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e) Registrationresult

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Quick 3D-2D Registrationbased on linearizationof rotation matrix

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to deal with more general shapes, it is taken into consider-ation to use the occluding contour as the feature for 3D-2Dmatching. To quickly calculate the contour generator thatis the 3D line on the object’s surface corresponding to theoccluding contour in the observed image, we take a 3Dgraphics system like OpenGL into our 3D-2D registrationmethod and effectively use the depth image supplied by it.In the following sections, first we briefly explain our basicstrategy, and then describe the details of these improvementswith lots of experiments showing their effects.

2. Basic strategy

The basic strategy is shown in Fig. 1. The details canbe found in [1]. Fig. 1b is a X-ray image of the rightinternal carotid circulation (the right side of the cerebralvessels) in Fig. 1a taken after injecting contrast mediuminto only the right part. The acquisition angle is givenfrom the graduations of the X-ray system but includes someerrors. As a result, the projection of the 3D model pointssampled from the skeleton of the 3D model vessel at thestate is deviated from the observed vessel as shown bythe black points in Fig. 1b. First, the projected shape istwo-dimensionally translated on the image to the positionwhich gives optimal overlap on the dark regions (possiblevessel regions) as shown by the black points in Fig. 1c. Thecorresponding features on the image, that is the skeletonsof 2D vessels in this case, are extracted in the neighbor-hood of the projected shape in a model-based way(whitelines). At this stage, perfect feature extraction is not re-quired. Lack of corresponding features can be allowed,due to the characteristics of the following territory-based3D-2Dmatching. The territory-based 3D-2Dmatching usesanisotropic search regions determined from the projectedshape of the model when making 3D-2D correspondingpairs based on the closeness in the image. As shown inFig. 2, this territory-based search restriction well removesout the model point whose corresponding 2D feature has

Closest pairs Magnification ofa part of Fig.1d

not been extracted in the observed image. Finally, takingadvantage of the high-ratio of viable 3D-2D point pairs ob-tained by the previous processes, the 3D transformation ofthe model is quickly calculated by separating the translationeffect out and linearizing the rotation matrix[2]. Althoughthe correct position and pose of the model is not obtained atonce, because of inaccurate matching pairs and linearizationerrors, the 3D model quickly converges to the correct stateby iterating the point matching and model transformationprocesses. In this example, the processes were iterated 30times for convergence and the total processing time was 2.7sec on a Pentium II(333MHz) machine.

3. Generalization of the camera coordinates

To calculate the 3D transformation of the model from3D-2D corresponding point pairs, we take the strategypresented in [2]. Supposed that we have correspondingpairs of the observed point and a 3D modelpoint . First, the following minimizationcriteria based on only rotation is built by separating outthe translation effect and by linearizing the rotation matrixrepresented with quaternions 0 1 2 3 .

Proceedings of the International Conference on Pattern Recognition (ICPR'00)1051-4651/00 $10.00 @ 2000 IEEE





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To examine the effect of the new formulae, calibrationof a 3D vessel model with a X-ray system using two X-rayimages were conducted. The position and pose of the 3Dmodel relative to the system known from its graduationsinclude about 20 degrees error in rotation and about( 100, 100, 200)(mm) in translation, since the positionand pose of the head is not calibrated to the X-ray system.Although the 3D relative geometry between the two X-rayimages given by the graduations also includes small errorsbecause of the deflection of the arm of the X-ray system,the errors are enough small to ignore comparing to theerrors in the relation between the 3D model and the X-raysystem. Fig. 3 shows a result of registration of the leftinternal carotid circulation in Fig. 1a. In Fig. 3a the blackand white points respectively represent the projection of the3D model skeleton at its initial state and after translatingit on the image so as to overlap the dark regions. In Fig.3b, correspondences using territory-based search restrictionat the first iteration are shown. After 27 iterations usingEquations (3) and (4), the 3D model is converged to theposition and pose so as to produce the projection as shownas white points in Fig. 3c. The transformation of the 3Dmodel is a 13.0 degree rotation around the axis (0.98, -0.06, 0.19) and a (19.6, 3.3, -10.1)(mm) translation.In this case, results using one image only are not very

different since the vessel has enough complexity in shape todetermine its position and pose from one view. However,in the case that an object has a simple shape as shown inFig. 4, it is fairly effective to use multiple images. Theblack and white points in Fig. 4a show the projection ofthe 3D model in the same way as Fig. 3a. The resultwhen we use only one view is shown in Fig. 4b. In theleft-hand image, the projection is largely deviated from theobserved vessels after a 63.6 degree rotation around the axis(0.63, 0.47, 0.61). Although, in the right-hand image, theprojection is on the observed vessels after a 62.5 degree

rotation around the axis (0.78, -0.44, -0.43), the rotationangle is clearly far from the actual value, which is known upto 20 degree. On the other hand, in Fig. 4c, the result usingtwo views simultaneously gives reasonable projections onthe both images. Although, unfortunately, we do not havethe ground truth data for the experiments, a 16.9 degreerotation around the axis (0.81, -0.01, 0.58) seems proper.

