# Extended Smoothing Methods for Sparse Test Data Based on Zero-Padding
## 1 Introduction  
In engineering practice, affected by the test conditions, often only sparse test data can be obtained. For example, the geographical environment test cannot fully test the environmental information of each point, and it is necessary to smooth the test data based on a limited number of observations to get the overall terrain data [1]. These problems also arise in the fields of sound [2] and image processing [3]. As a result, obtaining more complete results based on sparse test data has become a key research priority. Smoothing sparse test data can not only solve the problem of too little test data but also, to a certain extent, speed up the processes of data acquisition, processing, and practical application. Interpolation methods have been proposed to smooth sparse test data for the purpose of enhancing the characteristics of the data under study [4; 5]. The main interpolation methods include Lagrange fitting [6; 7], piecewise cubic Hermite fitting [8; 9; 10; 11], least squares [2; 3; 12], cubic spline curve method [13; 14] and others. To expand on this, [PERSON] first established the least squares regression model using the measurement data of the two-dimensional laser sensor in order to accurately extract the wheel contour, and then applied the Lagrange multiplication to obtain the least squares solution of the parameters, which gave a smooth and continuous contour [6]. Data from second-order systems can be more easily fitted using the algebraic hyperbolic cubic Hermite spline interpolation approach provided by [PERSON] et al. [8]. The above two interpolation methods have a narrow application range, and there are some problems such as non-convergence and non-smoothness of piecewise linear interpolation function at nodes, while the cubic spline function can solve the above problems and the calculation is simple. The cubic spline function can pass through all sampling points and has a second continuous derivative, which is an ideal interpolation tool [15]. [PERSON] et al. improved the cubic spline interpolation method, solved the problem of cubic spline interpolation of dynamic curves, and enabled the measured discrete signals to form a smooth curve [16]. [PERSON] et al. studied the advantages and disadvantages of ordinary Kriging interpolation, cubic spline interpolation and inverse distance weight reciprocal interpolation, aiming to guide researchers to choose appropriate spatial interpolation methods according totheir actual measurement needs [17]. Each of these curve smoothing methods has its own advantages, but they also have some limitations. The advantage is that they greatly improve the smoothness and accuracy of the curve; the limitation is that they can only be used under specific conditions and are not universal. In addition, zero-padding of the angular spectrum allows for an increase in data resolution [18; 19; 20]. Some scholars discussed the zero-padding problem of one-dimensional Fourier transform, and obtained the spectrum value by the zero-padding of one-dimensional test data. In contrast, this paper focuses on the zero-padding of the wavenumber domain data (i.e., the angular spectrum) after the two-dimensional Fourier transform. Furthermore, a number of other scholars are also working on such data processing issues [21]. They optimize the traditional FFT/IFFT method to get better imaging performance when studying 3D imaging. [PERSON] performs a phase shift to compensate the spherical wave effect after an FFT, and the efficiency and accuracy are comparable in the case of full sampling [22]. [PERSON] applies zero-padding to interpolate the spectrum of the sampled signal and finally obtains the three-dimensional human image [23]. In addition, in the field of frequency estimation, [PERSON] et al. can estimate the frequency of complex sinusoidal signals flexibly and accurately by using DFT interpolation and zero-padding on the original input data [24].