I'm read book about Computer Graphics with OpenGL (4th ed.) these formula come from Normalized Perspective-Projection
there are three equations:
and the book gives the parametric answer directly
there are four parameters:
I can only figure out S_z,I can't solve the equation for the other three parameters, who can give the detailed procedure of solving the equation
Related
In "Multiple View Geometry in Computer Vision" Chapter 12. Structure Computation, page 310-313, triangulation is used for point 3D reconstruction. There are three methods mentioned:
Midpoint method that "finds the midpoint of the common perpendicular to the two rays in space".
Linear triangulation methods that uses SVD to solve homogeneous equation Ax = 0. In this method algebraic error is minimized.
Nonlinear triangulation method that uses iterative optimization algorithms e.g. LM to solve a nonlinear least square equation Ax = b. The "gold standard" reprojection error is minimized.
My question is:
What are the geometric interpretations of the three methods?
What are the differences between the three triangulation methods above?
I did some research and found some useful information:
This post explains that algebraic error assumes noise comes from 3D points, while reprojection error assumes noise comes from 2D image
These slides introduce three methods but did not provide explicit geometric interpretations.
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 am trying to find the intersection point of a curve and a 3D surface with no luck. The surface is in the shape of a cone, and the curve is hyperbolic, as are shown in the figure.
CONE AND THE CURVE
This simulates a ray hits a certain surface. I tried to use bisection method, but it doesn't seem to work. then I tried newton's algorithm, but the results are still not good.
Is there any other good algorithms out there which are suitable for solving this kind of problem?
With the curve given in parametric form
x = fx(t)
y = fy(t)
z = fz(t)
and the surface by one equation of the form
g(x,y,z) = 0
just plug in the curve functions and bisection should work:
g(fx(t), fy(t), fz(t)) = 0
The only problem is to find suitable starting points t1 and t2 where g has opposite sign.
Problem
You are searching for a curve-surface intersection algorithm. Note that both curves and surfaces can be represented in either implicit form or in parametric form. Surface in implicit form is defined by equation F(x, y, z) = 0, which is a quadratic polynomial of x, y, z in case of conic surface. Surface in parametric form is defined by point-valued function S(u, v) of its parameters (e.g. you can use distance along cone axis and polar angle as parameters of conical surface). Curve is usually described only in parametric form, as a function C(t) with parameter t, which could be quadratic for a hyperbolic curve.
Implicit surface
The simplest cases of all is to treat your problem as an intersection of parametric curve against implicit surface. In this case you can write down a single equation q(t) = F(C(t)) = 0 with single variable t. Of course, Newton's iteration is not guaranteed to find all solutions in general case, bisection can only surely find one solution if you find two points with different sign of q(t).
In your case q(t) is a quartic polynomial (after putting quadratic curve parametrization into quadratic surface equation). It can be theoretically solved with Ferrari's analytic formula, but I strongly advise against it, because it is quite unstable numerically. You can apply any popular polynomial solver here, like Jenkins-Traub algorithm or eigenvalues algorithm for companion matrix (also see this question). You can also use methods of interval mathematics: for example, you can recursively subdivide the domain interval of parameter t into smaller pieces, while pruning all the pieces that surely do not contain zeros (interval arithmetic would help you to detect such pieces).
Parametric surface
Now we can move on to the case when both the curve and the surface are represented parametrically. I do not know any solutions that could benefit from the fact the your surface is conical and your curve is hyperbolic, so you have to apply the general curve-surface intersection algorithm. Alternatively, you can fit an implicitly-defined cone into your parametric surface, then use the solution above for quartic polynomial roots.
A lot of reliable general intersection algorithms are based on the subdivision method (which is actually interval mathematics again). The general idea is to continiously divide the curve and the surface into smaller and smaller pieces. The pairs of pieces which surely do not intersect are dropped as soon as possible. At the end you'll have a set of small piece pairs, tightly bounding your intersection points. Yoy might want to run Newton's iteration from them in order to make intersection points precise.
Here is the outline of a sample algorithm:
Start with a single curve piece (the whole input curve) and a single surface
piece (whole surface), and one potentially intersection pair (PIP) of these pieces.
Subdivide each curve piece into two halves (by parameter), subdivide each surface piece into four quadrants (by both parameters).
For each old PIP check all 8 pairs of curve half vs surface quadrant. If they surely do not intersect, forget them. If they can intersect, save them as a new PIP.
Unless all pieces are small enough, repeat from step 2 with new pieces and PIP-s.
For each pair of curve piece and surface piece, you have to check whether they can potentially intersect, which can be easily done by checking their axis-aligned bounding boxes. Also, you can represent your curves and surfaces as NURBS, in which case you can use convex hulls as tighter bounding volumes.