5. Visual feedback for active camera headWhen a 3D model of the environment surrounding a

camera is given, the position and pose of the camera can bedetermined using registration of the 3D model to the imagetaken by the camera. Based on this principle, we apply ourproposed method to visual feedback for an active camerahead. An active camera head can usually know its positionand pose from its control driving modules, but typically thevalues include some errors. Our 3D-2D registration methodcan be used for estimating the error.In the case of the registration of the blood vessel, its 3D

skeleton can be used as the matching feature independentlyof the view direction. In general, the occluding contouris typically the observed feature. The contour generator,that is the 3D line on the object’s surface corresponding tothe occluding contour, is largely changed depending on theview direction. That is, the 3D model points correspondingto the observed features should be calculated at each stateof the model. To quickly obtain such 3D model points evenfor complex 3D scene model, we propose to combine a 3Dgraphics system like OpenGL with our 3D-2D registrationmethod as shown in Fig. 5, since it can rapidly supply boththe view and depth image of a scene.The basic flow for obtaining the estimation error is

follows:i) The edges in the observed image are extracted in a model-based way using the view image (predicted view) suppliedby the 3D graphics system.ii) 3D model points on the contour generators are derivedfrom the differentiation of the depth image supplied by the



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V�#Â@zÁÃđđđ�12/05/31 10:15これなら分かる最適化数学: 紀伊國屋書店BookWeb

1/3 ページhttp://bookweb.kinokuniya.co.jp/htm/4320017862.html

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コレナラワカルサイテキカスウガク　キソゲンリカラケイサンシュホウマデ

これなら分かる最適化数学―基礎原理から計算手法まで金谷　健一【著】共立出版（2005/09/25 出版）

249p / 21cm / A5判ISBN： 9784320017863NDC分類： 417

価格：￥3,045 (税込) ポイント： 29 pt ？ポイントについて

在庫がございます。通常1-3日以内に発送致します。この商品は、国内送料無料でお届けします。※在庫情報は、前日の営業終了時のものです。？在庫と納期について

震災の影響により、福島県の一部地域では、ヤマト運輸営業所でのお引渡しとなります。＊最新の詳細情報はヤマト運輸のホームページを参照下さい。

詳細近年最適化は経営学やＯＲを越えてあらゆる工学の分野で応用されるようになった。その最大の理由は、計算機技術の進歩によって過去には不可能と思われた多変数の複雑な最循化問題が実際的な時間で解けるようになったことである。特に今日では、以前は机上の空論と思われていたベイズ推定を始めとする統計的最適化、サポートベクトルマシンやＥＭアルゴリズムを始めとする機械学習法、ニューラルネットワークなど多くの手法が実際の問題に適用されている。本書はそのような背景を考慮して、経営学やＯＲから離れ、多くの工学分野で用いられている各種の最適化手法の原理を説明することを目的とした。

第１章　数学的準備第２章　関数の極値第３章　関数の最適化第４章　最小二乗法第５章　統計的最適化第６章　線形計画法第７章　非線形計画法第８章　動的計画法

●詳細目次第1章数学的準備 1.1 曲線と曲面 1.2 1次形式と2次形式 1.3 2次形式の標準形第2章関数の極値

冊数： 1

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5

Simon Baker and Iain Matthews, Lucas-Kanade 20 Years On: A Unifying Framework, International Journal of Computer Vision, Vol. 56, No. 3, March, 2004, pp. 221 - 255. Simon Baker and Iain Matthews, Lucas-Kanade 20 Years On: A Unifying Framework: Part 1, tech. report CMU-RI-TR-02-16, Robotics Institute, Carnegie Mellon University, July, 2002

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T.Tamaki, T. Yamamura and N. Ohnishi

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Figure 1: Observation I2 of the calibration pattern I1 modeled by three transfor-mations.

2.1. Modeling view change

Given two images of the same planar object from di!erent viewpoints, the rela-tionship between them is described by a planar perspective motion model with eightparameters.2,20 As shown in Fig.2, I1 and I2 can be considered as the di!erent viewsof the same plane because of the following reason. Since I1 is just a digital image,I1 is a plane exactly identical to the image plane. The printed sheet is regarded asa plane transformed from the plane of I1, and I2 is the projection of the sheet ontothe image plane having a slight displacement due to the distortion.

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2.2. Modeling distortion

The relationships between undistorted and distorted coordinates in an image(shown in Fig.3) are often modeled by five intrinsic camera parameters21,6: the ra-dial distortion parameters #1 and #2, the coordinates of image center (cx, cy)T , andthe horizontal scale factor sx. We write these parameters as !d = (!d

1 , . . . , !d5) =

(#1, #2, cx, cy, sx)T . Although we consider only the radial distortion, the followingdiscussion can also be applied when another model involving decentering distortion22,23

is employed.Distortion is represented with respect to the image center (cx, cy)T . Let pu =

(xu, yu)T be a point in I2 without considering distortion, that is, pu = u(p); pu

is moved to pd = (xd, yd)T by the radial distortion. Here we have two functionsbetween pu and pd.

pd = d(pu, !d) (3)

T.Tamaki, T. Yamamura and N. Ohnishi

I1 I2fu

d

H

Figure 1: Observation I2 of the calibration pattern I1 modeled by three transfor-mations.

2.1. Modeling view change

Given two images of the same planar object from di!erent viewpoints, the rela-tionship between them is described by a planar perspective motion model with eightparameters.2,20 As shown in Fig.2, I1 and I2 can be considered as the di!erent viewsof the same plane because of the following reason. Since I1 is just a digital image,I1 is a plane exactly identical to the image plane. The printed sheet is regarded asa plane transformed from the plane of I1, and I2 is the projection of the sheet ontothe image plane having a slight displacement due to the distortion.

