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Areas Within Irregular Boundaries

Areas Within Irregular Boundaries

Calculating areas within irregular boundaries, such as along rivers or curved roads, is crucial in various fields, including surveying, engineering, and environmental management. Surveyors often begin by creating a traverse, a connected series of straight lines approximating the area's boundary. The coordinates of each traverse point are essential for calculating the enclosed area. The double meridian distance formula is a widely used technique for this purpose. This method utilizes the...

Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

Area Computation by the Alternative Coordinate Method

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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...

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Updated: Jun 23, 2026

Scanning Light Scattering Profiler (SLPS) Based Methodology to Quantitatively Evaluate Forward and Backward Light Scattering from Intraocular Lenses

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Regularizing method for the determination of the backscatter cross section in lidar data.

Yanfei Wang¹, Jianzhong Zhang, Andreas Roncat

¹Key Laboratory of Petroleum Geophysics, Institute of Geology and Geophysics, Chinese Academy of Sciences, PO Box 9825, Beijing 100029, China. yfwang_ucf@yahoo.com

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|May 5, 2009

Summary

Retrieving lidar backscatter cross sections is crucial for remote sensing. A new regularizing method improves accuracy by avoiding invalid Gaussian assumptions and suppressing noise in ill-posed deconvolution problems.

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Area of Science:

Remote Sensing
Geophysics
Signal Processing

Background:

Accurate retrieval of backscatter cross section from lidar data is essential for remote sensing applications.
Conventional methods like Gaussian decomposition rely on assumptions about laser pulses and scatterers, which may not hold for complex land surfaces.
The determination of the backscatter cross section involves deconvolution, an inherently ill-posed problem susceptible to noise propagation.

Purpose of the Study:

To propose a novel regularizing method for solving the ill-posed problem of backscatter cross section retrieval in lidar data.
To address the limitations of existing methods that make potentially invalid assumptions about land surface characteristics.
To enhance numerical computation stability and noise suppression during deconvolution.

Main Methods:

Development of a regularizing method for deconvolution of lidar waveforms.
Implementation of an a posteriori approach for selecting the regularization parameter.
Numerical validation of the proposed method against established techniques.

Main Results:

The proposed regularizing method effectively recovers the backscatter cross section from lidar data.
The method demonstrates robustness against noise, unlike traditional approaches.
It alleviates computational difficulties associated with ill-posed deconvolution problems.

Conclusions:

The presented regularizing method offers a more reliable approach for backscatter cross section retrieval in remote sensing.
This technique overcomes the limitations of Gaussian decomposition for complex scattering environments.
The a posteriori parameter choice enhances the practical applicability and stability of the lidar data processing.