Parameter estimation and error analysis of range data

Study on Extraction and Expression Methods of Bulging Diseases in Ancient City Walls

“…Plane fitting refers to the process of calculating the required plane from the acquired discrete points (Bo, 2018), usually using eigenvalue method, least squares method, etc. (Acharya et al, 1988;He et al, 2003;Wang et al,2001;Guan et al, 2008). Due to the surface of the ancient city wall, there are both an inflated portion and a non-bulging portion.…”

In Situ Measurement of 3D Crystal Size Distribution by Double-View Image Analysis with Case Study onl-Glutamic Acid Crystallization

“…Suppose a fitting plane expressed by ax + by + cz = d , where a , b , and c are unit normal vectors of the plane, satisfying a 2 + b 2 + c 2 = 1 and d ≥ 0. For four space points denoted by { P n ( x n , y n , z n ), n = 1,2,3,4}, a recognized optimization program for determining the fitting plane parameters ( a , b , c , d ) can be used, To solve the above optimal program, let s n = | ax n + by n + cz n – d | and a penalty function with the Lagrange multiplier is defined by The derivative of eq with respect to d is obtained as By letting eq be zero, it yields Similarly, by letting the derivative of eq with respect to a , b , c be zero, respectively, there follows where Δ x n = x n – x̅ n , Δ y n = y n – y̅ n , and Δ z n = z n – z̅ n .…”

“…Suppose a fitting plane expressed by ax + by + cz = d, where a, b, and c are unit normal vectors of the plane, satisfying a 2 + b 2 + c 2 = 1 and d ≥ 0. For four space points denoted by {P n (x n , y n , z n ), n = 1,2,3,4}, a recognized optimization program 39 for determining the fitting plane parameters (a, b, c, d) can be used,…”