The model warps a point p = (x, y)T on I1 into the corresponding point on I2,pu = (xu, yu)T , using the function u of !u = (!u

1 , . . . , !u8 )T as follows.2

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The relationships between undistorted and distorted coordinates in an image(shown in Fig.3) are often modeled by five intrinsic camera parameters21,6: the ra-dial distortion parameters #1 and #2, the coordinates of image center (cx, cy)T , andthe horizontal scale factor sx. We write these parameters as !d = (!d

1 , . . . , !d5) =

(#1, #2, cx, cy, sx)T . Although we consider only the radial distortion, the followingdiscussion can also be applied when another model involving decentering distortion22,23

is employed.Distortion is represented with respect to the image center (cx, cy)T . Let pu =

(xu, yu)T be a point in I2 without considering distortion, that is, pu = u(p); pu

is moved to pd = (xd, yd)T by the radial distortion. Here we have two functionsbetween pu and pd.

pd = d(pu, !d) (3)

Correcting distortion of image by image registration

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In addition to the Jacobian of u, the Jacobian of d is also needed for a gradientmethod. Here we introduce the implicit function theorem17 for systems18. Thistheorem can represent the Jacobian of d as an explicit form through f . Let F bea function of q = (pu, !d) and pd represented by

F (q, pd) = pu ! f(pd, !d) (5)

If F (q, d(q)) = 0 is satisfied for "q, then pd = d(q) is called an implicit functiondetermined by F (q, pd) = 0. In our case, the condition is theoretically alwayssatisfied because we defined d as the inverse of f , and numerically Eq.(5) is almost0 (it can be less than 10!10).

According to the theorem, the Jacobian is given by the following equations.

"d

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Correcting distortion of image by image registration

estimateall !u, !d, !h

10

20

30

40

50

60

70

the number of iterations

estimate only !u

cost

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tion

/ num

ber o

f poi

nts

0 5 10 15 20 25

Figure 6: Convergence of the estimation. Horizontal axis is the number of iterationsto update the estimates; vertical axis represents the sum of squares divided by thenumber of points in I1.

on the grid pattern in the distorted image are corrected to straight lines, so theproposed method works well. The computational time was about 20 minutes ona PC (866MHz CPU, GNU C++ and CLAPACK). However, the optimization hadalmost converged after fewer than 30 iterations.

We can see the convergence in Fig.6, which shows the sum of the squares of theintensity residuals of the first 25 iterations. As we mentioned in section 3.3, only!u was estimated in the early stage of the iteration while !d and !h were fixed totheir initial value. After the estimation of !u had converged (16 iterations), theminimization using all parameters began and converged after several iterations.

To visualize the convergence, we produced a synthetic image which is trans-formed from I1 by using u, d and H with current estimates at every step. Eachimage of Fig.7 illustrates the di!erence between I2 and the synthesized image. Atthe first iteration, the two images were quite di!erent and the di!erence image hadmany dark pixels. After 25 iterations, the estimation had converged and the sub-traction image had few dark pixels, which means that the synthetic image and I2

became quite similar to each other and the estimation result was good.

4.2. Distortion parameters while changing zoom

The advantage of the proposed method is convenience for the human operator.The requirements are just a printed pattern and one captured image of it; a batchprocess is then called without any manual operations. This simplicity enables usto see the distortion parameter change that arises due to changing the zoom of thecamera, while point correspondence-based conventional methods require an enor-

Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Correcting Distortion of Image by Image Registration with the Implicit Function Theorem," International Journal on Image and Graphics, World Scientific Publishing Co., Vol.2, No.2, pp.309-330 (2002 4).

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Toru Tamaki : "Unified Approach to Image Distortion: D-U and U-D models," IEICE Transactions on Information & Systems, Vol.E88-D, No.5, pp.1086-1090 (2005 05). Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Correcting Distortion of Image by Image Registration with the Implicit Function Theorem," International Journal on Image

and Graphics, World Scientific Publishing Co., Vol.2, No.2, pp.309-330 (2002 4). Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Unified Approach to Image Distortion,” ICPR2002, Vol.2, pp.584-587 (2002 8). Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Correcting Distortion of Image by Image Registration,” ACCV 2002, Vo.2, pp.521-526 (2002 1). Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "A Method for Compensation of Image Distortion with Image Registration Technique," IEICE Transactions on Information

& Systems, Vol.E84-D, No.8, pp.990-998 (2001 8). Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "An Automatic Camera Calibration Method with Image Registration Technique,” SCI2000, Vol.5, pp.317-322 (2000 7). ńƠū3429280hǳȘŎ4ȼʔʗɬĦɇȼƓĤ÷Įș US Patent 6,791,616 “Image lens distortion correcting method”

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Toru Tamaki, Takahiko Sugino, Masanobu Yamamoto: "Measuring Ball Spin by Image Registration," Proc. of FCV2004 ; the 10th Korea-Japan Joint Workshop on Frontiers of Computer Vision, pp.269-274 (2004 2), Kyushu University, Fukuoka, Japan, 2004/2/3-4.

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Tamaki Toru, Yukihiko Ushiyama, Bisser Raytchev, Kazufumi Kaneda, Inverse Composite Alignment of a sphere under orthogonal projection for ball spin estimation, Ø�Cŉ�"ŝŦ�p, ɥʗɾʎʙɯɽɪʐʗȷəʊʙɪʊɵɘɗ, Vol. 2011-CVIM-178, No. 13, pp. 1-5 (2011 08). Tamaki Toru, Haoming Wang, Bisser Raytchev, Kazufumi Kaneda, Yukihiko Ushiyama, Estimating the spin of a table tennis ball using inverse compositional image alignment, Proc. of ICASSP 2012, pp. 1457-1460 (2012 03).

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Tamaki Toru, Haoming Wang, Bisser Raytchev, Kazufumi Kaneda, Yukihiko Ushiyama, Estimating the spin of a table tennis ball using inverse compositional image alignment, Proc. of ICASSP 2012, pp. 1457-1460 (2012 03).

