## 2 Related Work  
In general, there are many ways to perform elastography, each with their own strengths, weaknesses, and applications. One of the highest quality methods of elastography is magnetic resonance elastography (MRE) [23; 24; 25; 26; 27]. While this method offers high quality elastography images measured by its contrast-to-noise ratio (CNR), unfortunately, it suffers from pitfalls including low spatial resolution and low specificity. Moreover, broad application of MRI for breast cancer screening or assessment is not clinically feasible due to its high cost. USE on the other hand is cost-effective and easy to use, while it relatively has high spatial resolution. However, it does not have desirable overall quality compared to MRE. To enhance the quality of tissue displacement estimates which is necessary to ramp up the quality of USE images, 3D USE has been performed with reasonable success [28; 29; 30]. While high quality, 3D USE is far less accessible than 2D USE. Unfortunately, 2D USE introduces complications due to the simplification of a 3D problem into a 2D counterpart, leading to compromise in generated USE image quality. Recently, a 2D tissue motion tracking method was developed which has been shown to produce high quality displacement field [19]. This method starts with an initial rough estimate of the displacement field and applies regularization derived from an approximate 3D tissue mechanical model, leading to accurate estimation of both axial and lateral tissue displacement fields.  
In 2D USE, there are major methods, including quasi-static elastography and shear wave elastography. Quasi-static elastography includes two methodologies: strain imaging and elastic modulus imaging. Both methodologies follow the same fundamental approach, where the tissue is mechanically stimulated through external application of compression by ultrasound probe or intrinsically through blood pulsation [31; 32] before generated tissue displacement data are employed to create a strain or elastic modulus image. Unlike strain imaging, which estimates the tissue strain as a measure of stiffness, elastic modulus elastography utilizes an inversion framework to process the estimated displacement field before elastic modulus reconstruction [33; 34; 35; 36]. The advantage of this methodology is that the hardware required is relatively simple, while the image reconstruction involves only a few assumptions. Shear wave elastography generates shear waves propagating through the tissue by producing an acoustic force or push impulse before the tissue shear modulus is reconstructed based on the estimated shear wave velocity [37; 38]. One very important advantage of elastic modulus imaging techniques is that they provide a quantitative measurement of the underlying tissue stiffness, whereas strain imaging can only provide a qualitative image of stiffness at best. This makes these methods attractive to the diagnosis of diseases like breast cancer, liver cancer, fatty liver disease, and cirrhosis where an objective measurement is required [26; 39; 40]. This investigation focuses on quasi-static ultrasound elastography (QUSE), where the tissue is mechanically stimulated via very low-frequency tissue compression by the operator.