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| In mathematics, the '''structure [[tensor]]''', also referred to as the '''second-moment matrix''', is a [[matrix (mathematics)|matrix]] derived from the [[gradient]] of a [[function (mathematics)|function]]. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent. The structure tensor is often used in [[image processing]] and [[computer vision]].<ref name=bigun86>
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| J. Bigun and G. Granlund (1986), ''Optimal Orientation Detection of Linear Symmetry''. Tech. Report LiTH-ISY-I-0828, Computer Vision Laboratory, Linkoping University, Sweden 1986; Thesis Report, Linkoping studies in science and technology No. 85, 1986.
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| </ref><ref name=bigun87>
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| {{cite conference|author=J. Bigun and G. Granlund|title=Optimal Orientation Detection of Linear Symmetry|location=Piscataway|booktitle=First int. Conf. on Computer Vision, ICCV, (London) |publisher=IEEE Computer Society Press, Piscataway|pages=433–438|year=1987 }}
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| </ref><ref name=knutsson89>
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| {{cite conference|author=H. Knutsson|title=Representing local structure using tensors|location=Oulu|booktitle=Proceedings 6th Scandinavian Conf.
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| on Image Analysis|publisher=Oulu University|pages=244–251|year=1989}}
| |
| </ref>
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| | |
| ==The 2D structure tensor==
| |
| | |
| ===Continuous version===
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| For a function <math>I</math> of two variables ''p''=(''x'',''y''), the structure tensor is the 2×2 matrix
| |
| | |
| :<math>
| |
| S_w(p) =
| |
| \begin{bmatrix}
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| \int w(r) (I_x(p-r))^2\,d r & \int w(r) I_x(p-r)I_y(p-r)\,d r \\[10pt]
| |
| \int w(r) I_x(p-r)I_y(p-r)\,d r & \int w(r) (I_y(p-r))^2\,d r
| |
| \end{bmatrix}
| |
| </math>
| |
| where <math>I_x</math> and <math>I_y</math> are the [[partial derivative]]s of <math>I</math> with respect to ''x'' and ''y''; the integrals range over the plane <math>\mathbb{R}^2</math>; and ''w'' is some fixed "window function", a [[distribution (mathematics)|distribution]] on two variables. Note that the matrix ''S''<sub>''w''</sub> is itself a function of ''p''=(''x'',''y'').
| |
| | |
| The formula above can be written also as <math>S_w(p)=\int w(r) S_0(p-r)\,d r</math>, where <math>S_0</math> is the matrix-valued function defined by
| |
| :<math>
| |
| S_0(p)=
| |
| \begin{bmatrix}
| |
| (I_x(p))^2 & I_x(p)I_y(p) \\[10pt]
| |
| I_x(p)I_y(p) & (I_y(p))^2
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| If the [[gradient]] <math>\nabla I = (I_x,I_y)</math> of <math>I</math> is viewed as a 1×2 (single-row) matrix, the matrix <math>S_0</math> can be written as the [[matrix product]] <math>(\nabla I)'(\nabla I)</math>, where <math>(\nabla I)'</math> denotes the 2×1 (single-column) [[transpose]] of the gradient. (Note however that the structure tensor <math>S_w(p)</math> cannot be factored in this way.)
| |
| | |
| ===Discrete version===
| |
| In image processing and other similar applications, the function <math>I</math> is usually given as a discrete [[array data structure|array]] of samples <math>I[p]</math>, where ''p'' is a pair of integer indices. The 2D structure tensor at a given [[pixel]] is usually taken to be the discrete sum
| |
| | |
| :<math>
| |
| S_w[p] =
| |
| \begin{bmatrix}
| |
| \sum_r w[r] (I_x[p-r])^2 & \sum_r w[r] I_x[p-r]I_y[p-r] \\[10pt]
| |
| \sum_r w[r] I_x[p-r]I_y[p-r] & \sum_r w[r] (I_y[p-r])^2
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| Here the summation index ''r'' ranges over a finite set of index pairs (the "window", typically <math>\{-m..+m\}\times\{-m..+m\}</math> for some ''m''), and ''w''[''r''] is a fixed "window weight" that depends on ''r'', such that the sum of all weights is 1. The values <math>I_x[p],I_y[p]</math> are the partial derivatives sampled at pixel ''p''; which, for instance, may be estimated from by <math>I</math> by [[finite difference]] formulas.