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Visual validation�

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Radius: 61 pixels

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ď�ªĚ��ŝŦ�Research Project, ɰəʔɢɶɫɴʔɝəʊʙɪɗʑəʊʗɶȺɎɑǰű¶ʋʙɩʐʗì�, http://www.ok.ctrl.titech.ac.jp/res/PPE/ ċĊƊĠ, ?ōŠǠ, ��,Ĵi, ��Ĥñʞ“ɫɴʔɝýŲGŎ4ɕŌȝȰŗëĮȺɎɑ�ýǗʋʙɩʐʗì�”, MIRU2009. ċĊƊĠ, ��,Ĵi, ��Ĥñʞ“ɰəʔɢɶɫɴʔɝəʊʙɪɗʑəʊʗɶȺɎɑ3 ģ5�ýǗì�”, SSII2010.� ǷǷǷ�

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ƌǳē�, ¦ƬǳƮ�ǳ& Dcǳ7�ǋ. Ŏ4ȼĝĊXǆťɕƃÜȪȰǰű¶ȡȴǩ2ȹ±ǦȼƿƵ. In Ŏ4ȼƤƫʘŉƜɩʗʆɪɚʉ(MIRU2011)Ʃôǡ, Vol. 2011, pp. 440-447, 2011.

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M. Yamamoto, "A General Aperture Problem for Direct Estimation of 3-D Motion Parameters," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 5, pp. 528-536, May 1989, doi:10.1109/34.24785

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Masanobu Yamamoto, Katsutoshi Yagishita: Scene Constraints-Aided Tracking of Human Body. CVPR 2000.

CGʋɵʓ ʛʖəʍɿʔʙʉʜ Ŏ4�

Yamamoto, M.; Koshikawa, K.; , "Human motion analysis based on a robot arm model,” CVPR '91.�

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Active Appearance Model (AAM)�

The Robotics Institute, Mellon University, AAM Fitting Algorithms, http://www.ri.cmu.edu/research_project_detail.html?project_id=448&menu_id=261�

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Active Appearance Model (AAM)�

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AAMModel InstanceM(W(x;p))

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Figure 3: An example of AAM instantiation. The shape parameters p = (p1, p2, . . . , pn)T are used tocompute the model shape s and the appearance parameters ! = (!1,!2, . . . ,!m)T are used to compute themodel appearance A. The model appearance is defined in the base mesh s0. The pair of meshes s0 and s

define a (piecewise affine) warp from s0 to s which we denote W(x;p). The final AAM model instance,denoted M(W(x;p)), is computed by forwards warping the appearance A from s0 to s usingW(x;p).

parameters ! is then created by warping the appearance A from the base mesh s0 to the model

shape s. This process is illustrated in Figure 3 for concrete values of p and !.

In particular, the pair of meshes s0 and s define a piecewise affine warp from s0 to s. For each

triangle in s0 there is a corresponding triangle in s. Any pair of triangles define a unique affine

warp from one to the other such that the vertices of the first triangle map to the vertices of the

second triangle. See Section 4.1.1 for more details. The complete warp is then computed: (1) for

any pixel x in s0 find out which triangle it lies in, and then (2) warp x with the affine warp for that

triangle. We denote this piecewise affine warp W(x;p). The final AAM model instance is then

computed by warping the appearance A from s0 to s with warp W(x;p). This process is defined

by the following equation:

M(W(x;p)) = A(x) (4)

where M is a 2D image of the appropriate size and shape that contains the model instance. This

equation, describes a forwards warping that should be interpreted as follows. Given a pixel x in

6

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6

ǫ�ÁȼŹƞʋɵʓ (Active Shape Model) ȕȕ+ǫ:(ȼŤU

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LK�

AMM�

Iain Matthews and Simon Baker, Active Appearance Models Revisited, International Journal of Computer Vision, Vol. 60, No. 2, November, 2004, pp. 135 - 164. T.F.Cootes, G.J. Edwards and C.J.Taylor. "Active Appearance Models”, ECCV98, Vol. 2, pp. 484-498, 1998.

T.F. Cootes, D. Cooper, C.J. Taylor and J. Graham, "Active Shape Models - Their Training and Application." Computer Vision and Image Understanding. Vol. 61, No. 1, Jan. 1995, pp. 38-59. T.F.Cootes, C.J.Taylor, Active Shape Models - `Smart Snakes'. in Proc. British Machine Vision Conference. Springer-Verlag, 1992, pp.266-275.

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¦Beyond ICIA: CoDe and LinCoDe�

Brian Amberg, Andrew Blake and Thomas Vetter, On Compositional Image Alignment with an Application to Active Appearance Models, CVPR2009. http://www.cs.unibas.ch/personen/amberg_brian/aam/

1981 Lucas-Kanade 1992 Active Shape Model 1998 Active Appearance Model 2002 ICIA 2009 Code/LinCode

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•  ICIAɗʁʕʙɱ •  `ÈȼƞŮǑȢ�ȩȝ •  ǘòȼkàɕJŌ •  ɴʗʁʔʙɶŎ4ȼÉEɕJŌ

•  ÏŌ •  ʔʗɬĦɇƓĤʞɞʊʑĔĤ •  [ňʅʙʓxƸǑƞķ •  ±ǦƿƵʞAAM, Code/LinCoDe

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■  Iterative Closest Point (ICP) !  Ľǡkʇɲɱʗɣȼ�ƑŔȹâĮ !  �òȼıŋâĮȢŒ�

■  ʇɲɱʗɣ !  ʫO(ʔɪɫɶʔʙɩʐʗ !  xƸƍGRȷǄʃɢɶʓtɕì�Ȭɑ

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Chen, Y. and Medioni, G. “Object Modeling by Registration of Multiple Range Images,” Proc. IEEE Conf. on Robotics and Automation, 1991. Besl, P. and McKay, N. “A Method for Registration of 3-D Shapes,” Trans. PAMI, Vol. 14, No. 2, 1992.

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3.  �ȼ�ÏʪRX+tȼjĽȺăɊƽȝYȼĽɕĬɉɑǳ(closest point)

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Shinji Umeyama. Least-squares estimation of transformation parameters between two point patterns. IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 13, No. 4, pp. 376–380, 1991.

