Pierluigi Taddei

I have presented at NORDIA Workshop in Anchorage (Alaska) a work done in conjuction with Adrien Bartoli. Here is the related paper

• Template-Based Paper Reconstruction from a Single Image is Well Posed when the Rullings are Parallel
  @INPROCEEDINGS{author = {Pierluigi Taddei and Adrien Bartoli}, title = {Template-Based Paper Reconstruction from a Single Image is Well Posed when the Rullings are Parallel}, booktitle = {NORDIA Workshop}, year = {2008}, month = {June}}

Here is the presentation of the work:

The following papers have been presented the 20th of October at Omnivis 2007 in conjunction with the ICCV 2007 conference in Rio de Janeiro. I will add a brief description of the two works as time will permit.

• Single-Image Calibration of Off-Axis Catadioptric Cameras Using Lines
  @INPROCEEDINGS{author = {Vincenzo Caglioti and Pierluigi Taddei and Giacomo Boracchi and Simone Gasparini and Alessandro Giusti}, title = {Single-Image Calibration of Off-Axis Catadioptric Cameras Using Lines}, booktitle = {OMNIVIS Workshop}, year = {2007}, month = {October}}
• Methods for space line localization from single catadioptric images: new proposals and comparisons
  @INPROCEEDINGS{author = {Vincenzo Caglioti and Simone Gasparini and Pierluigi Taddei}, title = {Methods for space line localization from single catadioptric images: new proposals and comparisons}, booktitle = {OMNIVIS Workshop}, year = {2007}, month = {October}}
  

I made some small bug fixes to the conic intersection package. Thanks to Dany for pointing the bug out. The matlab code that you can download here can be used to detect the (up to) four intersections of two conics. The usage is quite straightforward but anyway, here is an example:

%%the homogeneous representation of a conic is a matrix 
%% m = [A C D; C B E; D E F] that represents the equation 
%% A x^2 + B y^2 + 2C xy + 2D x + 2Ey + F = 0 

%a circle centered in the origin 
E1 = [1 0 0; 0 1 0; 0 0 -3] 

%an ellipse centered in the origin 
E2 = [1 0 0; 0 3 0; 0 0 -6] 

%get the four homogeneous intersections 
P = intersectConics(E1, E2) 
%plot the normalized points 
plot(P(1,:) ./ P(3,:) , P(2,:) ./ P(3,:), 'ro');

• Conics intersections v.1.02

Algorithm description

The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic section. In particular two conics may possess none, two, four possibly coincident intersection points. The best method to locate these solutions is to exploit the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.

The procedure to locate the intersection points follows these steps:

• given the two conics C₁ and C₂ consider the pencil of conics given by their linear combination ?C₁ + ?C₂
• identify the homogeneous parameters (?,?) which corresponds to the degenerate conic of the pencil. This can be done by imposing that det(?C₁ + ?C₂) = 0, which turns out to be the solution to a third degree equation.
• given the degenerate cone C₀, identify the two, possibly coincident, lines constituting it
• intersects each identified line with one of the two original conic; this step can be done efficiently using the dual conic representation of C₀
• the points of intersection will represent the solution to the initial equation system

Matlab implementation

In the library you will find a couple of matlab function which implements the above steps to intersect two conics. In particular the main functions are:

• intersectConics : this is the main function to perform the intersection
• decomposeDegenerateConic: this function allow to split a degenerate conic into two lines
• intersecLineConic: this function recover the, possibly two, points of intersection of a line and a conic

If you have any comments please refer to me by email!

In my last work I have extensively used multivariate b spline to perform data interpolation ad to exploit a parametric model for image mappings. In particular my work required periodic and continuous b-spline. This version of a b-spline requires two less parameters for each dimension since we imposes C²continuity on the period boundary. 
In this article I present a rather smart and compact implementation for R -> R² periodic b-spline with evenly spaced knots using cubic spline as atomic support (i.e each b-spline covers 5 subsequent knots).

You may use this function as in the following  example

a = linspace(0,2*pi, 12); a = a(1:end-1);
p = [cos(a); 2*sin(a)]
 
 cs = closedSpline(a, p, 8); %creates the periodic spline 
 y  = cs.eval( linspace(0,2*pi,100) ); %evaluate the spline in 100 points 

 plot(p(1,:),p(2,:), 'b.');
plot(y(1,:), y(2,:), '-r');

I am going to post shortly the link to the two required function and possibly to extends this post to add more details!

Here is the matlab function used to create a closed 2D spline.

 % Compute a 2D closed spline 
 %              - Pierluigi Taddei (
  This e-mail address is being protected from spambots. You need JavaScript enabled to view it
 )
 %
 % Usage:  sp =  closedSpline(a, p, n)
 %               
 % Arguments:
 %          a - domain values (i.e linspace(0, 2*pi, k) )
 %          p - image values on the plane (the value of the spline in a)
 %          n - number of spline knots (should be less than number of
 %               values)
 % 27.03.2009 : Created
 function ps = closedSpline(a, p, n)
 
 	if(n > length(a))
 	n = length(a);
 	end
 
 	ps = struct();
 
 	%atomic spline element 
 	knots = linspace(0,2*pi, n+1); knots = knots(1:end-1);
 	baseSpline = spmak(knots(1:5),1);
 
 
  	%%move values to a single bspline equivalent
 	Afull = repmat(a', 1,length(knots)) - repmat(knots, length(a),1);
 	Afull2 = mod(Afull, 2*pi); %remove periodicity
 
 	A = fnval(baseSpline, Afull2); %evaluate bspline in all points
  
 	%least square minimization
 	ku = A \ p(1,:)';
 	kv = A \ p(2,:)';
 
 	%file the structure
 	ps.knots = knots;
 	ps.baseSpline = baseSpline;
 	ps.paramU = ku;
 	ps.paramV = kv;
 
 	%add the evaluation method to the spline itself
 	ps.eval = @(x)csval(ps,x);
 end

Given a set of initial points the function creates a linear sistem wich will be solved in order to detect the  best parameter value in the least square sense.

Once the parameters are known you can call the csval function to evaluate the spline in any point of the domain (thanks to its periodicity). You can both call this function directly or use an object oriented call style on the spline structure.

% Evaluate a closed 2D spline for values a
 %              - Pierluigi Taddei (
  This e-mail address is being protected from spambots. You need JavaScript enabled to view it
 )
 %
 % Usage:  y = csval(cs,a)
 %               
 %          
 % Arguments:
 %          cs - the closed spline
 %          a - domain values vector
 % 27.03.2009 : Created
 function y = csval(cs,a)
	 k = cs.knots;
 
 	aVal = repmat(a, length(k),1) - repmat(k', 1, length(a));
 	aVal2 = mod(aVal, 2*pi);
 	bVal = fnval(cs.baseSpline, aVal2);
 
 	u = sum( bVal .* repmat(cs.paramU, 1,length(a)),1 );
 	v = sum( bVal .* repmat(cs.paramV, 1,length(a)),1 );
 
 	y = [u;v];
 end