| |
| | |
| The formula of the structure tensor can be written also as <math>S_w[p]=\sum_r w[r] S_0[p-r]</math>, where <math>S_0</math> is the matrix-valued array such that
| |
| :<math>
| |
| S_0[p] =
| |
| \begin{bmatrix}
| |
| (I_x[p])^2 & I_x[p]I_y[p] \\[10pt]
| |
| I_x[p]I_y[p] & (I_y[p])^2
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| ===Interpretation===
| |
| The importance of the 2D structure tensor <math>S_w</math> stems from the fact that its [[eigenvalue]]s <math>\lambda_1,\lambda_2</math> (which can be ordered so that <math>\lambda_1 \geq \lambda_2\geq 0</math>) and the corresponding [[eigenvector]]s <math>e_1,e_2</math> summarize the distribution of the [[gradient]] <math>\nabla I = (I_x,I_y)</math> of <math>I</math> within the window defined by <math>w</math> centered at <math>p</math>.<ref name=bigun86/><ref name=bigun87/><ref name=knutsson89/>
| |
| | |
| Namely, if <math>\lambda_1 > \lambda_2</math>, then <math>e_1</math> (or <math>-e_1</math>) is the direction that is maximally aligned with the gradient within the window. In particular, if <math>\lambda_1 > 0, \lambda_2 = 0</math> then the gradient is always a multiple of <math>e_1</math> (positive, negative or zero); this is the case if and only if <math>I</math> within the window varies along the direction <math>e_1</math> but is constant along <math>e_2</math>.
| |
| | |
| If <math>\lambda_1 = \lambda_2</math>, on the other hand, the gradient in the window has no predominant direction; which happens, for instance, when the image has [[rotational symmetry]] within that window. In particular, <math>\lambda_1 = \lambda_2 = 0</math> if and only if the function <math>I</math> is constant (<math>\nabla I = (0,0)</math>) within <math>W</math>.
| |
| | |
| More generally, the value of <math>\lambda_k </math>, for ''k''=1 or ''k''=2, is the <math>w</math>-weighted average, in the neighborhood of ''p'', of the square of the [[directional derivative]] of <math>I</math> along <math>e_k</math>. The relative discrepancy between the two eigenvalues of <math>S_w</math> is an indicator of the degree of [[isotropy|anisotropy]] of the gradient in the window, namely how strongly is it biased towards a particular direction (and its opposite).<ref name="Jahne1993">
| |
| {{cite book|author=B. Jahne|title=Spatio-Temporal Image Processing: Theory and Scientific Applications|location=Berlin|publisher=Springer-Verlag|volume=751|year=1993}}
| |
| </ref><ref name=MedioniEA>
| |
| {{cite book|author=G. Medioni, M. Lee and C. Tang|title=A Computational Framework for Feature Extraction and Segmentation|publisher=Elsevier Science|date=March 2000}}
| |
| </ref> This attribute can be quantified by the '''coherence''', defined as
| |
| | |
| :<math>c_w=\left(\frac{\lambda_1-\lambda_2}{\lambda_1+\lambda_2}\right)^2</math>
| |
| | |
| if <math>\lambda_2>0</math>. This quantity is 1 when the gradient is totally aligned, and 0 when it has no preferred direction. The formula is undefined, even in the [[limit]], when the image is constant in the window (<math>\lambda_1=\lambda_2=0</math>). Some authors define it as 0 in that case.
