## 1 Introduction  
An inverse problem refers to any problem related to finding the desired information using an indirect method such as observation when some information cannot be found directly in the field of mathematics or science. In other words, it indicates all problems related to obtaining the values of model parameters from observational data. This inverse problem has been widely used and studied not only in radiology but also in fields closely related to our lives, such as earth science, astronomy, engineering, and non-destructive inspection. Related works can be found in [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14] and references therein. The inverse scattering problem, which is one of the inverse problems, happens when electromagnetic waves whose information such as frequency or wavelength are known emitted into a scatterer included in the arbitrary space. Then, the problem is to find out the location, size, shape, and characteristics of the scatterers from the scattered waves. Although this inverse scattering problem is a significant field for research, it is known to be a difficult and complicated problem due to the fundamental nonlinearity and ill-posedness. Thus, no particular solutions have been developed other than a numerical method based on an iterative algorithm such as [PERSON]'s method; refer to [15; 16; 17; 18; 19; 20; 21; 22]. It is well-known that these iterative methods may not converge or may converge to a shape different from the ground truth unless a good initial value is given. In addition, although it converges to the desired shape, it may take a very long time. Therefore, there is a need for a non-iterative method that can directly obtain approximate information such as shape, location, etc., of a scatterer from the collected scattering wave despite the absence of an initial value. Motivated by this, various non-iterative methods have already been developed for the inverse scattering problem. Recently, the linear sampling method (LSM), which is based on the structure of the range of a self-adjoint operator, was developed and applied for determining the locations and shapes of unknown scatterers [23; 24; 25]. However, the LSM requires a large number of directions of the incident and corresponding scattered fields, and most of the research deals with the full-view inverse scattering problem. MUltiple Slgnal Classification (MUSIC), which is a method of characterizing the range of a self-adjoint operator (see [26] for instance), is also applied to various inverse scattering problems and microwave imaging [27; 28; 29]. Let us mention that the location of small scatterers can be identified clearly through the MUSIC in the full-view problem, but, in the limited-view problem, incorrect locations are retrieved, refer to [30]. Moreover, similar to the LSM, MUSIC requires a significant number of directions of the incident and scattered field data. Direct and orthogonality sampling methods are fast, effective, and robust techniques for retrieving small scatterers with single source [31; 32; 33], but the imaging performance is significantly dependent on the location of source or direction of incident field. Moreover, similar to the MUSIC, incorrect locations of small scatterers are retrieved in a limited-view inverse scattering problem. Topological derivative (TD) based imaging techniques have been successfully applied to various inverse scattering problems [34; 35; 36]. One of the advantages of the TD is that one can obtain good results even with a small number of incident field data, but most of research is applied to the full-view inverse scattering problem. As shown above, the inverse scattering problem can be divided into a full-view situation and a limited-view situation. We refer to Figure 1 for an illustration of the full- and limited-view inverse scattering problems. In situation such as exploring the ground, only limited-view data can be collected, so research on the limited-view situation is required.