Say we are making a program to render the plot of a function (black box) provided by the user as a sequence of line segments. We want to get the minimum number of samples of the function so the resulting image "looks" like the function (the exact meaning of "looks" here is part of the question). A naive approach might be to just sample at fixed intervals but we can probably do better than that eg by sampling the "curvy bits" more than the "linear bits". Are there systematic approaches/research on this problem?
This reference can be helpful which is using the combined sampling method. Before that its related works explain more about other methods of sampling:
There are several strategies for plotting the function y = f(x) on interval Ω = [a, b]. The
naive approach based on sampling of f in a fixed amount of the equally spaced points is
described in [20]. The simple functions suffer from oversampling, while the oscillating curves
are under-sampled; these issues are mentioned in [14]. Another approach based on the interval
constraint plot constructing a hull of the curve was described in [6], [13], [20]. The automated
detection of a useful domain and a range of the function is mentioned in [41]; the generalized
interval arithmetic approach is described in [40].
A significant refinement is represented by adaptive sampling providing a higher sampling
density in the higher-curvature regions. The are several algorithms for the curve interpolation preserving the speed, for example: [37], [42], [43]. The adaptive feed rate technique
is described in [44]. An early implementation in the Mathematica software is presented in
[39]. By reducing data, these methods are very efficient for the curve plotting. The polygonal approximation of the parametric curve based on adaptive sampling is mentioned in the
several papers. The refinement criteria, as well as the recursive approach, are discussed in
[15]. An approximation by the polygonal curves is described in [7], the robust method for
the geometric and spatial approximation of the implicit curves can be found in [27], [10], the
affine arithmetic working in the triangulated models in [32]. However, the map projections
are never defined by the implicit equations. Similar approaches can be used for graph drawing
[21].
Other techniques based on the approximation by the breakpoints can be found in many
papers: [33], [9], [3]; these approaches are used for the polygonal approximation of the closed
curves and applied in computer vision.
Hence, these are the reference methods that define some measures for a "good" plot and introduce an approach to optimize the plot base on the measure:
constructing a hull of the curve
automated detection of a useful domain and a range of the function
adaptive sampling: providing a higher sampling density in the higher-curvature regions
providing a higher sampling density in the higher-curvature regions
approximation by the polygonal curves
affine arithmetic working in the triangulated models
combined sampling: providing the polygonal approximation of the parametric curve involving the discontinuities will be presented. The modified method will be used for the function f(x) reconstruction and plot. Based on the ideas of splitting the domain into the subintervals without the discontinuities, it represents a typical problem solvable by the recursive approach.
Related
This question is not directly related to a particular programming language but is an algorithmic question.
What I have is a lot of samples of a 2D function. The samples are at random locations, they are not uniformly distributed over the domain, the sample values contain noise and each sample has a confidence-weight assigned to it.
What I'm looking for is an algorithm to reconstruct the original 2D function based on the samples, so a function y' = G(x0, x1) that approximates the original well and interpolates areas where samples are sparse smoothly.
It goes into the direction of what scipy.interpolate.griddata is doing, but with the added difficulty that:
the sample values contain noise - meaning that samples should not just be interpolated, but nearby samples also averaged in some way to average out the sampling noise.
the samples are weighted, so, samples with higher weight should contrbute more strongly to the reconstruction that those with lower weight.
scipy.interpolate.griddata seems to do a Delaunay triangulation and then use the barycentric cordinates of the triangles to interpolate values. This doesn't seem to be compatible with my requirement of weighting samples and averaging noise though.
Can someone point me in the right direction on how to solve this?
Based on the comments, the function is defined on a sphere. That simplifies life because your region is both well-studied and nicely bounded!
First, decide how many Spherical Harmonic functions you will use in your approximation. The fewer you use, the more you smooth out noise. The more you use, the more accurate it will be. But if you use any of a particular degree, you should use all of them.
