Can I estimate hessian matrix without using of curve fitting

Hi I have a lot of points(3d). The goal is calculation hessian matrix in each point. So I must fit a curve for each point first. In my field, for each point, a curve is estimated based on local neighbors(not a general curve for all points!). So, first I select neighbors but I don't want use polyfit or polyfitn for curve fitting,they have high errors because type of each curve is general. can anybody, any suggestion for discrete calculation of hessian matrix(So I am n't forced to fit a curve! and I can estimate hessian matrix directly with out using curve fitting), or any suggestion for a curve fitting that doesn't want type of curve! and have low error! Thanks a lot

Answer 1

[Walter Roberson]

You can use knn to find the local neighbors. There is a specific knn routine that is apparently fairly efficient.

But once you have the local neighbors you can not estimate the curve based on the local neighbors without having any idea of what the equation for the curve might be. If you have no idea what the curve might be then you have to assume that between the point and the located neighbors that the curve "really" behaves arbitrarily badly, that the "real" surface might have massive peaks or valleys or might have holes or might have a 6.318 yottahertz sine wave.

[masyrezaei]

Thanks, But I don't know type of curve so what is the most general curve fitting? But really ideally I want a method for estimating hessian matrix that it doesn't curve fitting! do you have any suggestion for it?

[Walter Roberson]

The most general curve fitting would have to involve at least multinomials, exp(), ln(), and periodic functions such as sin().

Remember, that for any given finite number of points, there is a multinomial that algebraically fits the points perfectly -- but that does not mean that the multinomial correctly represents that real surface. For example, given any finite number of points generated from sin() you can perfectly fit a polynomial of degree at most (number of points minus 1), but any finite polynomial goes off to +infinity or -infinity as the parameter approaches infinity, which sin() does not do. You need an infinite polynomial to fit sin() correctly.

For any given finite number of points, there are an infinite number of functions that algebraically fit the points perfectly. For example for any function whose values were known at positive integers, you could add in sin(2*pi*x*F) where F is an integer and x is the independent variable: as sin(2*pi*AnInteger) is 0, this would not affect the value of the function at those known points, but it is clearly a very different function between the known points.

Given any finite number of points and no information about what really happens between the points, it is impossible to say that what the function "really" is. I would say that the task gets even harder when you allow for the possibility that the function values are only known to finite precision or that the known function values might have noise, but that would be incorrect, as even without those aspects it is already the continuous infinity of possibilities and there is no larger known infinity.

In short, in order to estimate the Hessian, you need to assume that the values can be approximated by some model... and that gets you to what you refer to as curve fitting. A curve fit on the basis of the closest 1023 points is still curve fitting.

[masyrezaei]

Thanks a lot So what is your suggestion for curve fitting function in matlab?

[Walter Roberson]

Hessian implies connectivity. Do you even have that, or are you using a point cloud?

[masyrezaei]

No, I have point cloud

[Walter Roberson]

Is the idea that the point cloud is a sampling of something that is the surface of a single continuous function? No sharp edges? For example if a whole hen's egg had been scanned (see analytic formula) ? But not, for example, a face (that would require a composition of functions to model the nose and ears and various other parts.)

If the point cloud does not represent a single continuous smooth function, then a hessian has no real meaning, as you do not know that the points you are examining are part of the same function.

If you are looking for ways to determine the surface of a point cloud, consider using the new boundaries() function, which uses alpha shapes.

[Walter Roberson]

https://en.wikipedia.org/wiki/Not_even_wrong

[masyrezaei]

I don't understand the last response, but thanks for all responses