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2101:{\displaystyle {\begin{array}{lcl}{\dfrac {\partial SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right){\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i})}{\partial \mu }}\\&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)\left({\dfrac {\partial {T}_{\mu }(x_{i})}{\partial \mu }}\right)^{t}{\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i}))}{\partial x}}\\\end{array}}}
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1361:{\displaystyle NCC(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)}{\sqrt {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)^{2}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)^{2}}}},}
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is far from being CPU efficient. Cachier et al. demonstrated that the problem of minimizing image and mechanical energy can be reformulated in solving the energy image then applying a
Gaussian filter at each iteration. We use this strategy in Yadics and we add the median filter as it is massively used in PIV. One notes that the median filter avoids local minima while preserving discontinuities. The filtering process is illustrated in the figure below :
22:
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The metrics is often called image energy; people usually add energy that comes from mechanics assumptions as the
Laplacian of displacement (a special case of Tikhonov regularization ) or even finite element problems. As one decided not to solve the Gauss-Newton problem for most of cases this solution
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None of these optimization methods can succeed directly if applied at the last scale as the gradient methods are sensitive to the initial guests. In order to find a global optimum one has to evaluate the transformation on a filtered image. The figure below illustrates how to use the pyramidal filter
351:
The common idea of image registration and digital image correlation is to find the transformation between a fixed image and a moving one for a given metric using an optimization scheme. While there are many methods to achieve such a goal, Yadics focuses on registering images with the same modality.
300:
is a tool that allows to identify the displacement field to register a reference image (called herein fixed image) to images during an experiment (mobile image). For example, it is possible to observe the face of a specimen with a painted speckle on it in order to determine its displacement fields
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functions and in solid mechanics with finite element basis. The global transformations are defined on the whole picture using rigid body or affine transformation (which is equivalent to homogeneous strain transformation). More complex transformations can be defined such as mechanically based one.
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on a 12×12×12 mesh), dealing with such a problem is more a matter of numerical scientists and required specific development (using libraries like Petsc or MUMPS) so we don't use Gauss-Newton methods to solve such problems. One has developed a specific steepest gradient algorithm with a specific
282:
on 2D and 3D tomographic images. The program was designed to be both modular, by its plugin strategy and efficient, by it multithreading strategy. It incorporates different transformations (Global, Elastic, Local), optimizing strategy (Gauss-Newton, Steepest descent), Global and/or local shape
352:
The idea behind the creation of this software is to be able to process data that comes from a μ-tomograph; i.e.: data cube over 1000 voxels. With such a size it is not possible to use naive approach usually used in a two-dimensional context. In order to get sufficient performances
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Contrary to image registration, Digital Image
Correlation targets the transformation, one wants to extracted the most accurate transformation from the two images and not just match the images. Yadics uses the whole image as a sampling grid: it is thus a total sampling.
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There are three categories of parametrization: elastic, global and local transformation. The elastic transformations respect the partition of unity, there are no holes created or surfaces counted several times. This is commonly used in Image
Registration by the use of
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F. Hild, S. Roux, N. Guerrero, M. Marante, and J. Flórez-Llópez, "Calibration of constitutive models of steel beams subject to local buckling by using digital image correlation," European journal of mechanics - a/solids, vol. 30, iss. 1, pp. 1–10,
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These transformations have been used for stress intensity factor identification by and for rod strain by. The local transformation can be considered as the same global transformation defined on several Zone Of
Interest (ZOI) of the fixed image.
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T. S. Yoo, M. J. Ackerman, W. E. Lorensen, W. Schroeder, V. Chalana, S. Aylward, Dimitris
Metaxas, and R. Whitaker, "Engineering and algorithm design for an image processing api: a technical report on itk - the insight toolkit", pp. 586–592,
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This metric is only used to find local translation in Yadics. This metric with translation transform can be solved using cross-correlation methods, which are non iterative and can be accelerated using Fast
Fourier Transform .
