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tensor

2011. 7. 7. 23:20 Computer Vision
http://en.wikipedia.org/wiki/Tensor

http://mathworld.wolfram.com/Tensor.html
 
kipid@다음: 텐서란? 

Joseph C. Kolecki @NASA: An Introduction to Tensors for Students of Physics and Engineering

Tensors_TM2002211716.pdf





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posted by maetel

pinhole camera model

2010. 6. 14. 22:13 Computer Vision
ref.
Learning OpenCV
Chapter 11: Camera Models and Calibration


Al-Hytham, Book of Optics, 1038

Descartes
Kepler
Galileo
Newton
Hooke
Euler
Fermat
Snell

J. J. O'Connor and E. F. Roberson, "Light through the ages: Ancient Greece to Maxwell," http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Light_1.html

E. Trucco and A. Verri, Introductory Techniques for 3-D Computer Vision, Englewood Cliffs, NJ: Prentice-Hall, 1998.

B. Jaehne, Digital Image Processing, 3rd ed., Berlin: Springer-Verlag, 1995.

B. Jaehne, Practical Handbook on Image Processing for Scientific Applications, Boca Raton, FL: CRC Press, 1997

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, Cambridge, UK: Cambridge University Press, 2006.

D. Forsyth and J. Ponce, Computer Vision: A Modern Approach, Englewood Cliffs, NJ: Prentice-Hall, 2003.

L. G. Shapiro and G. C. Stockman, Computer Vision, Englewood Cliffs, NJ: Prentice-Hall, 2002

G. Xu and Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition, Dordrecht: Kluwer, 1996




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김준식 & 권인소 "동심원 패턴을 이용한 카메라 내부변수 보정 시스템 및 카메라 보정 방법"  (3) 2010.06.11
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posted by maetel

Three-dimensional computer vision: a geometric viewpoint By Olivier Faugeras

2010. 3. 15. 15:56 Computer Vision
Three-dimensional computer vision: a geometric viewpoint 
By Olivier Faugeras

googleBooks
mitpress

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Seong-Woo Park & Yongduek Seo & Ki-Sang Hong <Real-Time Camera Calibration for Virtual Studio>

2010. 2. 10. 15:47 Computer Vision
Seong-Woo Park, Yongduek Seo, Ki-Sang Hong: Real-Time Camera Calibration for Virtual Studio. Real-Time Imaging 6(6): 433-448 (2000)
doi:10.1006/rtim.1999.0199
swPark_2000rtim.pdf

Real-Time Camera Calibration for Virtual Studio

Seong-Woo Park, Yongduek Seo and Ki-Sang Hong1

Dept. of E.E. POSTECH, San 31, Hyojadong, Namku, Pohang, Kyungbuk, 790-784, Korea


Abstract

In this paper, we present an overall algorithm for real-time camera parameter extraction, which is one of the key elements in implementing virtual studio, and we also present a new method for calculating the lens distortion parameter in real time. In a virtual studio, the motion of a virtual camera generating a graphic studio must follow the motion of the real camera in order to generate a realistic video product. This requires the calculation of camera parameters in real-time by analyzing the positions of feature points in the input video. Towards this goal, we first design a special calibration pattern utilizing the concept of cross-ratio, which makes it easy to extract and identify feature points, so that we can calculate the camera parameters from the visible portion of the pattern in real-time. It is important to consider the lens distortion when zoom lenses are used because it causes nonnegligible errors in the computation of the camera parameters. However, the Tsai algorithm, adopted for camera calibration, calculates the lens distortion through nonlinear optimization in triple parameter space, which is inappropriate for our real-time system. Thus, we propose a new linear method by calculating the lens distortion parameter independently, which can be computed fast enough for our real-time application. We implement the whole algorithm using a Pentium PC and Matrox Genesis boards with five processing nodes in order to obtain the processing rate of 30 frames per second, which is the minimum requirement for TV broadcasting. Experimental results show this system can be used practically for realizing a virtual studio.