| |
| | |
| Note that the average of the gradient <math>\nabla I</math> inside the window is '''not''' a good indicator of anisotropy. Aligned but oppositely oriented gradient vectors would cancel out in this average, whereas in the structure tensor they are properly added together.<ref>
| |
| {{cite journal|author=T. Brox, J. Weickert, B. Burgeth and P. Mrazek|title=Nonlinear Structure Tensors|booktitle=Universitat des Saarlandes, Tech. Report|issue=113|pages=1–32|year=2004}}
| |
| </ref>
| |
| | |
| By expanding the effective radius of the window function <math>w</math> (that is, increasing its variance), one can make the structure tensor more robust in the face of noise, at the cost of diminished spatial resolution.<ref name=MedioniEA /><ref name=lin94book>T. Lindeberg (1994), ''[http://www.nada.kth.se/~tony/book.html Scale-Space Theory in Computer Vision]''. Kluwer Academic Publishers, (see sections 14.4.1 and 14.2.3 on pages 359-360 and 355-356 for detailed statements about how the multi-scale second-moment matrix/structure tensor defines a true and uniquely determined multi-scale representation of directional data).
| |
| </ref> The formal basis for this property is described in more detail below, where it is shown that a multi-scale formulation of the structure tensor, referred to as the [[Structure tensor#The multi-scale structure tensor|multi-scale structure tensor]], constitutes a ''true multi-scale representation of directional data under variations of the spatial extent of the window function''.
| |
| | |
| ==The 3D structure tensor==
| |
| | |
| ===Definition===
| |
| The structure tensor can be defined also for a function <math>I</math> of three variables ''p''=(''x'',''y'',''z'') in an entirely analogous way. Namely, in the continuous version we have <math>S_w(p) = \int w(r) S_0(p-r)\,d r</math>, where
| |
| :<math>
| |
| S_0(p) =
| |
| \begin{bmatrix}
| |
| (I_x(p))^2 & I_x(p)I_y(p) & I_x(p)I_z(p) \\[10pt]
| |
| I_x(p)I_y(p) & (I_y(p))^2 & I_y(p)I_z(p) \\[10pt]
| |
| I_x(p)I_z(p) & I_y(p)I_z(p) & (I_z(p))^2
| |
| \end{bmatrix}
| |
| </math>
| |
| where <math>I_x,I_y,I_z</math> are the three partial derivatives of <math>I</math>, and the integral ranges over <math>\mathbb{R}^3</math>.
| |
| | |
| In the discrete version,<math>S_w[p]=\sum_r w[r] S_0[p-r]</math>, where
| |
| :<math>
| |
| S_0[p] =
| |
| \begin{bmatrix}
| |
| (I_x[p])^2 & I_x[p]I_y[p] & I_x[p]I_z[p] \\[10pt]
| |
| I_x[p]I_y[p] & (I_y[p])^2 & I_y[p]I_z[p]\\[10pt]
| |
| I_x[p]I_z[p] & I_y[p]I_z[p] & (I_z[p])^2
| |
| \end{bmatrix}
| |
| </math>
| |
| and the sum ranges over a finite set of 3D indices, usually <math>\{-m..+m\}\times\{-m..+m\}\times\{-m..+m\}</math> for some ''m''.
| |
| | |
| ===Interpretation===
| |
| As in the two-dimensional case, the eigenvalues <math>\lambda_1,\lambda_2,\lambda_3</math> of <math>S_w[p]</math>, and the corresponding eigenvectors <math>e_1,e_2,e_3</math>, summarize the distribution of gradient directions within the neighborhood of ''p'' defined by the window <math>w</math>. This information can be visualized as an [[ellipsoid]] whose semi-axes are equal to the eigenvalues and directed along their corresponding eigenvectors.<ref name="Medioni"/>
| |
| | |
| [[Image:STgeneric.png|thumb|center|240px|Ellipsoidal representation of the 3D structure tensor.]]
| |
| | |
| In particular, if the ellipsoid is stretched along one axis only, like a cigar (that is, if <math>\lambda_1</math> is much larger than both <math>\lambda_2</math> and <math>\lambda_3</math>), it means that the gradient in the window is predominantly aligned with the direction <math>e_1</math>, so that the [[isosurface]]s of <math>I</math> tend to be flat and perpendicular to that vector. This situation occurs, for instance, when ''p'' lies on a thin plate-like feature, or on the smooth boundary between two regions with contrasting values.
