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| {{expert-subject}}
| | I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is where her home is but her spouse desires them to move. My working day occupation is a travel agent. What me and my family adore is to climb but I'm considering on beginning something new.<br><br>Here is my blog :: psychics online ([http://www.publicpledge.com/blogs/post/7034 enquiry]) |
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| '''Thin plate splines (TPS)''' are an [[interpolation]] and [[smoothing]] technique, the generalisation of [[splines]] so that they may be used with two or more dimensions. They were introduced to [[geometric design]] by Duchon (Duchon, 1976).
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| ==Physical analogy==
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| The name ''thin plate spline'' refers to a physical analogy involving the bending of a thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the <math>z</math> direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the <math>x</math> or <math>y</math> coordinates within the plane. In 2D cases, given a set of <math>K</math> corresponding points, the TPS warp is described by <math>2(K+3)</math> parameters which include 6 global affine motion parameters and <math>2K</math> coefficients for correspondences of the control points. These parameters are computed by solving a linear system, in other words, TPS has [[closed-form solution]].
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| ==Smoothness measure==
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| The TPS arises from consideration of the integral of the square of the second derivative -- this forms its smoothness measure. In the case where <math>x</math> is two dimensional, for interpolation, the TPS fits a mapping function <math>f(x)</math> between corresponding point-sets <math>\{y_i\}</math> and <math>\{x_i\}</math> that minimises the following energy function:
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| :<math>
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| E = \iint\left[\left(\frac{\partial^2 f}{\partial x_1^2}\right)^2 + 2\left(\frac{\partial^2 f}{\partial x_1 \partial x_2}\right)^2 + \left(\frac{\partial^2 f}{\partial x_2^2}\right)^2 \right] \textrm{d} x_1 \, \textrm{d}x_2
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| </math>
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| The smoothing variant, correspondingly, uses a tuning parameter <math>\lambda</math> to control how non-rigid is allowed for the deformation, balancing the aforementioned criterion with the measure of goodness of fit, thus minimising:
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| :<math>
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| E_{tps}(f) = \sum_{i=1}^K \|y_i - f(x_i) \|^2 + \lambda \iint\left[\left(\frac{\partial^2 f}{\partial x_1^2}\right)^2 + 2\left(\frac{\partial^2 f}{\partial x_1 \partial x_2}\right)^2 + \left(\frac{\partial^2 f}{\partial x_2^2}\right)^2 \right] \textrm{d} x_1 \, \textrm{d}x_2
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| </math>
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| For this variational problem, it can be shown that there exists a unique minimizer <math>f</math> (Wahba,1990).The [[finite element]] discretization of this variational problem, the method of [[elastic map]]s, is used for [[data mining]] and [[nonlinear dimensionality reduction]].
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| ==Radial basis function==
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| {{Main|Radial basis function}}
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| The Thin Plate Spline has a natural representation in terms of radial basis functions. Given a set of control points <math>\{w_{i}, i = 1,2, \ldots,K\}</math>, a radial basis function basically defines a spatial mapping which maps any location <math>x</math> in space to a new location <math>f(x)</math>, represented by,
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| :<math>
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| f(x) = \sum_{i = 1}^K c_{i}\varphi(\left\| x - w_{i}\right\|)
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| </math>
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| where <math>\left\|\cdot\right\|</math> denotes the usual [[Norm (mathematics)|Euclidean norm]] and <math>\{c_{i}\}</math> is a set of mapping coefficients. The TPS corresponds to the radial basis kernel <math>\varphi(r) = r^2 \log r</math>.
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| <!--Commenting out this section... I'm not sure this is correct!
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| ===Spline===
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| Suppose the points are in 2 dimensions (<math>D = 2</math>). One can use ''homogeneous coordinates'' for the point-set where a point <math>y_{i}</math> is represented as a vector <math>(1, y_{ix}, y_{iy})</math>. The unique minimizer <math>f</math> is parameterized by <math>\alpha</math> which comprises two matrices <math>d</math> and <math>c</math> (<math>\alpha = \{d,c\}</math>).
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| :<math>
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| f_{tps}(z, \alpha) = f_{tps}(z, d, c) = z\cdot d + \sum_{i = 1}^K \phi(\| z - x_i\|)\cdot c_i
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| </math>
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| where d is a <math>(D+1)\times(D+1)</math> matrix representing the affine transformation (hence <math>z</math> is a <math>1\times (D+1)</math> vector) and c is a <math>K\times (D+1)</math> warping coefficient matrix representing the non-affine deformation. The kernel function <math>\phi(z)</math> is a <math>1\times K</math> vector for each point <math>z</math>, where each entry <math>\phi_i(z) = \|z - x_i\|^2 \log \|z - x_i\|</math> for each (<math>D</math>) dimensions. Note that for TPS, the control points <math>\{w_i\}</math> are chosen to be the same as the set of points to be warped <math>\{x_i\}</math>, so we already use <math>\{x_i\}</math> in the place of the control points.