876:
623:{\displaystyle SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {1}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)^{2},}
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J. Réthoré, T. Elguedj, P. Simon, and M. Correct, "On the use of nurbs functions for displacement derivatives measurement by digital image correlation," Experimental mechanics, vol. 50, iss. 7, pp. 1099–1116,
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S. Klein, M. Staring, K. Murphy, M. A. Viergever, and J. P. W. Pluim, "Elastix: a toolbox for intensity-based medical image registration," Medical imaging, IEEE transactions on, vol. 29, issue 1, pp. 196–205,
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P. Cachier, E. Bardinet, D. Dormont, X. Pennec, and N. Ayache, "Iconic feature based nonrigid registration: the \PASHA\ algorithm," Computer vision and image understanding, vol. 89, issue 2?3, pp. 272–298,
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G. Besnard, F. Hild, and S. Roux, "Finite-element displacement fields analysis from digital images: application to portevin-le châtelier bands," Experimental mechanics, vol. 46, iss. 6, pp. 789–803, 2006.
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to measure the mechanical state of the material but strain gauges only measure the strain on a point and don't allow to understand material with an heterogeneous behavior. One can obtain a full in plane
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Many different methods exist (e.g. BFGS, conjugate gradient, stochastic gradient) but as steepest gradient and Gauss-Newton are the only ones implemented in Yadics these methods are not discussed here.
892:(NCC) is used when one cannot assure the optical flow conservation; it happens in case of change of lighting or if particles disappear from the scene can occur in particle images velocimetry (PIV).
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325:(PIV); the algorithms are similar to those of DIC but it is impossible to ensure that the optical flow is conserved so a vast majority of the software used the normalized cross correlation metric.
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J. Réthoré, S. Roux, and F. Hild, "From pictures to extended finite elements: extended digital image correlation (x-dic)," Comptes rendus mécanique, vol. 335, iss. 3, pp. 131–137, 2007.
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A. N. Tikhonov and V. B. Glasko, "Use of the regularization method in non-linear problems," \USSR\ computational mathematics and mathematical physics, vol. 5, iss. 3, pp. 93–107, 1965.
754:
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F. Hild and S. Roux, "Digital image correlation: from displacement measurement to identification of elastic properties ? a review," Strain, vol. 42, iss. 2, pp. 69–80, 2006.
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F. Hild and S. Roux, "Measuring stress intensity factors with a camera: integrated digital image correlation (i-dic)," Comptes rendus mécanique, vol. 334, iss. 1, pp. 8–12, 2006.
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R. Hamam, F. Hild, and S. Roux, "Stress intensity factor gauging by digital image correlation: application in cyclic fatigue," Strain, vol. 43, iss. 3, pp. 181–192, 2007.
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The Gauss-Newton method is a very efficient method that needs to solve a {U}={F}. On 1000 voxels μ-tomographic image the number of degrees of freedom can reach 1e6 (
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using the same algorithms (on monomodal images) but where the goal is to register images and thereby identifying the displacement field is just a side effect.
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In mechanics the displacement or velocity fields are the only concern, registering images is just a side effect. There is another process called
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This metric is the main one used in the YaDICs as it works well with same modality images. One has to find the minimum of this metric
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parallelism is used and data are not globally stored in memory. As an extensive description of the different algorithms is given in.
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2296:{\displaystyle d_{k}=-\gamma _{k}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}}
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YaDICs uses the general principle of image registration with a particular attention to the displacement fields basis.
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tuning of the αk scalar parameter at each iteration. The Gauss-Newton method can be used in small problems in 2D.
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for the grey level evaluation at non integer coordinates. The bi-cubic interpolation is the recommended one.
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The first step is to compute the gradient of the metric regarding the transform parameters
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871:{\displaystyle T_{\mu }(x)=x+\left\{\Phi (x)\right\}^{t}\left\{\mu \right\}.}
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The YaDICs optimization process follows a gradient descent scheme.
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First-order quadrangular finite elements Q4P1 are used in Yadics.
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allows one to choose between the following methods :
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are the mean values of the fixed and mobile images.
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46:. Unsourced material may be challenged and removed.