| |
| | |
| <center>
| |
| <table cellborder=0px border=0px>
| |
| <tr valign=top>
| |
| <td>[[Image:STsurfel.png|thumb|180px|The structure tensor ellipsoid of a surface-like neighborhood ("[[surfel]]"), where <math>\lambda_1 >\!> \lambda_2 \approx \lambda_3</math>.]]</td>
| |
| <td>[[Image:StepPlane3D.png|thumb|180px|A 3D window straddling a smooth boundary surface between two uniform regions of a 3D image.]]</td>
| |
| <td>[[Image:StepPlane3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| If the ellipsoid is flattened in one direction only, like a pancake (that is, if <math>\lambda_3</math> is much smaller than both <math>\lambda_1</math> and <math>\lambda_2</math>), it means that the gradient directions are spread out but perpendicular to <math>e_3</math>; so that the isosurfaces tend to be like tubes parallel to that vector. This situation occurs, for instance, when ''p'' lies on a thin line-like feature, or on a sharp corner of the boundary between two regions with contrasting values.
| |
| | |
| <center>
| |
| <table cellborder=0px border=0px>
| |
| <tr valign=top>
| |
| <td>[[Image:STcurvel.png|thumb|180px|The structure tensor of a line-like neighborhood ("curvel"), where <math>\lambda_1 \approx \lambda_2 >\!> \lambda_3</math>.]]</td>
| |
| <td>[[Image:curve3D.png|thumb|180px|A 3D window straddling a line-like feature of a 3D image.]]</td>
| |
| <td>[[Image:curve3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| Finally, if the ellipsoid is roughly spherical (that is, if <math>\lambda_1\approx\lambda_2\approx\lambda_3</math>), it means that the gradient directions in the window are more or less evenly distributed, with no marked preference; so that the function <math>I</math> is mostly isotropic in that neighborhood. This happens, for instance, when the function has [[spherical symmetry]] in the neighborhood of ''p''. In particular, if the ellipsoid degenerates to a point (that is, if the three eigenvalues are zero), it means that <math>I</math> is constant (has zero gradient) within the window.
| |
| | |
| <center>
| |
| <table cellborder=0px border=0px>
| |
| <tr valign=top>
| |
| <td>[[Image:STball.png|thumb|180px|The structure tensor in an isotropic neighborhood, where <math>\lambda_1\approx\lambda_2\approx\lambda_3</math>.]]</td>
| |
| <td>[[Image:Sphere3D.png|thumb|180px|A 3D window containing a spherical feature of a 3D image.]]</td>
| |
| <td>[[Image:Sphere3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]</td>
| |
| </tr>
| |
| </table>
| |
| </center>
| |
| | |
| ==The multi-scale structure tensor==
| |
| The structure tensor is an important tool in [[scale space]] analysis. The '''multi-scale structure tensor''' (or '''multi-scale second moment matrix''') of a function <math>I</math> is in contrast to other one-parameter scale-space features an image descriptor that is defined over ''two'' scale parameters.
| |
| One scale parameter, referred to as ''local scale'' <math>t</math>, is needed for determining the amount of pre-smoothing when computing the image gradient <math>(\nabla I)(x; t)</math>. Another scale parameter, referred to as ''integration scale'' <math>s</math>, is needed for specifying the spatial extent of the window function <math>w(\xi; s)</math> that determines the weights for the region in space over which the components of the outer product of the gradient by itself <math>(\nabla I)(\nabla I)^T</math> are accumulated.
| |
| | |
| More precisely, suppose that <math>I</math> is a real-valued signal defined over <math>\mathbb{R}^k</math>. For any local scale <math>t > 0</math>, let a multi-scale representation <math>I(x; t)</math> of this signal be given by <math>I(x; t) = h(x; t)*I(x)</math> where <math>h(x; t)</math> represents a pre-smoothing kernel. Furthermore, let <math>(\nabla I)(x; t)</math> denote the gradient of the [[scale space representation]].