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Berthold K. P. Horn, Hugh M. Hilden, and Shahriar Negahdaripour. Closed-form solutions of absolute orientation using orthonormal matrices. Journal of the Optical Society of America, Vol. 5, pp. 1127–1135, 1988.�Berthold K. P. Horn. Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America, Vol. 4, pp. 629–642, 1987.�

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K. S. Arun, T. S. Huang, and S. D. Blostein. Least-squares fitting of two 3-D point sets. PAMI, Vol. 9, No. 5, pp. 698–700, 1987. Peter H. Schönemman. A generalized solution of the orthogonal procrustes problem. Psychometrika, Vol. 31, No. 1, pp. 1–10, 1966. �

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Kenichi Kanatani. Analysis of 3-D rotation fitting. PAMI, Vol. 16, No. 5, pp. 543–549, 1994. Shinji Umeyama. Least-squares estimation of transformation parameters between two point patterns. PAMI, Vol. 13, No. 4, pp. 376–380, 1991. ǒƬ2�, Ŏ4ŉƜ—ʣģ5Ƥƫȼòŉ—, 3.4ů, ėYDŁ, 1990.�

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Kenichi Kanatani. Analysis of 3-D rotation fitting. PAMI, Vol. 16, No. 5, pp. 543–549, 1994. Shinji Umeyama. Least-squares estimation of transformation parameters between two point patterns. PAMI, Vol. 13, No. 4, pp. 376–380, 1991. ǒƬ2�, Ŏ4ŉƜ—ʣģ5Ƥƫȼòŉ—, 3.4ů, ėYDŁ, 1990.�

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K. S. Arun, T. S. Huang, and S. D. Blostein. Least-squares fitting of two 3-D point sets. PAMI, Vol. 9, No. 5, pp. 698–700, 1987. Kenichi Kanatani. Analysis of 3-D rotation fitting. PAMI, Vol. 16, No. 5, pp. 543–549, 1994. ǒƬ2�, Ŏ4ŉƜ—ʣģ5Ƥƫȼòŉ—, 3.4ů, ėYDŁ, 1990.�


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■  Orthogonal Procrustes Problem !  Ìŉ�EǐʛPsychometrikaȹȸʜ

■  Wahba’s Problem !  �ô�Eǐ

■  Absolute Orientation !  BřķǑ�ʘ7�ǳʛPhotogrammetria, J. Optical Soc. of America ȹȸʜ

■  Fitting corresponding point sets !  ɥʗɾʎʙɯɽɪʐʗʛIEEE PAMIȹȸʜ

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Orthogonal Procrustes Problem�

Procrustes. "Now then, you fellows; I mean to fit you all to my little bed!”�

The Modern Bed of Procrustes - Cartoon from the Project Gutenberg eBook of Punch, Volume 101, September 19, 1891, by John Tenniel. http://commons.wikimedia.org/wiki/File:The_Modern_Bed_of_Procustes_-_Punch_cartoon_-_Project_Gutenberg_eText_13961.png

Orthogonal Procrustes�

Extended Orthogonal Procrustes�

Generalized Orthogonal Procrustes�

Peter H. Schönemmanand Robert M. Carroll. Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, Vol. 35, No. 2, pp. 245–255, 1970.�Devrim Akca. Generalized procrustes analysis and its applications in photogrammetry. Technical report, ETH, Swiss Federal Institute of Technology Zurich, Institute of Geodesy and Photogrammetry, 2003.�

Peter H. Schönemman. A generalized solution of the orthogonal procrustes problem. Psychometrika, Vol. 31, No. 1, pp. 1–10, 1966. �

John R. Hurley, Raymond B. Cattell, The procrustes program: Producing direct rotation to test a hypothesized factor structure, Behavioral Science, Volume 7, Issue 2, pages 258–262, April 1962.�

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Wahba’s Problem�

http://www.stat.wisc.edu/~wahba/public/jpg/jsm.05/noether.html�

Prof. Grace Wahba (University of Wisconsin-Madison)�

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Grace Wahba, “Problem 65–1: A Least Squares Estimate of Spacecraft Attitude,” SIAM Review, Vol. 7, No. 3, July 1965, p. 409. Malcolm D. Shuster. The generalized Wahba problem. The Journal of the Astronautical Sciences, Vol. 54, No. 2, pp. 245–259, 2006. http://home.comcast.net/Ȕmdshuster2/PUB 2006c J GenWahba AAS.pdf. F. Landis Markley. 30 years of Wahba’s problem. 1999 Flight Mechanics Symposium, 1999. http://airex.tksc.jaxa.jp/ pl/dr/19990063982.�

İʪɗʆʕãȲ�ȦȽ1966²ȡɏ

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Absolute Orientation�Berthold K. P. Horn, Hugh M. Hilden, and Shahriar Negahdaripour. Closed-form solutions of absolute orientation using orthonormal matrices. Journal of the Optical Society of America, Vol. 5, pp. 1127–1135, 1988. Berthold K. P. Horn. Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America, Vol. 4, pp. 629–642, 1987.�

E. H. Thompson. An exact linear solution of the problem of absolute orientation. Photogrammetria, Vol. 15, pp. 163-179, 1958-1959. G.H. Schut, On exact linear equations for the computation of the rotational elements of absolute orientation, Photogrammetria, Volume 17, pp. 34-37, 1960–1961.�

E. H. Thompson, 1958-1959

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■  Orthogonal Procrustes Problem !  Ìŉ�EǐʛPsychometrikaȹȸʜ

■  Wahba’s Problem !  �ô�Eǐ

■  Absolute Orientation !  BřķǑ�ʘ7�ǳʛPhotogrammetria, J. Optical Soc. of America ȹȸʜ

■  Fitting corresponding point sets !  ɥʗɾʎʙɯɽɪʐʗʛIEEE PAMIȹȸʜ

■  +ʪ !  �Ǖst, Bisser Raytchev, ņĈË, ǒōsô, Procrustes AnalysisɕŌȝȰ�Vì�, MIRU2011

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ABSTRACT DEMO MOVIES REFERENCES CODE SVN DOWNLOAD

ABSTRACTThis demonstration provides CUDA-based implementations of two 3D point sets registration algorithms:Softassign [Gold 1998] and EM-ICP [Granger 2002]. Both algorithms are known for being time demanding,even on modern multi-core CPUs. Our GPU-based implementations vastly outperform CPU ones. Forinstance, our CUDA EM-ICP aligns 5000 points in less than 7 seconds on a GeForce 8800GT, while thesame implementation in OpenMP on an Intel Core 2 Quad would take 7 minutes.

