## 1 Introduction  
Machined surface topography refers to the microgeometry shape left on the workpiece surface by the tools and chips during the process of machining [1], which not only has a significant impact on the function of the mechanical product, but also affects the surface characteristics of the mechanical parts, including wear, lubrication, friction, corrosion resistance, and reflection [2; 3; 4]. Accurate and effective surface topography information is of importance to the modern engineering field. At present, surface topographic information is mainly obtained by the contact measurement method and non-contact measurement method. The main equipment used in contact measurement method is the stylus profilometer. However, due to the limitations of measurement conditions and stylus, the measurement results of linear sampling points cannot represent the whole surface topography, and the workpiece surface cannot avoid being scratched [5]. Moreover, there are many types of non-contact measurement methods, including optical methods [6], ultrasonic methods [7], and three-dimensional (3D) reconstruction methods. Optical and ultrasonic methods have high requirements for their measurement environment, and they also have a limited measurement range. With the development of 3D reconstruction, 3D reconstruction measurement methods have been widely used for non-contact measurement. [PERSON] et al. [8] proposed the 3D reconstruction method of wear particle surfaces based on a photometric stereo, and obtained the surface topographies of wear particles for further feature-based wear particle identification. [PERSON] et al. [9] used the 3D reconstruction method based on the shape from a focus to determine the grinding wheel surface topography. Three-dimensional reconstruction is the process of recovering 3D shapes of objects from single or multiple images [10; 11; 12], and the shape from shading (SFS) method is considered to be one of the fastest and most efficient 3D reconstruction methods [13; 14; 15]. Therefore, the SFS-based reconstruction method has attracted attention from numerous researchers. [PERSON] et al. [16] used a method that combined SFS and stereo to establish the functional energy with a detail-preserving anisotropic second-order smoothness term, and estimated the depth of the object surface. [PERSON] et al. [17] proposed a Lambert-Phong hybrid model algorithm, and obtained the coordinate information of the highlighted region of the droplet surface by the mask regions with convolutional neural network (R-CNN). Furthermore, they used the Taylor expansion and Newton iteration method in the reflection model to obtain the heights of all the positions. [PERSON] et al. [18] proposed a new roughness evaluation method based on the complex wavelet enhanced SFS transform, and recovered the 3D topography of a milling surface using the 3D models that were developed from the digital image. [PERSON] et al. [19] proposed a novel method that uses Fourier transform to fuse the depth from focus (DFF) and SFS, and achieved the reconstruction of the tool wear image in addition to ensuring the continuity of the depth map. [PERSON] et al. [20] used the SFS algorithm to reconstruct the 3D shapes of the welding seam, and extracted the curvature information as the feature vector of the welds from the reconstructed topography. The SFS algorithm is essentially an ill-conditioned problem. Therefore, the SFS algorithm is generally combined with other methods in studies, which adds many constraints when reconstructing the surface topography. However, it easily affects the accuracy and smoothness of the reconstructed surface. Our analytical model is effective, simple, and easy to use, and has been widely used in the machining field. By using the machining variables such as spindle speed, feed rate, depth of cut, and step over, [PERSON] and [PERSON] [21] developed a statistical model for surface roughness estimation in a high-speed flat end milling process under wet cutting conditions.