6

Two systematic ways of smoothing a function $h$ would be: 1. Join the piecewise smooth parts of your function using Hermite interpolation so that the derivatives are matched to your satisfaction. 2. Convolve your function $h(x)$ with a heat kernel of the form $f(x) = \frac{\exp\left\{-\frac{x^2}{2 \sigma^2}\right\}}{\sqrt{2 \pi \sigma^2}}$ so that instead ...


6

To complement the two answers from Daniel Shapero and Nicoguaro: Basically, there are two ways of smoothing a mesh, subdivision (generate new vertices) and smoothing (move the points in such a way that the obtained shape is smoother). Subdivision To grasp the intuition, imagine you want to "smoothen" a 2D square. The 2D square is not smooth because it has ...


5

As mentioned in the answer by @DanielShapero, you can follow an approach based on local approximations of the curvature for your nodes. In the post he suggest, there is an article by Desbrun. I would suggest to check another article by him: Anisotropic Polygonal Remeshing [1]. Another option that comes to my mind is to use Catmull-Clark subdivision ...


4

For just mesh smoothing, you can start by looking at Laplacian smoothing and some of the references therein. The idea is to update the position of every vertex in the interior of the mesh by replacing it with the average of its neighbors. There are loads of more sophisticated ways of doing this by using different operators. If you're doing both surface mesh ...


4

Meshing algorithms can place the vertices of triangles and tetrahedra in a wide variety of ways, but they are us usually essentially constructive (i.e new vertices are introduced, existing vertices stay where they are). Disappointingly, this can cause the meshes which are generated to be unsuitable for numerical calculation. Mesh smoothing, at least in the ...


3

In microwave imaging, a great chunk of literature is devoted to regularization and its effect on the solution process and inversion results. One of the common methods for microwave imaging is the Contrast Source Inversion (CSI) method, which is essentially a gradient-based optimization. In CSI, one would formulate the inverse problem as the optimization of ...


3

Surprisingly, Lloyd smoothing hasn't come up here yet. Check out Du, Qiang; Faber, Vance; Gunzburger, Max (1999), "Centroidal Voronoi tessellations: applications and algorithms", SIAM Review, 41 (4): 637–676 (and perhaps voropy, a small project of mine, if you're interested to see Lloyd smoothing in action).


3

Is the white part NaNs? If so, then you will need to use some sort of extrapolation to smooth that region. The function inpaint_nans may be appropriate (it smoothly fills in NaN regions, essentially by solving a Laplace equation). If more smoothing is needed, you could then follow Juan's approach (i.e. gaussian blur). Another thing you might consider: I do ...


2

You can regard your original matrix as an image (i.e., a real-valued function $I = I(x, y)$), then you can apply a Gaussian filter (convolve $I$ with a Gaussian kernel), and finally you can get the contours of the filtered image/matrix. MATLAB's Image Processing toolbox already has the functionality available.


2

Yes. You need to choose the radius $h$ large enough that for each particle, there is always a significant number of other particles within the first particle's radius.


2

Separately, but it does depend. Not very strongly, however: A very large number of pre- and post-smoothing steps only improves the convergence rate a little bit over a large number of steps. The difference is most between using one, two, or three pre- and post-smoothing steps. And you need both pre- and post-smoothing steps. You cannot compensate for no pre-...


2

The least squares optimization problem in your question has a penalty term that only penalizes 1st derivatives. However, for a Gaussian kernel, the corresponding penalty term would have to penalize all derivatives because Gaussians are infinitely smooth. A mathematically precise way to link various penalty terms to smoothing kernel functions would require ...


1

I believe these issues arise because you are solving the transient form of the heat equation which, being parabolic, can lead to some instability for continuous galerkin method. There are ways to circumvent that. GGLS approaches (Galerkin Gradient Least Square) introduce stabilization term that dampen these oscillations by minimizing the Error on ...


1

You have to normalize by the number of elements in the FFT. That is if the size of your matrix is $(NxM)$ you must normalize your FFT by $(NxM)$. You can check this is valid by looking at the energy content of the two matrices.


1

Fitting together B-splines with a given continuity is a hard problem. This was actually the motivation behind developing a generalization of NURBS called T-splines. There are many articles on T-splines and T-NURCCs that could be helpful to you if you're interested in going that route.


1

If there's some model for the signal for $t < t_0$ and $t > t_f$, perhaps it would be possible to try to fit the full signal using a Metropolis-Hastings or Goodman-Weare algorithm. This could then also yield some information about the probability distributions for the parameters involved.


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