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| If one substitutes the solution for <math>f</math>, <math>E_{tps}</math> becomes:
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| :<math>
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| E_{tps}(d,c) = \|Y - Xd - \Phi c\|^2 + \lambda \textrm{Tr}(c^T\Phi c)
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| </math>
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| where <math>Y</math> and <math>X</math> are just concatenated versions of the point coordinates <math>y_i</math> and <math>x_i</math>, and <math>\Phi</math> is a <math>(K\times K)</math> matrix formed from the <math>\phi (\|x_i - x_j\|)</math>. Each row of each newly formed matrix comes from one of the original vectors. The matrix <math>\Phi</math> represents the TPS kernel. Loosely speaking, the TPS kernel contains the information about the point-set's internal structural relationships. When it is combined with the warping coefficients <math>c</math>, a non-rigid warping is generated.
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| A nice property of the TPS is that it can always be decomposed into a global affine and a local non-affine component. Consequently, the TPS smoothness term is solely dependent on the non-affine components. This is a desirable property, especially when compared to other splines, since the global pose parameters included in the affine transformation are not penalized.
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| ===Solution===
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| The separation of the affine and non-affine warping space is done through a [[QR decomposition]] (Wahba,1990).
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| :<math>
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| X = [Q_1 | Q_2] \left(
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| \begin{array}{cc}
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| R \\
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| 0
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| \end{array}
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| \right)
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| </math> | |
| where Q1 and Q2 are <math>K \times (D+1)</math> and <math>K \times (K-D-1)</math> orthonormal matrices, respectively. The matrix <math>R</math> is upper triangular.
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| With the QR decomposition in place, we have
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| :<math> | |
| E_{tps}(\gamma,d) = \|Q_2^T Y - Q_2^T\Phi Q_2 \gamma\|^2 + \|Q_1^T Y -Rd - Q_1^T\Phi Q_2 \gamma\|^2 + \lambda \textrm{trace}( \gamma^T Q_2^T \Phi Q_2 \gamma)
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| </math>
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| where <math>\gamma </math> is a <math>(K-D-1)\times (D+1)</math> matrix. Setting <math>c=Q_2\gamma</math> (which in turn implies that <math>X^T c = 0</math>) enables us to cleanly separate the first term in last third equation into a non-affine term and an affine term (first and second terms last equation respectively).
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| The least-squares energy function in the last equation can be first minimized w.r.t <math>\gamma</math> and then w.r.t. <math>d</math>. By applying [[Tikhonov regularization]] we have
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| :<math>
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| \hat{c} = Q_2(Q_2^T\Phi Q_2 + \lambda I_{(k-D-1)})^{-1}Q_2^T Y
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| </math>
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| :<math> | |
| \hat{d} = R^{-1}Q_1^T (Y - \Phi \hat{c})
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| </math>
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| The minimum value of the TPS energy function obtained at the optimum <math>(\hat{c},\hat{d})</math> is
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| :<math>
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| E_{bending} = \lambda\,\textrm{trace}[Q_2(Q_2^T\Phi Q_2 + \lambda I_{(k-D-1)})^{-1}Q_2^T Y Y^T]
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| </math>
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| -->
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| ==Application==
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| TPS has been widely used as the non-rigid transformation model in image
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| alignment and shape matching.
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| The popularity of TPS comes from a number of advantages:
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| #The interpolation is smooth with derivatives of any order.
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| #The model has no free parameters that need manual tuning.
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| #It has closed-form solutions for both warping and parameter estimation.
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| #There is a physical explanation for its energy function.
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| ==See also==
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| *[[Inverse distance weighting]]
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| *[[Radial basis function]]
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| *[[Subdivision surface]] (emerging alternative to spline-based surfaces)
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| *[[Elastic map]] (a discrete version of the thin plate approximation for [[manifold learning]])
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| *[[Spline (mathematics)|Spline]]
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| *[[Polyharmonic spline]] (the thin-plate-spline is a special case of a polyharmonic spline)
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| ==References==
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| *Haili Chui: Non-Rigid Point Matching: Algorithms, Extensions and Applications. PhD Thesis, Yale University, May 2001.
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| *G. Wahba, 1990, Spline models for observational data. Philadelphia: Society for Industrial and Applied Mathematics.
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| *J. Duchon, 1976, Splines minimizing rotation invariant semi-norms in Sobolev spaces. pp 85–100, In: Constructive Theory of Functions of Several Variables, Oberwolfach 1976, W. Schempp and K. Zeller, eds., Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977
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| ==External links==
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| *[http://www-cse.ucsd.edu/classes/fa01/cse291/hhyu-presentation.pdf Explanation for a simplified variation problem]
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| *[http://mathworld.wolfram.com/ThinPlateSpline.html TPS at MathWorld]
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| *[http://elonen.iki.fi/code/tpsdemo/index.html TPS in C++]
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| *[http://launchpad.net/templatedtps TPS in templated C++]
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| [[Category:Splines]]
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| [[Category:Multivariate interpolation]]
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I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is where her home is but her spouse desires them to move. My working day occupation is a travel agent. What me and my family adore is to climb but I'm considering on beginning something new.
Here is my blog :: psychics online (enquiry)