# Practical Guide to Ultrasound Beam Forming: Beam Pattern and Image Reconstruction Analysis
## 1 Introduction  
In ultrasound medical imaging, beam forming in essence deals with the shaping of the spatial distribution of the pressure field amplitude in the volume of interest, and the consequent recombination of the received ultrasound signals for the purpose of generating images. One can thus navigate through the different techniques using the following question as a compass: which imaging features are important to my application of interest, and which features can I sacrifice? There is, in fact, no ultimate beam-forming approach, and the answer to the previous question strongly depends on what one wants to see in the images. Below are the key imaging features that will be considered in this paper to review the different beam-forming techniques, along with their descriptions:  
**Spatial resolution:** the smallest spatial distance for which two scatterers can be distinguished in the final image. Spatial resolution can be either axial (along the direction of propagation of the ultrasound wave), lateral, or elevation resolution (along the plane to which the direction of propagation is perpendicular). This feature is normally expressed in mm. **Temporal resolution:** the time interval between two consecutive images. This feature is normally expressed in Hz. **Contrast:** the capability to visually delineate different objects, e.g., different tissue types, in the generated images. This feature is generally expressed in dB, and it is a relative measure between image intensities. **Penetration depth:** the larger depths for which a sufficiently high signal-to-noise ratio (SNR) level can be maintained. This feature is normally expressed in cm. **Array aperture:** the physical sizes of the surface representing the combined distribution of active and passive ultrasound sensors: in other words, the array footprint. The array aperture is definedby the number of ultrasound sensors (elements), their sizes, and their distribution. This feature is generally expressed in cm\({}^{2}\). **Field of view (FOV):** the sizes of the area represented by the obtained images. This feature is generally expressed in cm\({}^{2}\) or cm\({}^{3}\). Although introduced individually, these features are strongly related. For example, a decrease in temporal resolution can be traded to achieve a higher spatial resolution or a larger FOV; a deeper penetration depth can be achieved by lowering the transmitted center frequency, thus deteriorating spatial resolution, or a broader insonification area can be achieved by widening the transmitted beam, which will result in lower pressure levels being generated, thus lowering the SNR compared to a focused beam. To help the reader become more familiar with these concepts, a simple model can be used. Assuming linear propagation, the following wave equation can be applied to model the pressure field generated by an arbitrary source which propagates in a homogeneous medium [1]:  
\[\partial_{x}^{2}p(\mathbf{x},t)-\frac{1}{c_{0}^{2}}\partial_{t}^{2}p(\mathbf{x},t)=S( \mathbf{x},t) \tag{1}\]  
Here, \(\partial_{x}^{2}\) and \(\partial_{t}^{2}\) represent the second-order derivative as regards space and time, respectively, \(p(\mathbf{x},t)\) is the pressure field, \(t\) is time, \(\mathbf{x}=(x,y,z)\) is the three-dimensional spatial coordinate in a Cartesian system, \(c_{0}\) is the small signal speed of sound, and \(S(\mathbf{x},t)\) is the source. For a monochromatic point source, i.e., \(S(\mathbf{x},t)=\delta(\mathbf{x})\cos(2\pi f_{0}t)\), the solution to this equation is known [1], and can be expressed as:  
\[p_{MPs}(\mathbf{x},t)=\frac{P_{0}}{4\pi|\mathbf{x}|}\cos\left[2\pi f_{0}\left(t-\frac{ |\mathbf{x}|}{c_{0}}\right)\right] \tag{2}\]  
In this equation, \(p_{MPs}(\mathbf{x},t)\) is the pressure field generated by the monochromatic point source, \(P_{0}\) is the source amplitude, and \(f_{0}\) is the source frequency.