Registration (alignment) of 3D point sets is one of the most important problems in computer vision andseveral methods have been developed over the last two decades. The widely used Iterative Closest Point(ICP) algorithm [Besl 1992] provides quick registration, but requires a good initial alignment in order toprevent local minima and produce a plausible match. Softassign and EM-ICP represent efforts to overcomesuch limitations: instead of looking for "hard" correspondences between points (each point in one of the setshas to uniquely map to another point in the other set), the latter two algorithms focus on "soft"correspondences (each point in one of the sets corresponds somehow to every point in the other set bysome weight). Although these algorithms can handle any initial arrangement, their associated computationalcost has been preventing their practical usefulness even for moderately large number of points.

Recent advances in graphics hardware and software have motivated us to implement Softassign and EM-ICP on a GPU and evaluate their corresponding behavior and performance. Our contribution is twofold: weintroduce the GPU implementations and also demonstrate that most steps of these algorithms are GPU-friendly, consisting of either vector-matrix multiplications or element-wise operations. The registrationprocess is iteratively tracked in a window with interactive manipulation.

Connection to the main conferenceThis demonstration deals with 3D geometry, making it well-suited to the main conference, workshops, andtutorials:

CVPR2010 Workshop on Three Dimensional Information Extraction for Video Analysis and MiningCVPR2010 Tutorial on 3D Shape Reconstruction from Photographs: a Multi-View Stereo ApproachCVPR2009 Course on Computer Vision on GPUs

This demonstration is also related to the following workshops:

ICCV2009 Workshop on 3-D Digital Imaging and ModelingECCV2010 Workshop on Reconstruction and Modeling of Large Scale 3D Virtual EnvironmentsECCV2010 Workshop on Computer Vision on GPUs

MOVIESThe videos presented below consist of the registration of sets of 1000, 3000 and 5000 points, sequentiallyspaced in each video. The starting point-set (white) is progressively matched (cyan) to the target one(yellow).

CUDA-based implementations of Softassign and EM-ICPToru Tamaki, Miho Abe, Bisser Raytchev, Kazufumi Kaneda, Marcos SlompHiroshima University, JapanContact address: [email protected]

CVPR2010 DEMOSUBMISSION

15th June 2010The content of this page waspresented at CVPR201 demo.

SOURCE CODE

7th July 2010Source code is now available!Download tarball(cuda_emicp_softassign.0.1.tar.gz)or brouse SVNSee README.txt for usege.

The demo is written in C++ andCUDA and is Linux/GCC compliant(Fedora 11), but should port easily toWindows or Mac platforms.Additional dependencies includefreeglut (a modern GLUT alternative)for visualization and OpenMP for theCPU-based implementations.

PAPER

19th Nov. 2010The content of this page waspresented at UPDAS2010.

Toru Tamaki, Miho Abe, BisserRaytchev, Kazufumi Kaneda:"Softassign and EM-ICP on GPU",Proc. of The 2nd Workshop on UltraPerformance and DependableAcceleration Systems (UPDAS), CD-ROM, 5 pages, 2010.

Slide (pptx, pdf) | paper PDF (will beavailable on IEEE digital library)

ALIGN TWO 3D SETS OF 5000 POINTS WITHIN 7 SECONDShttp://home.hiroshima-u.ac.jp/tamaki/study/cuda_softassign_emicp/�

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■  Survey and Review: !  O. Fluck, C. Vetter, W. Wein, A. Kamen, B. Preim, and R. Westermann. A survey of

medical image registration on graphics hardware. Computer Methods and Programs in Biomedicine, 104, 3, e45-e57 (2011).

!  Shams, Ramtin, Parastoo Sadeghi, Rodney A Kennedy, and Richard I Hartley. A Survey of Medical Image Registration on Multicore and the GPU. IEEE Signal Processing Magazine 27, no. 2, pp. 50-60 (2010).

!  Derek L G Hill, Philipp G Batchelor, Mark Holden and David J Hawkes, Medical image registration, TOPICAL REVIEW, Physics in Medicine and Biology, Vol. 46, No. 3, pp.R1–R45 (2001).

!  Pluim, J.P.W.; Maintz, J.B.A.; Viergever, M.A.; , "Mutual-information-based registration of medical images: a survey," Medical Imaging, IEEE Transactions on , vol.22, no.8, pp.986-1004, Aug. 2003.

!  P. Markelj, D. Tomaževič, B. Likar, F. Pernuš, A review of 3D/2D registration methods for image-guided interventions, Medical Image Analysis, Volume 16, Issue 3, April 2012, Pages 642-661, ISSN 1361-8415, 10.1016/j.media.2010.03.005.

!  W. R. Crum, T. Hartkens, D. L. G. Hill, Non-rigid image registration: theory and practice, The British Journal of Radiology, 77 (2004), S140–S153.

■  Japanese: !  M�ÇÖ, »ãf��U!��×��{©�A4G, PRMU2012, 2012K5V. !  �y�Í, Ã}�áj: 3��Ï�Ù�)E;$3(/�¢�¦k�i�b��,�<%~]�zdskLÞ�D-II, J87-DII, No.10pp.1887-1920 , 2004.


