Ultrasonic Propagation Hyperbola Algorithm

TELKOMNIKA ISSN: 1693-6930  Ultrasonic Tomography of Immersion Circular Array by Hyperbola Algorithm Liu Yang 67

2.2. Ultrasonic Propagation

When the ultrasonic is propagating in the detection-zone, the total pressure field in the water is calculated by solving the transient pressure wave equation [15]:                     2 2 2 1 1 t t d m p p q Q c t 1 where  the fluid density, c is the sound speed, t p is the total acoustic pressure,  d q is the dipole source,  m Q is the monopole source. If there is no dipole source or monopole source, the variables  d q and  m Q should be 0. The total acoustic pressue:   t b p p p 2 where p stands for the propagation pressure field, b p stands for the background pressure field. In the circular array model, the cylindrical wave radiation is adopted to obtain a wide range covering of the ultrasonic field and more reconstruction data. Then, the cylindrical wave radiation equation is:                            1 1 1 2 t d i p p n p q Q c t r 3 where  n means the normal vector of the radiation source, and                 1 1 1 2 i i i i p p Q n p c t r 4 where i p is the incident pressure field which is a function of space.

2.3. Hyperbola Algorithm

A hyperbola may be defined equivalently as the locus of points where the absolute value of the difference between the distances and the two foci is a constant. Figure 2 shows the geometric parameters of a hyperbolic curve. In this figure, 1 F and 2 F are two focal points of the hyperbola with the coordinates of    ,0 c and   ,0 c . O is the origin of the axes and the center symmetric point for the hyperbolic curve. 1 V and 2 V with the coordinates of    ,0 a and   ,0 a are two points of the intersection of the horizontal axis and the two branches of the hyperbola, respectively. Line 2 V Q is perpendicular to the horizontal axis and it meets the asymptotes at point Q whose coordinates are   , a b . P stands for any possible point on each side of the curve.  ISSN: 1693-6930 TELKOMNIKA Vol. 13, No. 1, March 2015 : 65 – 75 68 Figure 2. Geometric parameters of a hyperbola Figure 3. Schematic diagram of hyperbola algorithm imaging According to the definition of the hyperbola:   1 2 2 PF PF a 5 where a is the distance between the vertex of a hyperbola and the origin of the axes, which is obviously a constant. Supposing 1 F and 2 F can be replaced by two receiving transducers, then P can be considered as a certain point on the tested scatterer which may reflect the ultrasonic pulse. Therefore, it is not necessary to care about the position of the transmitting transducer because the distance from the transmitter to the scatterer is the same for each receiver [16]. Once the parameter a is obtained, the parameter b can be easily induced from the equation:   2 2 2 a b c 6 where b is the length of line 2 V Q . Since parameter c has also been obtained from the coordinates of 1 F and 2 F , the slope of the asymptotic line of the hyperbola is b a . So far, a determined hyperbola can be drawn through the following function see Figure 2: TELKOMNIKA ISSN: 1693-6930  Ultrasonic Tomography of Immersion Circular Array by Hyperbola Algorithm Liu Yang 69      2 2 2 2 1 0, b0 x y a a b 7 The tested scatterer is supposed on the hyperbolic curve, however only one hyperbola is insufficient to locate the tested scatterer. Since fan-shaped ultrasonic waves are reflected by the scatterer in many directions, hyperbolas with different parameters and coordinates can be drawn through one transmitting transducer and different receiving transducer pairs. Figure 3 shows the two hyperbolic curves intersecting at P and P . If the hyperbolic curve on the right is extended, one more intersection will appear. In fact, two hyperbolas may have at most 4 points of intersection and theoretically all of them are likely to be the reflection points. With the changing of the transmitting transducer in sequence and the permutation and combination of the receiving transducer pairs, a sufficient number of the hyperbola is obtained and superimposed together. Then, the image of the tested scatterer is reconstructed. Hyperbola approach considers the transducers in groups of three, one acting as the transmitter and two as the receivers. The differences between the arrival times of the scattered signals at two receivers are given by:                          2 2 2 2 1 , ij p p p i p i p j p j t x y x x y y x x y y c 8 where   , i i x y are the coordinates of the first receiver and   , j j x y the second;   , p p x y stand for the coordinates of the scatterer. In the circular array model, the transducer array cannot be rotated because they are all embedded in the wall of the water container. So, the number of received signal pairs for each time or the hyperbolas can be drawn is given by:     1 L N N 9 When the process is repeated for each transmitting transducer, the number of final receiving signals should be:      1 L N N N 10 The intensity of pixels at the coordinates   , p p x y in the reconstructed image can be calculated by the cross correlations of receiving signal pairs and the summation operation of the cross correlations:       1 , 1 1 1 , , N N N p p ni nj ij p p n i j k i n j n I x y R t x y            11 where , ni nj R is the cross correlation calculation;     , , ni nj ij p p R t x y  is the cross correlation value of the receiving sensor pair   , i j 3. Results and Analysis 3.1. Propagation Simulation