High Frame Rate Ultrasonic Imaging through Fourier Transform using an Arbitrary Known Transmission Field

Based on the study of limited array diffraction beams, a High Frame Rate (HFR) imaging method which uses a broadband pulsed plane wave, or array beams transmission field from a linear transducer array to illuminate the area to be imaged has been developed by Jian-yu LU. Echoes from the objects are received with the same transducer as is used in transmission. For each array beam parameter in a certain range, the received signals are weighted with that array beam and are summed up. The summations are Fourier transformed from time domain to frequency domain, and then processed further with the so called ‘‘parameter match’’ to produce the spectrum of the imaging. 2D and 3D images are constructed with inverse Fourier transformer respectively. In this way, the frame per transmission imaging rate is achieved. Despite its advantages of high frame rate and high signal to noise ratio, the original HFR method has several drawbacks. It can only use the plane wave or the array beam transmission field, and is difficult to be ported for a non-array beam field, such as a cylindrical or spherical wave. Moreover, since the plane wave transmission field illuminates only a narrow area of its own width, the imaged area is quite small, and the only way to widen it up is to steer the transmission beams several times from different angles, which lowers the frame rate. Besides, the array beam field demands a linear transducer and a very complex weighting process. Therefore, the HFR method needs to allow diffraction wave transmission fields in order to be practically useful. For example, it may use a cylindrical or spherical field and output sector format images like the conventional sector B mode ones, which have contributed a lot in diagnosing myocardial diseases. In this chapter, an extended HFR method for 2D imaging is proposed. It allows all kinds of transmission field, including the cylindrical one and the spherical one, as well as the plane wave one. It is more general than the original HFR method. The extended HFR method works mostly like the original one, except that 1, it implements the weight-and-sum process through the Fourier transform; and 2, it iterates for each frequency in a certain range to obtain firstly a coarse image component at that frequency and then the refined one with the information of the transmission field removed. After the iteration the image components are summed up and that is the final image.


Introduction
Based on the study of limited array diffraction beams, a High Frame Rate (HFR) imaging method which uses a broadband pulsed plane wave, or array beams transmission field from a linear transducer array to illuminate the area to be imaged has been developed by Jian-yu LU. Echoes from the objects are received with the same transducer as is used in transmission. For each array beam parameter in a certain range, the received signals are weighted with that array beam and are summed up. The summations are Fourier transformed from time domain to frequency domain, and then processed further with the so called ''parameter match'' to produce the spectrum of the imaging. 2D and 3D images are constructed with inverse Fourier transformer respectively. In this way, the frame per transmission imaging rate is achieved. Despite its advantages of high frame rate and high signal to noise ratio, the original HFR method has several drawbacks. It can only use the plane wave or the array beam transmission field, and is difficult to be ported for a non-array beam field, such as a cylindrical or spherical wave. Moreover, since the plane wave transmission field illuminates only a narrow area of its own width, the imaged area is quite small, and the only way to widen it up is to steer the transmission beams several times from different angles, which lowers the frame rate. Besides, the array beam field demands a linear transducer and a very complex weighting process. Therefore, the HFR method needs to allow diffraction wave transmission fields in order to be practically useful. For example, it may use a cylindrical or spherical field and output sector format images like the conventional sector B mode ones, which have contributed a lot in diagnosing myocardial diseases. In this chapter, an extended HFR method for 2D imaging is proposed. It allows all kinds of transmission field, including the cylindrical one and the spherical one, as well as the plane wave one. It is more general than the original HFR method. The extended HFR method works mostly like the original one, except that 1, it implements the weight-and-sum process through the Fourier transform; and 2, it iterates for each frequency in a certain range to obtain firstly a coarse image component at that frequency and then the refined one with the information of the transmission field removed. After the iteration the image components are summed up and that is the final image.
In ultrasonic imaging systems, the cylindrical transducer, circle or curve transducer and linear transducer are commonly used. The advantages and disadvantages among them are different. One characteristic of the cylindrical or circle transducer is they can illuminate a sector or pyramidal area of the object. Therefore, in the following section, we extend the HFR method by using a cylindrical wave to illuminate an object. Mathematical formulas are derived and computer simulations are performed to verify the method. The method allows to increase the illumination area by using a transducer of a small footprint, which is important for applications such as cardiac imaging where acoustic window sizes are limited. This Chapter is organized as follows. Firstly, the HFR ultrasonic imaging system based on the angular spectrum principle is introduced. In the flowing section, this system is extended. The extended HFR method allows all kinds of transmission field. Finally, a high frame rate 2D and 3D imaging system with a curved or cylindrical array is proposed.