And now you just impose the condition that the sum of the squares of the weighted errors should be minimized. That will lead to a system of linear equations, which you then solve to get the coefficients of each harmonic function.
Let's say I want to estimate the camera pose for a given image I and I have a set of measurements (e.g. 2D points ui and their associated 3D coordinates Pi) for which I want to minimize the error (e.g. the sum of squared reprojection errors).
My question is: How do I compute the uncertainty on my final pose estimate ?
To make my question more concrete, consider an image I from which I extracted 2D points ui and matched them with 3D points Pi. Denoting Tw the camera pose for this image, which I will be estimating, and piT the transformation mapping the 3D points to their projected 2D points. Here is a little drawing to clarify things:
My objective statement is as follows:
There exist several techniques to solve the corresponding non-linear least squares problem, consider I use the following (approximate pseudo-code for the Gauss-Newton algorithm):
I read in several places that JrT.Jr could be considered an estimate of the covariance matrix for the pose estimate. Here is a list of more accurate questions:
Can anyone explain why this is the case and/or know of a scientific document explaining this in details ?
Should I be using the value of Jr on the last iteration or should the successive JrT.Jr be somehow combined ?
Some people say that this actually is an optimistic estimate of the uncertainty, so what would be a better way to estimate the uncertainty ?
Thanks a lot, any insight on this will be appreciated.
The full mathematical argument is rather involved, but in a nutshell it goes like this:
The outer product (Jt * J) of the Jacobian matrix of the reprojection error at the optimum times itself is an approximation of the Hessian matrix of least squares error. The approximation ignores terms of order three and higher in the Taylor expansion of the error function at the optimum. See here (pag 800-801) for proof.
The inverse of the Hessian matrix is an approximation of the covariance matrix of the reprojection errors in a neighborhood of the optimal values of the parameters, under a local linear approximation of parameters-to-errors transformation (pag 814 above ref).
I do not know where the "optimistic" comment comes from. The main assumption underlying the approximation is that the behavior of the cost function (the reproj. error) in a small neighborhood of the optimum is approximately quadratic.
I was trying to make a application that compares the difference between 2 images in java with opencv. After trying various approaches I came across the algorithm called Demons algorithm.
To me it seems to give the difference of images by some transformation on each place. But I couldn't understand it since the references I found were too complex for me.
Even the demons algorithm does not do what I need I'm interested in learning it.
Can any one explain simply what happens in the demons algorithm and how to write a simple code to use that algorithm on 2 images.
I can give you an overview of general algorithms for deformable image registration, demons is one of them
There are 3 components of the algorithm, a similarity metric, a transformation model and an optimization algorithm.
A similarity metric is used to compute pixel based / patch based similarity between pixels/patches. Common similarity measures are SSD, normalized cross correlation for mono-modal images while information theoretic measures like mutual information are used in the case of multi-modal image registration.
In the case of deformable registration, they generally have a regular grid super-imposed over the image and the grid is deformed by solving an optimization problem which is formulated such that the similarity metric and the smoothness penalty imposed over the transformation is minimized. In deformable registration, once there are deformations over the grid, the final transformation at the pixel level is computed using a B-Spine interpolation of the grid at the pixel level so that the transformation is smooth and continuous.
There are 2 general approaches towards solving the optimization problem, some people use discrete optimization and solve it as a MRF optimization problem while some people use gradient descent, I think demons uses gradient descent.
In case of MRF based approaches, the unary cost is the cost for deforming each node in grid and it is the similarity computed between patches, the pairwise cost which imposes the smoothness of the grid, is generally a potts/truncated quadratic potential which ensures that neighboring nodes in the grid have almost the same displacement. Once you have the unary and pairwise cost, you feed it to a MRF optimization algorithm and get the displacements at the grid level, then you use a B-Spline interpolation to compute pixel level displacement. This process is repeated in a coarse to fine fashion over several scales and also the algorithm is run many times at each scale (reducing the displacement at each node every time).
