized holography

Different recording techniques of ultrasonic scattering data — either stationary or within a synthetic aperture — in either a bistatic or a monostatic arrangement lead to different classification and reconstruction algorithms which may be further distinguished by the structure of the incident field, either time harmonic or a broadband impulse. It turns out that one has to differentiate between the axial resolution between scattering centres for broadband excitation and the definite reconstruction of a scattering surface; in the latter case, specular reflection has to be included in the backscattered signal. This is shown with simulated data for the two-dimensional problem of an infinitely long cylinder with arbitrary elliptical cross-section and a soft boundary condition using the POFFIS-algorithm which, on the other hand, can be shown to contain a couple of widely used reconstruction algorithms as special cases; even though it starts with the high frequency assumption, the results for voids are identical to those obtained with the Inverse Born reconstruction scheme.

Inverse Scattering Algorithms

Inverse scattering, as applied to such different fields as radar, geophysical probing, medical diagnostics, or non-destructive testing of materials with ultrasound attempts to identify or reconstruct a geometrical obstacle and structure illuminated by an electromagnetic, acoustic or elastic wave from the information contained in the scattered field. As illustrated by the far-field behavior, this information is available through the radiation pattern of the obstacle, which is generally a function of direction and frequency uniquely determined by the scattering geometry. The inversion process, therefore, requires the variation of the mutual situation of the transmitter and receiver and/or the frequency of the exciting wave.

M. Fischer

Papers

Potential and limits of holographic reconstruction algorithms

Imaging performance of generalized holography

Inverse Scattering Algorithms

Remarks on Recent Developments in Inverse Scattering Theory

Transient Methods in Electromagnetic Imaging