High Frame Rate ultrasonic imaging system based on the angular spectrum principle
A kind of high frame rate (HFR) 2D and 3D imaging method was developed by Jianyu Lu in 1997. Because only one transmission is required to construct a frame of image, this method can reach an ultra high frame rate (about 3750 volumes or frames per second for biological soft tissues at a depth of 200 mm). In this section, a new HFR method is presented in the view of angular spectrum. Compared with conventional dynamic focusing method which uses delay-and-sum processing and Lu's HFR method, which uses a kind of special weighting on the received signal, the new method only use the Fourier transform algorithm to construct image. So the system implementation of the method could be greatly simplified. During constructing image, several array beams with different parameters are used as transmitted signals, and the spectra of a frame of image is obtained by synthesizing the image spectrums related to different transmit event. The simulation result shows that the solution not only suppresses the sidelobe of system greatly and obtains the high quality image but also still keeps high frame rate to some extent.

Theory
The HFR method is based on one transmission event. In order to get the image of the object, the transducer transmits the limited diffraction beams to the object then the same transducer receives the echo signals reflected by the scatters and constructs image by Fourier transform. The equation (2.6) means the signal, which is received by transducer located in the plane 0 z = , is come from echo signal produced by the scatters at the plane ' zz = . In fact the received signal comes from a lot of planes in the acoustic field. So it should be the summation of ( , , ) xy Rk k k over different depth, namely www.intechopen.com ,, Where the function (,,) p xyz is defined as From the equation (2.11) and (2.12), we can see that if the size of the aperture is infinite, the image is the result of the convolution between object reflection coefficient and the function (,,) p xyz . So (,,) p xyz is the point spread function (PSF) of the imaging system, which is determined by the excited signal and the frequency response of transducer. Obviously under the condition that k is infinite and ()() AkTk is equal to one, (,,) p xyz turns to be Dirac delta function and (,,) BL f xyz is the object (,,) f xyz . Generally, The bandwidth of () Tk and () Ak is limited and (,,) p xyz is pulse in three dimension. So (,,) f xyz only presents some useful information of the object.

Simulation results
In equation ( xy Rk k kcan be obtained from the modified Fourier transform results of the received signal ' (,,) rxy t under the condition below:   To verify the new process, a simulation in two dimension was performed to construct image by the HFR method. In the simulation, the phantom consists of eight point scatters objects. The linear array transducers is a 2.5 MHz array of 64 elements and a dimension of 38.4 mm with an inter-element space of 0.6 mm. Two-way (pulse-echo) spectra of the arrays are assumed to be proportional to the Blackman window function with a fractional bandwidth of about 81% that is typical for a modern array. The simulation results shows in Fig.2

Conclusion
This section presents a theory analysis, which is based on angular spectrum principle, to simplify the HFR imaging system presented by Lu. Besides a new imaging mode is proposed, which use several transmission events to synthesize the image. In every transmission event, array beam with different parameters is used as excited signal. The constructed image has very high resolution and contrast, and meanwhile the imaging system still hold high frame rate to some extent.

Construction of High Frame Rate ultrasonic images with Fourier transform in any kind of acoustic field
In HFR method, a plane wave was used to illuminate an area for either 2D or 3D imaging. The drawback of this method is that the area illuminated is only as wide as the size of the aperture of the array transducer. In this section, a generic HFR method is developed. 2D high frame rate images can be constructed using the Fourier transform with a single transmission of an ultrasound pulse from an array under different transmission filed as long as the transmission filed is known. To verify our theory, computer simulations have been performed in the non-plane wave field. The field is cylindrical field defined by zero order Hankle function and produced by a linear array. The image with sector format and lower sidelobes is obtained. The simulation results are consistent with our theory.