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2118:The gradient method principle is explained below:
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1435:{\displaystyle {\overline {\mathcal {I_{M}}}}}
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398:. The equation below defines the SSD metric:
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106:Learn how and when to remove this message
756:the number of pi(vo)xels (cardinal) and
749:{\displaystyle \left|\Omega _{F}\right|}
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2569:{\displaystyle \gamma _{k}=\left^{-1}}
788:The transformation can be written as:
785:the transformation parametrized by μ
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44:adding citations to reliable sources
2838:Multidimensional signal processing
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373:It is possible to choose between
347:Image registration general scheme
2588:{\displaystyle \Longrightarrow }
2372:{\displaystyle \Longrightarrow }
390:Sum of squared differences (SSD)
278:is a program written to perform
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884:Normalized cross-correlation
339:Image registration principle
2353:{\displaystyle \gamma _{k}}
2323:{\displaystyle \gamma _{k}}
714:{\displaystyle \Omega _{F}}
139:; 12 years ago
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2652:Particle Image Velocimetry
778:{\displaystyle {T}_{\mu }}
323:Particle Image Velocimetry
394:The SSD is also known as
298:digital image correlation
280:digital image correlation
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1475:Brazilian (Only in 2D),
895:The NCC is defined by:
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164:; 9 years ago
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296:In solid mechanics,
40:improve this article
2647:Displacement vector
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2803:Graphics libraries
2637:Image registration
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125:Original author(s)
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312:strain tensor
308:
307:strain gauges
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57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
2765:
2756:
2746:
2737:
2727:
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2700:
2691:
2681:
2642:Optical flow
2625:
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2613:
2610:
2604:
2602:
2598:
2190:
2120:
2117:
2114:
1505:
1502:
1499:Optimization
1494:
1486:
1467:
1453:
1444:
1369:
894:
887:
879:
790:
787:
631:
400:
393:
372:
369:Interpolator
363:
350:
342:
334:
327:
320:
316:optical flow
303:tensile test
295:
275:
274:
261:.univ-lille1
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2797:Categories
2658:References
265:/wordpress
197:Written in
183:Repository
169:2015-05-27
66:newspapers
2583:⟹
2559:−
2540:μ
2537:∂
2495:μ
2482:∂
2470:μ
2467:∂
2425:μ
2412:∂
2390:γ
2367:⟹
2342:γ
2312:γ
2287:μ
2284:∂
2242:μ
2229:∂
2216:γ
2212:−
2161:α
2148:μ
2129:μ
2085:∂
2059:μ
2032:∂
2011:μ
2008:∂
1985:μ
1975:∂
1937:μ
1910:−
1864:Ω
1860:∈
1846:∑
1829:Ω
1801:μ
1798:∂
1775:μ
1748:∂
1715:μ
1688:−
1642:Ω
1638:∈
1624:∑
1607:Ω
1582:μ
1579:∂
1537:μ
1522:∂
1428:¯
1392:¯
1336:¯
1319:−
1295:μ
1254:Ω
1250:∈
1236:∑
1219:¯
1202:−
1155:Ω
1151:∈
1137:∑
1124:¯
1107:−
1083:μ
1044:¯
1027:−
981:Ω
977:∈
963:∑
915:μ
859:μ
832:Φ
803:μ
771:μ
734:Ω
703:Ω
583:μ
556:−
509:Ω
505:∈
491:∑
474:Ω
420:μ
301:during a
2631:See also
1457:B-Spline
360:Sampling
249:or later
96:May 2015
55:"YaDICs"
1483:Elastic
385:Metrics
292:Context
254:Website
242:License
167: (
144:2012-01
142: (
80:scholar
1464:Global
1370:where
632:where
354:OpenMP
276:YaDICs
259:yadics
224:18.4MB
119:YaDICs
82:
75:
68:
61:
53:
2770:2003.
2751:2002.
2732:2011.
2686:2010.
1491:Local
1406:and
247:GPLv2
213:Linux
187:none
87:JSTOR
73:books
2676:2010
2605:i.e:
888:The
377:and
230:Type
220:Size
59:news
263:.fr
201:C++
42:by
2799::
2666:^
2303:,
318:.
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1062:I
1055:(
1050:)
1038:F
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1024:)
1019:i
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1011:(
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955:=
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941:I
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918:,
912:(
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866:.
862:}
856:{
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823:+
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618:,
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588:(
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553:)
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470:|
466:1
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450:M
446:I
440:,
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417:(
414:D
411:S
408:S
267:/
171:)
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109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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