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■  Survey !  Richard Szeliski. Image mosaicing for tele-reality applications. Technical Report 94/2, Digital

Equipment Corporation, Cambridge Research Lab, June 1994. !  Lisa Gottesfeld Brown. A survey of image registration techniques. ACM Computing Surveys.

24, 4 (December 1992), 325-376. !  Barbara Zitová, Jan Flusser, Image registration methods: a survey, Image and Vision

Computing 21 (2003) 977–1000 ■  Books

!  D. P. Capel, Image Mosaicing and Super-resolution, PhD thesis from University of Oxford, 2001. Springer, 2004.

!  A. Ardeshir Goshtasby, 2-D and 3-D Image Registration for Medical, Remote Sensing, and Industrial Applications, Wiley Publishers, 2005.

!  Richard Szeliski, Image Alignment and Stitching: A Tutorial, Foundations and Trends® in Computer Graphics and Vision,2(1), now publishers, 2006.

!  Richard Szeliski. Image Stiching, in Computer Vision: Algorithms and Applications. Chapter 9, Springer, 2010.

■  DP"i��2DËp>32I) !  Seiichi Uchida and Hiroaki Sakoe, A survey of elastic matching techniques for handwritten

character recognition, IEICE Transactions on Information & Systems, vol.E88-D, no.8, pp.1781-1790, Aug. 2005.

■  Procrustes !  Nick Higham, Matrix Procrustes Problems, The University of Manchester, 1993. !  ³yÔ: �¯�±\�`�RÊ�, SIS/SIP/IPSJ-AVM�z, SIP2009-48, SIS2009-23, Vol.

109, No.202, pp.59-64, (2009 09).


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■  ICP !  �W°, ICP#G+F0?, CVIM2009K8V. !  �W°, ICP#G+F0?, *I:C�19.DIm|É'%63, x2Ú, #6*?@4$#,

2010. !  Szymon Rusinkiewicz, Marc Levoy, Efficient Variants of the ICP Algorithm, 3DIM2001. !  How to use iterative closest point, PCL documentation,

http://pointclouds.org/documentation/tutorials/iterative_closest_point.php ■  LK, ICIA, 8E@5F3(§Ð«�Ï

!  Lucas-Kanade 20 Years On, http://www.ri.cmu.edu/research_project_detail.html?project_id=515&menu_id=261

!  À�å, *I:C�19.DI�l�ÏäÂg, *I:C�19.DI:�¢´��¹q¬, x16Ú, 1998.

!  WO�tR, Î£�tÝ: “�Äv�Ø"��¡�Ìv�Ï��«�Ï�¢�, ÅÏ�z@4$#kLÞ, 62, 3, pp.337-342 (2008) .

■  AMM !  AAM Fitting Algorithms

http://www.ri.cmu.edu/research_project_detail.html?project_id=448&menu_id=261 !  The AAM-API http://www2.imm.dtu.dk/~aam/ !  CoDe and LinCoDe http://www.cs.unibas.ch/personen/amberg_brian/aam/ !  aam-opencv http://code.google.com/p/aam-opencv/

■  m¼g !  S�°Q, �� X��m¼gvk�Q��ßnh��{�Ye��Q�, �cPÓ, 2005. !  ��¶½, 7I6G#.B/5@I5, *I:C�19.DIm|É'%63, x1Ú, #6*?@4$#, 2010.


Tutorials on registration�■  ICCV2011

!  Non-rigid registration and reconstruction http://users.isr.ist.utl.pt/~adb/tutorial/ !  Parts Based Deformable Registration http://ci2cv.net/Tutorials/ICCV_2011_Tutorial.html

■  CVPR2011 !  Tools and Methods for Image Registration http://www.imgfsr.com/CVPR2011/Tutorial6/

■  MICCAI2011 !  Flexible Algorithms for Image Registration

http://www.mic.uni-luebeck.de/research-and-projects/miccai-2011.html ■  MICCAI2010

!  Intensity-based Deformable Registration http://campar.in.tum.de/DefRegTutorial/WebHome ■  MICCAI2009

!  Image-Guided Interventions http://www.isis.georgetown.edu/CAIMR/Workshops/MICCAI2009.htm ■  ICIP2009

!  Introduction to Image Registration http://www.icip2009.org/Tutorial_09.asp !  Shape Representation and Registration using Different Implicit Spaces http://www.icip2009.org/Tutorial_02.asp

■  MICCAI2008 !  Image-Guided Interventions http://www.isis.georgetown.edu/CAIMR/Workshops/MICCAI2008.htm !  Numerical optimization for Medical Image Registration

http://web.archive.org/web/20090228095031/http://www.cas.mcmaster.ca/~modersit/2008-MICCAI/index.html

■  CVPR2008 !  Survey and Recent Advances in Image Registration and Fusion

http://www.cs.wright.edu/~agoshtas/cvpr08RegFusTut.html ■  CVPR2004

!  2-D and 3-D Image Registration: A Tutorial http://www.cs.wright.edu/~agoshtas/CVPR04_Registration_Tutorial.html

■  ICIP2005 !  Image Registration: A Survey and Recent Advances http://dar.site.cas.cz/download.php?bd=85


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Workshops on Non-Rigid Shape Analysis and Deformable Image Alignment (NORDIA) http://tosca.cs.technion.ac.il/nordia/�


Tamaki’s related publications 1/2�■  Radial distortion correction with registration

!  Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Correcting Distortion of Image by Image Registration with the Implicit Function Theorem," International Journal on Image and Graphics, World Scientific Publishing Co., Vol.2, No.2, pp.309-330 (2002 4).

