I am interested to know some ways to approximate discrete gradient if you have a function on point clouds in 2D or 3D.

If you have a function defined on a grid, it well known that you can use a standard difference formula to compute the partial derivative in each direction.

I came across a paper by Luo et al entitled Approximating Gradients for Meshes and Point Clouds via Diffusion Metric which uses diffusion metric. Another idea that came into mind is to defined a smoothed field using a specific kernel, then to take the gradient of the smoothed function.

Is there a standard way of doing this?


2 Answers 2


Very many numerical methods for partial differential equations compute derivatives from values on something that is not a regular grid (they use unstructured grids). Usually these are finite element methods and they use more than just point values (they have various local integrals of the function available).

A class of methods based on what you're asking for are the so-called meshfree methods. Typically, they use the point values to generate a continuous approximation by convolving with some smooth kernel (e.g., a Gaussian). Then they compute derivatives of the continuous function. Of course, if you have a lot of points this becomes expensive and there are lots of additional tricks you can play (e.g., using a kernel with compact support; employing the fast multipole method, etc.).


Certainly, there is not a standard method - and be aware that the calculation will be sensitive to noise.

A straightforward of calculating the gradient would be: Take some point $x$ and choose a numbers of neigbors (the nearest $k$, say). Now fit a hyperplane through the $k+1$ points (e.g. by a least squares fit) and take the slope of this plane as the gradient in $x$.


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