Theories
For simplicity, we discuss our new method in the two dimension (2D). Let us assume that there is a linear array at the position z=0 (Fig.3.1), and the transmitted field is (,,) p xzk where 2/ kf c π = . f means frequency and c is acoustic speed. If there is a scatter at the position which the coordination is (x,z), and the reflection coefficient is (,) f xz , The echo signal, which object scatter reflects, is as follow: where () Tk is spectra, which is related to the spectrum of excited signal and the frequency response of the transducer. We can prove that equation (3.9) is a good approximation of the object function (,) fxz.
Especially when the transmitted filed is plane wave, it is the original HFR method.

Simulation results
From the theoretical analysis above, the simulation process is divided into several steps as follows: 1. According to the transmitted signal and the boundary of the transducer, calculate the distribution of the acoustic filed (,,) p xzk ; 2. According to the equation ( The number of the transducer arrays is 128. The length of the transducer is 37mm. The central frequency of the transducer is 2.5MHz and the bandwidth is about 80 percent of the central frequency. The frequency response function T(k) is assumed to be Blackman window, which is adopted in most literature's simulation condition. The image is produced at the depth which z is equal to 50mm. Fig.3.2 is the result of the simulation for one scatters located at (0,50). Fig3.2.a is the images of the object, which is Log compressed over 40db. In order to observe the sidelobes, Fig3.2.b gives the plot line along lateral direction. Fig.3.2c gives the plot line along axial direction. From the figures, we can see that the sidelobes are about below -40db, which the resolution is about 1.2mm in the lateral direction, and 0.8 in the axial direction. Fig.3.3 is another result of the simulation for seven scatters. The scatters are on the part of a circle, among which the central point scatters is equal to 50mm far away from the surface of transducer. Fig3.3.a is the imaging of the object, which is Log compressed over 40db. Fig3.3.b shows the sidelobes along x direction. Though the size of array is only 37mm, the distance in the images along lateral direction from left point scatter to the right scatter is about 58mm, which is larger than the size of the transducer. Obviously the imaging is impossible to be obtained for original HFR method.

Conclusion
Though the method is analyzed in the two dimensions, it can be obviously used in the three dimensions. So the method gives an effective way to construct images with sector form (2D) and pyramidal form (3D) by the linear array based on the Fourier transformer. Like the original HFR method, the system can construct images with only one time transmission, and the quality of the imaging is high. Compared to the original Fourier transform method, it is effective in any kind of acoustic field, though the principle of the method and the original HFR method is the same. In original HFR method, the kernel function of the Fourier transform contains the information of the transmission field because the transmission waves and weighting waves are the same kind beams, which belong to array beams. In our new method we extend the kind of transmission field from plane wave or array beams to other www.intechopen.com High Frame Rate Ultrasonic Imaging through Fourier Transform using an Arbitrary Known Transmission Field 271 kind wave, such as cylindrical wave. As the weighting signal is not the same form as the transmission filed, it is difficult to combine the transmission filed and weighted signal together in the kernel function of the Fourier transform. As a result we have to repeat the Fourier transform process under different frequency component and make the summation over different frequency results. So the shortcoming of the method is obvious compared to the original HFR method, namely its quantity of the computation is high. If some kind of quick arithmetic is found, the method will be more effective in practice. Because the original HFR method assumes the transmission filed to be the plane wave or array beams although it is impossible actually due to diffraction property in physics, the assumption makes results obtained by original HFR method a little disturbed when the distance between object and transducer is some large. For our method if the transmission filed is pre-known exactly by some method, such as simulation or measurement, the better results can be obtained because the new method can cancel the effects of transmission filed.

High Frame Rate 2D and 3D imaging with a curved or cylindrical array
The cylindrical transducer, circle or curve transducer and linear transducer are commonly used in ultrasonic imaging system. The advantages and disadvantages between them are different. One characteristic of the cylindrical or circle transducer is the transducer can illuminate a sector or pyramidal area of the object. The scanning format is primarily useful for cardiac imaging to avoid interference from the ribs. Since this kind of transducer is nonlinear transducer, the method of constructed images is a little different from the linear transducer's method.

