# A Sibelobe Suppressing Beamformer for Coherent Plane Wave Compounding
## 2 Materials and Methods
### Singular Value Decomposition of CPWC Data  
For one CPWC, there are \(n_{x}\times n_{z}\times n_{a}\) samples where \(n_{x}\) and \(n_{z}\) are the number of spatial samples along \(x\)-direction and \(z\)-direction, and \(n_{a}\) the number of angles. For the 3D dataset, dimension reduction is done first by transforming the angle sequence into a spatio-angular matrix \(\mathbf{P}\) with dimensions \((n_{x}\times n_{z},n_{a})\) as shown in Figure 3. This technique is common in processing time series [17] or in other imaging modalities, such as computed tomography (CT) and magnetic resonance imaging (MRI) [20; 21; 22; 23; 24], but it is rarely used in angle compounding.  
The SVD of the matrix \(\mathbf{P}\) is then used to find decomposition matrices, such as:  
\[\mathbf{P}=\mathbf{U}\Delta\mathbf{V}^{\mathrm{H}}=\sum_{i}\lambda_{i}\mathbf{U}_{i}\times\mathbf{ V}_{i}, \tag{3}\]  
where \(\Delta\) is a diagonal matrix with dimension \((n_{x}\times n_{z},n_{a})\), \(\lambda_{i}\) are the sorted singular values inside \(\Delta\), \(\mathbf{U}\) and \(\mathbf{V}\) are orthonormal matrices with respective dimensions \((n_{x}\times n_{z},n_{x}\times n_{z})\) and \((n_{a},n_{a})\), \(\mathbf{U}_{i}\) and \(\mathbf{V}_{i}\) are the \(i\)th columns of \(\mathbf{U}\) and \(\mathbf{V}\), respectively, and \(\mathrm{.}^{\mathrm{H}}\) stands for the conjugate transpose.  
Note that each column \(\mathbf{U}_{i}\) corresponds to a spatial signal with the length of \(n_{x}\times n_{z}\) which describes, in fact, a 2D spatial image \(\mathbf{I}_{i}\) with dimensions \((n_{x},n_{z})\) and each column \(\mathbf{V}_{i}\) corresponds to an angular signal with the length of \(n_{a}\). From this perspective, the SVD of \(\mathbf{P}\) decomposes the field into a sum of separable images \(\mathbf{I}_{i}\), each characterized by a vector \(\mathbf{U}_{i}\), that are independently modulated by an angular signal \(\mathbf{V}_{i}\). Thus, owing to the SVD processing, the spatio-angular signal corresponding to ultrasonic beamformed RF data can be rewritten as:  
\[\mathbf{p}(x,z,a)=\sum_{i=1}\lambda_{i}\mathbf{I}_{i}(x,z)\mathbf{V}_{i}(a). \tag{4}\]  
Additionally, we can draw the conclusion that mainlobe components should be found mainly in the first singular values and singular vectors since their high spatio-angular coherence ensures the on-axis spatial pixels to exhibit the same angle profile. On the contrary, sidelobe signals should be described in the last singular values and singular vectors as they exhibit much lower spatio-angular coherence. Based on this, the sidelobe rejection is performed adaptively by using a threshold \(\beta\) on the number of singular vectors left from the original signal:  
\[\mathbf{p}_{mainlobe}(x,z,a)=\sum_{i=1}^{\beta}\lambda_{i}\mathbf{I}_{i}(x,z)\mathbf{V}_{ i}(a). \tag{5}\]  
The graphical explanation of this SVD filter is shown in Figure 4.  
Figure 3: The CPWC acquisitions form a 3D stack of images with two spatial dimensions (\(n_{x}\), \(n_{z}\)) and one angular dimension (\(n_{a}\)). It is reshaped in a 2D spatio-angular representation (\(P\)) where all pixels at one angle are arranged in one column (\(n_{x}\),\(n_{z}\)). As a consequence, all angle points for one pixel are arranged in one row (\(n_{a}\)).