| |
| Then, the ''multi-scale structure tensor/second-moment matrix'' is defined by
| |
| <ref name=lin94book/><ref name=lingar97>{{cite journal
| |
| | author=T. Lindeberg and J. Garding
| |
| | title=Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D structure
| |
| | journal=Image and Vision Computing
| |
| | year=1997
| |
| | volume=15
| |
| | pages=pp 415–434
| |
| | url=http://www.nada.kth.se/~tony/abstracts/LG94-ECCV.html
| |
| | doi=10.1016/S0262-8856(97)01144-X
| |
| | issue=6
| |
| }}</ref><ref name=garlin96>
| |
| J. Garding and T. Lindeberg (1996). ''[http://www.nada.kth.se/cvap/abstracts/cvap117.html "Direct computation of shape cues using scale-adapted spatial derivative operators]'', International Journal of Computer Vision, volume 17, issue 2, pages 163--191.
| |
| </ref>
| |
| :<math>
| |
| \mu(x; t, s) =
| |
| \int_{\xi \in \mathbb{R}^k}
| |
| (\nabla I)(x-\xi; t) \, (\nabla I)^T(x-\xi; t) \,
| |
| w(\xi; s) \, d\xi
| |
| </math>
| |
| Conceptually, one may ask if it would be sufficient to use any self-similar families of smoothing functions <math>h(x; t)</math> and <math>w(\xi; s)</math>. If one naively would apply, for example, a box filter, however, then non-desirable artifacts could easily occur. If one wants the multi-scale structure tensor to be well-behaved over both increasing local scales <math>t</math> and increasing integration scales <math>s</math>, then it can be shown that both the smoothing function and the window function ''have to'' be Gaussian.<ref name=lin94book/> The conditions that specify this uniqueness are similar to the [[scale-space axioms]] that are used for deriving the uniqueness of the Gaussian kernel for a regular Gaussian [[scale space]] of image intensities.
| |
| | |
| There are different ways of handling the two-parameter scale variations in this family of image descriptors. If we keep the local scale parameter <math>t</math> fixed and apply increasingly broadened versions of the window function by increasing the integration scale parameter <math>s</math> only, then we obtain a ''true formal [[scale space representation]] of the directional data computed at the given local scale'' <math>t</math>.<ref name=lin94book/> If we couple the local scale and integration scale by a ''relative integration scale'' <math>r \geq 1</math>, such that <math>s = r t</math> then for any fixed value of <math>r</math>, we obtain a reduced self-similar one-parameter variation, which is frequently used to simplify computational algorithms, for example in [[corner detection]], [[interest point detection]], [[texture analysis]] and [[image registration|image matching]].
| |
| By varying the relative integration scale <math>r \geq 1</math> in such a self-similar scale variation, we obtain another alternative way of parameterizing the multi-scale nature of directional data obtained by increasing the integration scale.
| |
| | |
| A conceptually similar construction can be performed for discrete signals, with the convolution integral replaced by a convolution sum and with the continuous Gaussian kernel <math> g(x; t)</math> replaced by the [[discrete Gaussian kernel]] <math>T(n; t)</math>:
| |
| :<math>
| |
| \mu(x; t, s) =
| |
| \sum_{n \in \mathbb{Z}^k}
| |
| (\nabla I)(x-n; t) \, (\nabla I)^T(x-n; t) \,
| |
| w(n; s)
| |
| </math>
| |
| When quantizing the scale parameters <math>t</math> and <math>s</math> in an actual implementation, a finite geometric progression <math>\alpha^i</math> is usually used, with ''i'' ranging from 0 to some maximum scale index ''m''. Thus, the discrete scale levels will bear certain similarities to [[pyramid (image processing)|image pyramid]], although spatial subsampling may not necessarily be used in order to preserve more accurate data for subsequent processing stages.
