## Introduction

## Materials and Methods

### Theory of SVD

*m*×

*n*matrix

*S*can be formulated as

*∑*is an

*m*×

*n*diagonal matrix, and

*U*and

*V*are

*m*×

*m*and

*n*×

*n*orthonormal matrices, respectively. The operator * in Eq. (1) presents the conjugate transpose. From Eq. (1), the matrix

*S*can be decomposed as

*σ*

*is the*

_{i}*i*

^{th}singular value, which is a diagonal element of

*∑*, and the

*i*

^{th}column of

*U*and

*V*are referred to as the left singular vector

*u*

*and right singular vector*

_{i}*v*

*, respectively.*

_{i}*A*

*is the*

_{i}*i*

^{th}component of

*S*decomposed by SVD. As these decomposed components contain independent features of the signals, each component can be selectively taken to exclude undesired characteristics in the signals.

### Adaptive SVD Filtering

*P*and

*N*were set to be 480 and 960, respectively, in this study.

*f*

*(*

_{mean}*i*) is the mean frequency of the

*i*

^{th}decomposed 1-D array of the SVD component,

*f*

*is the sampling frequency of the received data,*

_{s}*N*is the number of samples in the scanline, and

*x*

*(*

_{i}*n*) presents the

*n*

*element signal of the*

^{th}*i*

^{th}1-D array SVD component. For this study, the number of scanlines and the number of samples before demodulation were 128 and 1,920, respectively. The baseband data had 960 samples after decimation.

*k*) is used, where

*k*is the harmonic order of HIFU signals. For example,

*k*of 1 indicates the fundamental frequency and

*k*of 2, 3, … are the second and third harmonics, and so on. The attenuation factor αk determines the amount of HIFU fundamental and harmonic components that must be removed from the decomposed singular value components.

*f*

*is the fundamental center frequency of the HIFU signal,*

_{HIFU}*f*

*is the center frequency of the ultrasound image signal, and*

_{c}*α*

*is the attenuation factor of the*

_{k}*k*

*HIFU interference.*

^{th}*f*

*represents the bandwidth of the HIFU signals. The HIFU bandwidth was initially determined from the −3 dB operating frequency of the transducer measured by the manufacturer, which was tuned to be optimal in the experimental settings of this study. Eq. (4) searches for whether the signal of interest is placed within the HIFU frequency band and whether its singular value is greater than the attenuation factor. When both conditions are satisfied, the component is classified into the HIFU interference signal to be removed by simply excluding SVD components that are classified as HIFU signals prior to the inverse Hankel process. The interference-free ultrasound image can then be produced by adding the remaining components, excluding the undesired HIFU components. This process is repeated for all scan-lines to form a single frame of a 2-D ultrasound image.*

_{BW}### Experimental Setup

## Results

### HIFU Interference Classification

### Baseline Imaging Results with and without HIFU Interference

### Adaptive SVD-Based Filtering Results under HIFU Exposure

## Discussion

*in vivo*preclinical study targeting uterine fibroid treatment will focus on the practical, clinical use of this technique.