In case of gradient descent based methods, they formulate the problem with the similarity metric and the grid transformation computed over the image and then compute the gradient of the energy function which they have formulated. The energy function is minimized using iterative gradient descent, however these approaches can get stuck in a local minima and are quite slow.
Some popular methods are DROP, Elastix, itk provides some tools
If you want to know more about algorithms related to deformable image registration, I will recommend you to take a look to FAIR( guide book), FAIR is a toolbox for Matlab so you will have examples to understand the theory.
http://www.cas.mcmaster.ca/~modersit/FAIR/
Then if you want to specifically see some demon example,, here you have this other toolbox:
http://www.mathworks.es/matlabcentral/fileexchange/21451-multimodality-non-rigid-demon-algorithm-image-registration
Motivating example: I am trying to implement a land-only infection simulation in parallel based on the UK map.
I sample points uniformly spread over the land area and determine its infection status at each time step, which depends on the previous status of its neighbouring points (SIR model). The country is irregularly shaped, and so cartesian coordinates do not load-balance well - what are more efficient decomposition methods I should consider as standard?
Many thanks.
An excellent article (Seal & Aluru, 2001) outlining methods of
Orthogonal Recursive Bisection
Space Filling Curves
Octrees and Compressed Octrees
and a further paper (Aluru & Sevilgen) focussing on Space Filling Curves.
deLaunay meshes are another standard decomposition for irregular objects.
You should consider how such meshes are load balanced; here's a sample article.
I have a set points whose coordinates are given by the arrays x, y and z and the value of the density field in each point is stored in the array d.
I would like to reconstruct the density field on a uniform grid. What's the best algorithm to do that?
I know that in python, the scipy module come in handy with the griddata function but I would like to write my own code, I just need a hint.
If you have some sort of scalar field and the points are the origins of the field, you can implement a brute force approach by walking all lattice points and calculating the field intensity given the sources. There are both recursive methods that allow "blanking" wide volumes where the field is more or less constant, and techniques to save some CPU time by calculating the variations from one point to the next.
If the points you have are samplings of a value, then you will have to decompose your space in volumes and interpolate the values. You can employ a simple Voronoi decomposition - this is usually done in 2D for precipitation measurements - or a Delaunay tetrahedralization (you can look into TetGen's documentation). The first approach assumes that the function is constant throughout each Voronoi volume; the last allows rendering a trilinear interpolation.
If you need to smooth a 3D grid, the trilinear interpolation looks like the best approach.
There are also other methods used for fast visualization, that involve maintaining a list of 3D points in order of distance from any one given point in your regular grid. When moving through the grid, you recalculate distances using quadratic increments. Then, you perform a simple interpolation based on a subset of points of chosen cardinality (i.e., if you consider the four nearest points at distances d1..d4, you would calculate the value in P by proportionally weighing the values v1..v4). This approach is fast and easy to implement by yourself, but be warned that it underperforms wherever the minimum distance between points is less than the lattice step (you can compensate by considering more points where this happens; and the effect is less evident if the sampled function is smooth at the same scale).
If you want to implement a mathematical method yourself, you need to learn the theory, of course. In this case, it's 3D scattered data interpolation.
Wikipedia, MATLAB help and scipy help say there are at least half a dozen different methods. WP has a fairly good description of them and there's a comparison article but I strongly suggest you find something in your native language on such a terminology-intensive subject.
One approach is to form the Delaunay triangulation of the scattered points [x,y,z], (actually a tetrahedralisation in your 3d case!) and perform interpolation within each element using a linear representation of the density field, defined at the tetrahedron vertices.
To evaluate the density at each structured grid point you would (i) determine which tetrahedron the point lay within and (ii) evaluate the linear interpolant.
Forming the Delaunay triangulation is non-trivial, put there are a few good libraries that can be used for this, depending on your language of choice. One good option is CGAL.
Hope this helps.