Recently, work in this group has focused on the lateral cosine modulation method (LCM) which can be used for next-generation ultrasound (US) echo imaging and tissue displacement vector/strain tensor measurements (blood, soft tissues, etc.). For instance, in US echo imaging, a high lateral spatial resolution as well as a high axial spatial resolution can be obtained, and in tissue displacement vector measurements, accurate measurements of lateral tissue displacements as well as of axial tissue displacements can be realized. For an optimal determination of an apodization function for the LCM method, the regularized, weighted minimum-norm least squares (WMNLSs) estimation method is presented in this study. For designed Gaussian-type point spread functions (PSFs) with lateral modulation as an example, the regularized WMNLS estimation in simulations yields better approximations of the designed PSFs having wider lateral bandwidths than a Fraunhofer approximation and a singular-value decomposition (SVD). The usefulness of the regularized WMNLS estimation for the determination of apodization functions is demonstrated.

A beamformer and a transducer are used in applications such as medical ultrasound (US) imaging, blood flow measurement, tissue displacement/strain measurements, and sonar measurements. For these applications, US beamforming parameters such as US frequency, US bandwidth, pulse shape, effective aperture size, and the apodization function are chosen or selected, and appropriate values are set. In addition, US transducer parameters such as the size and materials used for the US array elements are also chosen. In choosing such settings, the US properties of the target are also considered (e.g., attenuation and scattering). Thus, all of the above parameters must be appropriately chosen and set when considering a system that involves the US properties of the target. In general, such parameters are chosen using the knowledge and experience of an engineer.

Recently, a cosine modulation (LCM) method [

For the LCM method, a lateral Gaussian envelope cosine modulation method (LGECM) [

Currently, efforts are being made to search for the optimal PSF for both US imaging and displacement vector measurements, and in order to construct a required or designed point spread function (PSF), it is proposed to select the aforementioned beamforming parameters on the basis of linear or nonlinear optimizations [

For LGECM used in the simulations in [

In the next trial, in this study, the weighted minimum-norm least squares (WMNLSs) estimation method [

For the optimization of target parameters, the beam properties of one element must be obtained in advance using analytical, numerical, or experimental methods as a function of the parameters [

Beamforming is performed during either the transmission or reception of US, or during both. Thus, conventionally, an apodization function can be obtained by dealing with either the transmission or the reception of US. Alternatively, apodization functions may be determined for both, the transmission and reception of US.

In this study, the simultaneous linear equations in [

However, note that because the independence of the rows of matrix

In this study, a regularized, weighted least squares estimation was used [

The minimization of (

In the next section, the same beam property consisting of one element calculated with Field II [

Here, as in [

In Figure

Apodization functions obtained using a regularized WMNLS estimation (solid line with circles), a Fraunhofer approximation (dashed line with triangles), and a SVD (dash-dot line with squares) for designed lateral Gaussian-type PSFs with (a)

For the respective values of

Designed Gaussian-type PSFs and PSFs obtained using apodization functions obtained with a regularized WMNLS estimation, a Fraunhofer approximation, and SVD for (a)

Lateral intensity profiles of a designed PSF (dash-dot line with square) and PSFs obtained using apodization functions obtained with a regularized WMNLS estimation (solid line with circle), a Fraunhofer approximation (dashed line with triangle) and a SVD (dotted line with cross) for (a)

Figure

Apodization functions obtained for

For Gaussian-type PSFs (i.e., LGECM), the regularized WMNLS estimation yielded better approximated PSFs having wider lateral bandwidths than the Fraunhofer approximation and the SVD method. The usefulness of the regularized WMNLS method for defining apodization functions was demonstrated. The effectiveness of the spatially variant weightings and regularization will be specifically reported elsewhere.

Moreover, it has recently been reported that PSFs having envelope shapes of Akaike window, power functions, and new windows which were developed by changing the Hanning window used in the Turkey window by the Akaike window or power functions are desirable in the sense that a wider bandwidth and a higher echo SNR can be obtained than with a Gaussian-type PSF [

In conventional US imaging (i.e., not modulation imaging) of a cyst, contrast resolution is optimized with another least squares estimation [

Specifically, for instance, in US imaging, a spatial resolution of less than 3 mm is currently required to overcome the clinical limitations in conventional digital US imaging equipment. Accurate 3D US imaging, 3D tissue motion measurements (3D blood flow vector, tissue strain tensors, etc.), and 3D shear modulus reconstructions [

Such LCM methods can also be used in US harmonic imaging and measurements as well as in radar applications [

Thus, efforts to develop new US diagnosis/treatment systems using proper beamforming and various methods of computational imaging are currently underway.