Theoretical preliminaries
In the section, a new imaging method (Fourier method and radial matched filter) for a pulse system will be developed and formulas for construction of 2D and 3D images will be derived with zero order Hankle function.

3D images construction
To simplify the analysis, we assume that the sampling of the array along each direction is regarded as continuous, our results, based on this assumption, should closely approximate that of a sampled aperture as long as the Nyquist criterion is met to avoid spatial aliasing . A sufficient condition for this criterion to be satisfied is a half-wavelength spacing of elements along the arrays. For simplicity, we will also neglect the diffraction patterns of the individual elements; they are assumed to be behaving as point sources and receivers. Although we assume continuously sampled, the simulation results shows similar principles can be applied to arrays of discrete elements of finite size. For the generality, we discuss the method in three-dimension in the cylindrical coordinate system. Fig.4.1 shows a cylindrical transducer. Though the filed produced by cylindrical transducer is much more complex than the plane wave, we still can get simple form under some reasonable assumption. The simplest mode of the filed form produced by the cylindrical transducer is zero order Hankle function. If the kr is relatively large, the acoustic pressure, which is presented by zero order Hankle function, can be estimated by: Where k is wave number, 22 rx y =+ represents radial coordinate, () Ak is related to the spectrum of the signal and the response of the transducer frequency and can be presented by the Blackman windows . Based on the Rayleigh-Sommerfeld diffraction theory, the received signal for an element for all scatterers is easily given by θ is azimuthal angle, z is axial axis, which is perpendicular to the plane defined by r and θ . The equation (4.5) is system transform function, which transforms the object function (, ,) f rz θ to the received signal (, , ) ee Rk z θ . In the study, the exact form is used to construct image instead of an approximate form. From equation (4.4), using Fourier transform theory, we have another expression in spectrum , z kk θ domain.
the surface of transducer is 50mm. To see the sidelobes of the constructed images, line plots of the single point scatter along r, z and θ direction in the z θ − plane, r θ − plane and zr − plane are shown in Fig.4.3. From the results it can be seen that imaging is very similar to the result of PSF. This means that the approximations (4.10) due to a finite temporal bandwidth and limited spatial Fourier-domain coverage that are typical in medical ultrasonic imaging do not significantly affect the equality of constructed images in terms of spatial resolutions, sidelobes, and contrast. Fig.4.5 shows the images for nine scatters. The nearest distance between surface of the transducer and the geometry center of the object is chosen to be 50mm ( 0 r =90mm), 100mm ( 0 r =140mm) and 200mm ( 0 r =240mm), respectively. An interesting phenomenon is that simulation shows the sidelobes and resolution of the images of the object, of which the geometry center is at different distance, is nearly the same (Fig.4.6). The reason is that the transmitted field keeps the same form as the theory prediction along r direction if the filed is cylindrical wave. Though the side lobe and resolution in the r and θ direction is the same for the larger area, the sidelobe rises and the resolution is lower in the regular coordinate system when r becomes larger because of the relationships cos( ), sin( ) zr xr θ θ = = .

Conclusion
In this section a new 3D images system in cylindrical coordinate has been developed with cylindrical wave beams (zero order Hankle function). This computation is much less than conventional delay and sum method, so the method has a potential to achieve a high image frame rate and can be implemented with relatively simple inexpensive hardware because the FFT and IFFT algorithm can be used. Computer simulation with the new method has been carried out to construct 3D images. Though the aperture geometry of the transducer is only part of a cylinder, and the transmitted filed is not exact zero order Hankle function, the results of the simulation still match theoretical prediction. So the new imaging method is robust and is not sensitive to various limitations imposed by practical system. In addition, though the discussion above is mainly for 2D cylindrical transducer in three-dimension in the cylindrical coordinate system, the method can be used directly for the 1D curve transducer in two-dimension in the polar coordinate system obviously.