Generally, there are tons of variations and improvements of this algorithms. I advise the following literature for deeper knowledge:
Shape interrogation for computer-aided design and manufacturing.
chapter 4: for root solvers
section 5.7: for curve-surface intersection
PhD of Michael Hohmeyer.
section 4.5: for curve-surface intersection
sections 4.1 and 4.2: for convex hulls intersection (if you are brave enough).
Bottom line
If you are seeking for a simple and working solution, and you are sure that hyperbolas and cones are the only things you have to worry about, then you'd better use implicit definition of cone and solve quartic equation with some standard numerical algorithm from a good library available to you.
I am trying to develop an algorithm that performs the following :
Given a 2D polygon and a 3D polyhedron, determine if the 2D polygon is a projection of the 3D polyhedron (a perspective projection to be precise) without knowing which transformation matrix we may have possibly used for the projection.
input
{2D Polygon}
{3D Polyhedron}
output
{bool} whether or not it's a perspective projection
I am not asking for code, but I would simply like to know if this is feasible in polynomial time.
Any help will be greatly appreciated.
A 3D to 2D perspective projection has 7 degrees of freedom (6 for the relative motion of the scene with respect to the camera, 1 for the focal length).
Select four vertices in the 2D projection and consider all possible correspondences with polyhedron vertices (there is a polynomial number of such associations). Then form a system of 7 equations in the 7 unknown parameters (unfortunately a nonlinear one; maybe the eighth equation can be useful to select among multiple solutions).
Knowing the parameters, you can check a solution by re-projecting the polyhedron and comparing to the polygon (with further search for correspondences with vertices and edges).
All of this will take polynomial time (quartic if I am right), if one admits that the solver takes bounded time (hence bounded precision).
If the focal length is known, then a better approach is possible. Indeed, with only 6 unknowns, you can find the projection parameters from the projection of just three points. This problem is known to have an analytical solution (actually up to 4 of them), as described at length in "New Algorithms for the Perspective-Three-Point Problem, GAO Xiaoshan & CHEN Hangfei, Vol.16 No.3 J. Comput. Sci. & Technol."
This should lead to an O(N³) exact procedure.
More generally speaking, you form putative correspondences between N pairs of points, solve the corresponding Perspective-N-point problem, and check the hypothesis by reprojecting the polyhedron and comparing to the known projection to validate the hypothesis.
Just an idea for an algorithm:
Take a triangle of the projection made of three points next to each other not on the same line. Iterate through all corresponding triangles of the original. For all possible projections that solve the pair of triangles, check if the rest matches.
I must admit I am not sure right now if there could be infinite solutions for triangles (which would be hard to iterate)? If so, start with four points.
I think it is possible but you have to do a fair amount of reverse engineering. A 2D sketch that represents a 3D object is known as an Orthographic Projection. The link shows you the transformation matrices you need apply to transform the 3D point onto its 2D projection. Now, how do you go the opposite way? Inverse matrices with a mix of some inverse transformations (translation, scaling, rotation...)? I think this is a good lead to follow.
Hi people i have 2 question related to Decasteljau algorithm,they are more of a general questions,but if im right it could help solving many problems.Here it is:
We have some sufrace: Ʃ(i=0,n) Ʃ(j=o,m) Bi,n(U),Bi,m(v) Pi,j analysis that i have found says that first we take some value for one parameter u=uo,then we itterate other parametar v -> 1 get a set of points,then increment u by one etc....for loop inside for loop in code language.My question is can we fix one parameter U=Uo for what ever value,and then just compute points on for parameter v?Because all points that are on one curve are also on the surface,and if distance between curves approaches to zero (which itteration really is) we can apply DeCasteljau algorithm only to one set of curves itterating only one parameter.Or i got something wrong?:)
Second question is i havent really figured out what do we really need DeCasteljau algorithm for,unless we are drawing curves by hand?If we know order of the curve we can easily form Bernstain polynoms for that curve order and compute point for given value of parametar.Because when you unwrap Decasteljau what you get is Bernstain polynom?
So like i said,please help have i got i wrong?
Yes you can fix one parameter (say U) and change the other (V) to generate an iso-U curve.
You can see the things as if you had an NxM array of control points. If you perform a first interpolation on U (actually M interpolations involving N control points), you get M new control points that define a Bezier curve. and by varying U, the curve moves in space.
The De Casteljau's algorithm is used for convenience: it computes the interpolant by using a cascade of linear interpolations between the control points. Direct evaluation of the Bernstein polynomials would require the precomputation of the coefficients, and would not be faster, even when implemented by Horner's scheme, and can be numerically less stable.
The De Casteljau's algorithm is also appreciated for its geometrical interpretation, and for its connection with the subdivision process: if you want to build the control points for just a part of a Bezier curve, De Calsteljau's provides them.