| |
| | |
| ==Applications==
| |
| The eigenvalues of the structure tensor play a significant role in many image processing algorithms, for problems like [[corner detection]], [[interest point detection]], and [[feature tracking]].<ref name="Medioni">
| |
| {{cite conference|author=M. Nicolescu and G. Medioni |title=Motion Segmentation with Accurate Boundaries — A Tensor Voting Approach|booktitle=Proc. IEEE Computer Vision and Pattern Recognition|volume=1|pages=382–389|year=2003}}
| |
| </ref><ref> | |
| {{cite journal
| |
| |author=W. Förstner|title=A Feature Based Correspondence Algorithm for Image Processing
| |
| |booktitle=International Archives of Photogrammetry and Remote Sensing|volume=26|pages=150–166|year=1986}}
| |
| </ref><ref>
| |
| {{cite conference|author=C. Harris and M. Stephens|title=A Combined Corner and Edge Detector
| |
| |booktitle=Proc. of the 4th ALVEY Vision Conference|pages=147–151|year=1988}}
| |
| </ref><ref>
| |
| {{cite journal|author=K. Rohr|title=On 3D Differential Operators for Detecting Point Landmarks
| |
| |booktitle=Image and Vision Computing|volume=15|issue=3|pages=219–233|year=1997}}
| |
| </ref><ref>
| |
| {{cite conference|author=I. Laptev and T. Lindeberg|title=Space-time interest points
| |
| |booktitle=International Conference on Computer Vision ICCV'03|url=ftp://ftp.nada.kth.se/CVAP/reports/LapLin03-ICCV.pdf|doi=10.1109/ICCV.2003.1238378|pages=432–439|volume=I|year=2003}}
| |
| </ref><ref>
| |
| {{cite conference|author=B. Triggs|title=Detecting Keypoints with Stable Position, Orientation, and Scale under Illumination Changes
| |
| |booktitle=Proc. European Conference on Computer Vision|volume=4|pages=100–113|year=2004}}
| |
| </ref><ref>
| |
| {{cite conference|author=C. Kenney, M. Zuliani and B. Manjunath, |title=An Axiomatic Approach to Corner Detection|booktitle=Proc. IEEE Computer Vision and Pattern Recognition|pages=191–197|year=2005}}
| |
| </ref> The structure tensor also plays a central role in the [[Lucas–Kanade Optical Flow Method|Lucas-Kanade optical flow algorithm]], and in its extensions to estimate [[affine shape adaptation]];<ref name=lingar97/> where the magnitude of <math>\lambda_2</math> is an indicator of the reliability of the computed result. The tensor has also been used for [[scale space]] analysis,<ref name=lin94book/> estimation of local surface orientation from monocular or binocular cues,<ref name=garlin96/> non-linear [[fingerprint enhancement]],<ref>
| |
| A. Almansa and T. Lindeberg (2000), ''[http://www.nada.kth.se/cvap/abstracts/cvap226.html Enhancement of fingerprint images using shape-adaptated scale-space operators]''. IEEE Transactions on Image Processing, volume 9, number 12, pages 2027-2042.
| |
| </ref> [[diffusion-based image processing]],<ref>[http://www.mia.uni-saarland.de/weickert/book.html J. Weickert (1998), Anisotropic diffusion in image processing, Teuber Verlag, Stuttgart.]</ref><ref>
| |
| {{cite journal|author=D. Tschumperle and Deriche|title=Diffusion PDE's on Vector-Valued Images|booktitle=IEEE Signal Processing Magazine|pages=16–25|date=September 2002}}
| |
| </ref><ref>
| |
| {{cite conference|author=S. Arseneau and J. Cooperstock|title=An Asymmetrical Diffusion Framework for Junction Analysis|booktitle=British Machine Vision Conference|volume=2|pages=689–698|date=September 2006}}
| |
| </ref><ref>
| |
| {{cite conference|author=S. Arseneau, and J. Cooperstock|title=An Improved Representation of Junctions through Asymmetric Tensor Diffusion|booktitle=International Symposium on Visual Computing|date=November 2006}}
| |
| </ref> and several other image processing problems.
