# Spatial decomposition of ultrasonic echoes

Abstract: The pulse-echo method is one of the most important in ultrasonic imaging. In many areas, including medical applications and nondestructive evaluation, it constitutes one of the fundamental principles for aquiring information about the examined object. An ultrasonic pulse is transmitted into a medium and the reflected pulse is recorded, often by the same transducer. In the area of 3-dimensional imaging, or surface profiling, the distance between the object and the transducer is estimated to be proportional to the time-of-flight (TOF) of the pulse. If the transducer is then moved in a plane parallell to the object, a surface profile can be obtained. Usually some sort of correlation between echoes is performed to estimate their relative difference in TOF. However, this assumes that the shape of the echoes are the same. This is not the case as the shape is dependent on the surface in the neighbourhood of the transducer's symmetry axis and this shape will vary as the transducer is moved across the surface. The change in signal shape will reduce the accuracy of the TOF estimation. A simple example is when the surface has a step. The resulting echo consists of the superposition of two echoes; one from the "top" and one from the "bottom". The TOF estimate will then be almost arbitrary. Another difficulty with pulse-echo imaging is the lateral resolution. The ultrasonic beam is not infinitesimally thin but has a non-neglectable spatial extent, even for focused transducers. This means that two point reflectors separated laterally with only a small distance can not be resolved by ultrasound. The spatial decompostion of the ultrasonic echoes suggested in this licentiate thesis can be used to extract information from the pulse deformation and to reduce the lateral resolution in the following way: ' In surface profiling, the surface is modelled as piecewise plane, i.e. the reflected pulse stems from a local plane and perpendicular object. If we instead model the part of the surface that reflects the ultrasonic pulse as a sloping plane there are two advantages. If we can estimate both the distance to, and the slope of, the surface, we can either increase the accuracy or decrease the number of scanning points while maintaining the same accuracy. ' To increase the lateral resolution we have to take into account how points off the symmetry axis contribute to the total echo. If we know this, some kind of inverse spatial filter or other method can be constructed in order to improve the resolution. This thesis is comprised of the following five parts: Part A1: (Magnus Sandell and Anders Grennberg) "Spatial decomposition of the ultrasonic echo using a tomographic approach. Part A: The regularization method" We conclude that since the pulse-echo system can be considered linear, i.e. the echo from an arbitrary object can be thought of as the sum of the echoes from the contributing points on the surface, it would be very useful to know the echo from a point reflector. By doing this spatial decomposition we can simulate the echo from any object. It is, however, not possible practically to measure the {em single point echo} (SPE) directly. If the reflector is to be considered pointlike, its size has to be so small that the echo will dissappear in the background noise. If it is increased, there will be spatial smoothing. Instead, we propose an indirect method that uses echoes from sliding halfplanes. This results in measurements with far better SNR and by modifying methods from tomography we can obtain the SPE. An error analysis is performed for the calculated SPE and simulated echoes from sloping halfplanes, using the obtained SPE, are compared with measured ones. Part A2 : (Anders Grennberg and Magnus Sandell) "Experimental determination of the single point echo of an ultrasonic transducer using a tomographic approach" The main ideas of Part A1 are presented in this conference paper. It was presented at the Conference of the IEEE Engineering in Medicine and Biology Society in Paris, France in October 1992. Part B1 : (Anders Grennberg and Magnus Sandell) "Spatial decomposition of the ultrasonic echo using a tomographic approach. Part B: The singular system method" In this part we continue the approach of spatially decomposing the ultrasonic echo. The SPE is again determined from echoes from sliding halfplanes. Here we interpret the SPE and the halfplane echoes to belong to two different weighted Hilbert spaces. These are chosen with regard to the properties of the SPE and the measured echoes. The SPE is supposed to belong to one of these spaces and is mapped by an integral operator to the other space. This is measured but the measurements also contain additive noise. A continuous inverse to this operator does not exist so the problem is ill-posed. A pseudo-inverse to this operator is constructed by using a singular value decomposition (SVD). By decomposing the halfplane echoes with N basis functions from the SVD, the SPE can be found. The spatial decomposition made in this part can be useful to obtain the long-term goals of estimating the slope of a tilted plane and to increase the lateral resolution. Part B2 : (Anders Grennberg and Magnus Sandell) "Experimental determination of the ultrasonic echo from a pointlike reflector using a tomographic approach" This is a contribution to the IEEE 1992 Ultrasonic Symposium in Tucson, USA. It is an extract of Part B1 and deals with the SVD-based inversion of the halfplane echoes. Part C : (Anders Grennberg and Magnus Sandell) "Estimation of subsample time delay differences in narrowbanded ultrasonic echoes using the Hilbert transform correlation" This part deals with a method for increased axial resolution. Using the fact that airborne ultrasonic pulses are narrowbanded, a new algorithm for estimating small time-delay is described. This method can be used in conjuction with a normal TOF-estimator. The latter can make a robust and rough (i.e. within a few samples) estimate and the remaining small time-delay is estimated using our proposed method. Another area of application is an improved averaging algorithm. Airborne ultrasound suffers from a jitter which is caused by air movement and temperature gradients. This jitter can be modelled as a small random time shift. A straightforward averaging will then be a summing of pulses that are not aligned in time which results in a pulse deformation. By estimating the time shift caused by the jitter, all echoes can be time aligned and no pulse deformation will occur when summing them.

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