!  Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "A Method for Compensation of Image Distortion with Image Registration Technique," IEICE Transactions on Information & Systems, Vol.E84-D, No.8, pp.990-998 (2001 8).

!  Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Unified Approach to Image Distortion," Proc. of ICPR2002 ; The 16th International Conference on Pattern Recognition, Vol.2, pp.584-587 (2002 8).

!  Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "Correcting Distortion of Image by Image Registration," Proc. of ACCV 2002 ; The Fifth Asian Conference on Computer Vision, Vo.2, pp.521-526 (2002 1).

!  Toru Tamaki, Tsuyoshi Yamamura, Noboru Ohnishi : "An Automatic Camera Calibration Method with Image Registration Technique," Proc. of SCI2000 ; The 4th World Multiconference on Systemics, Cybernetics and Informatics , by IIIS; International Institute of Informatics and Systemics, Vol.5, pp.317-322 (2000 7).

■  Calibration with registration !  Toru Tamaki, Masanobu Yamamoto : "Calibration Method by Image Registration with Synthetic Image of 3D

Model," IEICE Transactions on Information & Systems, Vol.E86-D, No.5, pp.981-985 (2003 05). ■  Human motion analysis with registration

!  Toru Tamaki : "Human Limb Extraction Based on Motion Estimation Using Optical Flow and Image Registration," Journal of Advanced Computational Intelligence and Intelligent Informatics, Fuji Technology Press Ltd., Vol.8, No.2, pp.150-155 (2004 3).

■  Non-rigid medical image registration !  Toru Higaki, Toru Tamaki, Kazufumi Kaneda, Nobutada Date, Shogo Azemoto: "Non-rigid Image Registration for

Medical Imaging using a Free-form Deformation", Proc. of IEVC2007; the IIEEJ Image Electronics and Visual Computing Workshop, Institute of Image Electronics Engineers of Japan (2007 11).

!  æÛÔ,â�Èw, Bisser Raytchev, ³yÔ, SWu��: �� 3��:LHE��!��=K+(57>'��, MIRU2010 �Ï��²�h�-I=.&?��, pp.894-897 (2010 07).


Tamaki’s related publications 2/2�■  Spin estimation with registraton

!  Tamaki Toru, Haoming Wang, Bisser Raytchev, Kazufumi Kaneda, Yukihiko Ushiyama: "Estimating the spin of a table tennis ball using inverse compositional image alignment", Proc. of ICASSP 2012 ; 2012 IEEE International Conference on Acoustics, Speech, and Signal Processing,pp. 1457-1460 (2012 03).

!  Tamaki Toru, Yukihiko Ushiyama, Bisser Raytchev, Kazufumi Kaneda: "Inverse Composite Alignment of a sphere under orthogonal projection for ball spin estimation", �z�hkL¥ªz�, *I:C�19.DI�%@�.@4$#, Vol. 2011-CVIM-178, No. 13, pp. 1-5 (2011 08).

!  ³yÔ, Üa¾Æ, ®µãº�: ��/)C<6%��5=KB2��031�, ~]�zdskL�¢z��81�I�²�@4$#h�¥ªL�PRMU2005-116, No.116, pp.13-18 (2005 11).

!  Toru Tamaki, Takahiko Sugino, Masanobu Yamamoto: "Measuring Ball Spin by Image Registration," Proc. of FCV2004 ; the 10th Korea-Japan Joint Workshop on Frontiers of Computer Vision, pp.269-274 (2004 2).

!  Yukihiko Ushiyama, Toru Tamaki, Osamu Hashimoto, Hisato Igarashi: "Measuring the spin of a ball by digital image analysis," in Science and Racket Sports III: The Proceedings of the Eighth International Table Tennis Federation Sports Science Congress and the Third World Congress of Science and Racket Sports, A. Lees, J.-F. Kahn, I. W. Maynard ed., Routledge, Taylor & Francis Inc., New York, pp.129-133 (2004).

!  Üa¾Æ, ³yÔ, M·, ��à¨N�: �$4N@/)�� #DI,�, _ÑJk�¤N[¸k^Áo, N��TL¸k¿(1), x5Ò, x2Â, pp.231-236 (2003 02).

!  Üa¾Æ, ³yÔ, M·, ��à¨N�: ��"=K9M� ��"��#DI,*.;8�, _Ñf¤k¥ª, x20Ò, pp.49-52 (2002).

■  3D point cloud registration !  Toru Tamaki, Miho Abe, Bisser Raytchev, Kazufumi Kaneda: "Softassign and EM-ICP on GPU", Proc. of

UPDAS2010; The 2nd Workshop on Ultra Performance and Dependable Acceleration Systems, 2010. !  Toru Tamaki, Miho Abe, Bisser Raytchev, Kazufumi Kaneda, Marcos Slomp: "CUDA-based implementations of

Softassign and EM-ICP," Demonstration presented at CVPR2010 ; IEEE Conference on Computer Vision and Pattern Recognition, June 15-17, 2010.

!  ³yÔ: �F?G,�-A&J�, SIS/SIP/IPSJ-AVM�z, SIP2009-48, SIS2009-23, Vol.109, No.202, pp.59-64, (2009 09).

■  2D-3D registration !  Toru Tamaki, Yuji Ueno, Shunsuke Tanigawa, Bisser Raytchev, Kazufumi Kaneda: "3D Keypoint Detection by

Embedding 2D Features for 3D-2D Matching", FCV2012, 18th Korea-Japan Joint Workshop on Frontiers of Computer Vision, pp. 62-65, 2012.

!  Yuji Ueno, Baowei Lin, Kouhei Sakai, Toru Tamaki, Bisser Raytchev, Kazufumi Kaneda: "Camera Position Estimation for Detecting 3D Scene Change", FCV2012, 18th Korea-Japan Joint Workshop on Frontiers of Computer Vision, pp. 344-350, 2012. �


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