| |
| | |
| ===Processing spatio-temporal video data with the structure tensor===
| |
| | |
| The three-dimensional structure tensor has been used to analyze three-dimensional video data (viewed as a function of ''x'', ''y'', and time ''t'').<ref name="Jahne1993" />
| |
| If one in this context aims at image descriptors that are ''invariant'' under Galilean transformations, to make it possible to compare image measurements that have been obtained under variations of a priori unknown image velocities <math>v = (v_x, v_y)^T</math>
| |
| :<math> \begin{bmatrix} x' \\ y' \\ t' \end{bmatrix} = G \begin{bmatrix} x \\ y \\ t \end{bmatrix} = G \begin{bmatrix} x - v_x \, t \\ y - v_y \, t \\ t \end{bmatrix} </math>,
| |
| it is, however, from a computational viewpoint more preferable to parameterize the components in the structure tensor/second-moment matrix <math>S</math> using the notion of ''Galilean diagonalization''<ref name=lin04icpr>
| |
| {{cite conference|author=T. Lindeberg, A. Akbarzadeh, and I. Laptev|title=Galilean-corrected spatio-temporal interest operators|booktitle=International Conference on Pattern Recognition ICPR'04|url=ftp://ftp.nada.kth.se/CVAP/reports/LinAkhLap04-ICPR.pdf|doi=10.1109/ICPR.2004.1334004|date=August 2004|volume=I| pages=57–62}}
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| </ref>
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| :<math> S' = R_{space}^{-T} \, G^{-T} \, S \, G^{-1} \, R_{space}^{-1} = \begin{bmatrix} \nu_1 & \, & \, \\ \, & \nu_2 & \, \\ \, & \, & \nu_3 \end{bmatrix} </math>
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| where <math>G</math> denotes a Galilean transformation of space-time and <math>R_{space}</math> a two-dimensional rotation over the spatial domain,
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| compared to the abovementioned use of eigenvalues of a 3-D structure tensor, which corresponds to an eigenvalue decomposition and a (non-physical) three-dimensional rotation of space-time
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| :<math> S'' = R_{space-time}^{-T} \, S \, R_{space -time}^{-1} = \begin{bmatrix} \lambda_1 & & \\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} </math>.
| |
| To obtain true Galilean invariance, however, also the shape of the spatio-temporal window function needs to be adapted,<ref name=lin04icpr/><ref>
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| {{cite conference|author=I. Laptev, and T. Lindeberg|title=Velocity adaptation of space-time interest points|booktitle=International Conference on Pattern Recognition ICPR'04|url=http://www.csc.kth.se/cvap/abstracts/LapLin04-ICPR.html|doi=10.1109/ICPR.2004.971|date=August 2004|volume=I| pages=52–56}}
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| </ref> corresponding to the transfer of [[affine shape adaptation]]<ref name=lingar97/> from spatial to spatio-temporal image data.
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| In combination with local spatio-temporal histogram descriptors,<ref>
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| {{cite conference|author=I. Laptev, and T. Lindeberg|title=Local descriptors for spatio-temporal recognition|booktitle=ECCV'04 Workshop on Spatial Coherence for Visual Motion Analysis (Prague, Czech Republic) Springer Lecture Notes in Computer Science|url=http://www.csc.kth.se/cvap/abstracts/LapLin04-ECCVWS.html|doi=10.1007/11676959|date=May 2004|volume=3667| pages=91–103.}}
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| </ref>
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| these concepts together allow for Galilean invariant recognition of spatio-temporal events.<ref>
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| {{cite conference|author=I. Laptev, B. Caputo, C. Schuldt, and T. Lindeberg|title=Local velocity-adapted motion events for spatio-temporal recognition|booktitle=Computer Vision and Image Understanding|url=http://www.csc.kth.se/cvap/abstracts/LapCapSchLin07-CVIU.html|doi=10.1016/j.cviu.2006.11.023|year=2007|volume=108| pages= 207–229}}</ref>
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| ==See also==
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| *[[Tensor]]
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| *[[Directional derivative]]
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| *[[Gaussian]]
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| *[[Corner detection]]
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| *[[Edge detection]]
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| *[[Lucas Kanade method|Lucas-Kanade method]]
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| *[[Affine shape adaptation]]
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| | |
| ==References==
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| <references/>
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| ==Resources==
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| *[http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=12362&objectType=FILE Download MATLAB Source]
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| *[http://www.cs.cmu.edu/~sarsen/structureTensorTutorial/ Structure Tensor Tutorial (Original)]
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| | |
| {{DEFAULTSORT:Structure Tensor}}
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| [[Category:Tensors]]
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| [[Category:Feature detection]]
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