전자공학회논문지 제36권 S편 제7호, 1999. 7 
가상스튜디오 구현을 위한 실시간 카메라 추적 ( Real-Time Camera Tracking for Virtual Studio )   
박성우 · 서용덕 · 홍기상 저 pp. 90~103 (14 pages)
http://uci.or.kr/G300-j12265837.v36n07p90

서지링크     한국과학기술정보연구원
가상스튜디오의 구현을 위해서 카메라의 움직임을 실시간으로 알아내는 것이 필수적이다. 기존의 가상스튜디어 구현에 사용되는 기계적인 방법을 이용한 카메라의 움직임 추적하는 방법에서 나타나는 단점들을 해결하기 위해 본 논문에서는 카메라로부터 얻어진 영상을 이용해 컴퓨터비전 기술을 응용하여 실시간으로 카메라변수들을 알아내기 위한 전체적인 알고리듬을 제안하고 실제 구현을 위한 시스템의 구성 방법에 대해 다룬다. 본 연구에서는 실시간 카메라변수 추출을 위해 영상에서 특징점을 자동으로 추출하고 인식하기 위한 방법과, 카메라 캘리브레이션 과정에서 렌즈의 왜곡특성 계산에 따른 계산량 문제를 해결하기 위한 방법을 제안한다.

DHJJIU_1999_v36Sn7_90.pdf


Practical ways to calculate camera lens distortion for real-time camera calibration
Pattern Recognition, Volume 34, Issue 6, June 2001, Pages 1199-1206
Seong-Woo Park, Ki-Sang Hong
swPark_2001jpr.pdf




generating virtual studio




Matrox Genesis boards
http://www.matrox.com/imaging/en/support/legacy/

http://en.wikipedia.org/wiki/Virtual_studio
http://en.wikipedia.org/wiki/Chroma_key

camera tracking system : electromechanical / optical
pattern recognition
2D-3D pattern matches
planar pattern


feature extraction -> image-model matching & identification -> camera calibration
: to design the pattern by applying the concept of cross-ratio and to identify the pattern automatically


영상에서 찾아진 특징점을 자동으로 인식하기 위해서는 공간 상의 점들과 영상에 나타난 그것들의 대응점에 대해서 같은 값을 갖는 성질이 필요한데 이것을 기하적 불변량 (Geometric Invariant)이라고 한다. 본 연구에서는 여러 불변량 가운데 cross-ratio를 이용하여 패턴을 제작하고, 영상에서 불변량의 성질을 이용하여 패턴을 자동으로 찾고 인식할 수 있게 하는 방법을 제안한다.


Tsai's algorithm
R. Y. Tsai, A Versatile Camera Calibration Technique for High Accuracy 3-D Maching Vision Metrology Using Off-the-shelf TV Cameras and Lenses. IEEE Journal of Robotics & Automation 3 (1987), pp. 323–344.

direct image mosaic method
Sawhney, H. S. and Kumar, R. 1999. True Multi-Image Alignment and Its Application to Mosaicing and Lens Distortion Correction. IEEE Trans. Pattern Anal. Mach. Intell. 21, 3 (Mar. 1999), 235-243. DOI= http://dx.doi.org/10.1109/34.754589

Lens distortion
Richard Szeliski, Computer Vision: Algorithms and Applications: 2.1.6 Lens distortions & 6.3.5 Radial distortion

radial alignment constraint
"If we presume that the lens has only radial distortion, the direction of a distorted point is the same as the direction of an undistorted point."

cross-ratio  http://en.wikipedia.org/wiki/Cross_ratio
: planar projective geometric invariance
 - "pencil of lines"
http://mathworld.wolfram.com/CrossRatio.html
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node25.html
http://www.cut-the-knot.org/pythagoras/Cross-Ratio.shtml
http://web.science.mq.edu.au/~chris/geometry/
chap04.pdf
pattern identification

 카메라의 움직임을 알아내기 위해서는 공간상에 인식이 가능한 물체가 있어야 한다. 즉, 어느 위치에서 보더라도 영상에 나타난 특징점을 찾을 수 있고, 공간상의 어느 점에 대응되는 점인지를 알 수 있어야 한다.

패턴이 인식 가능하기 위해서는 카메라가 어느 위치, 어느 자세로 보던지 항상 같은 값을 갖는 기하적 불변량 (Geometric Invariant)이 필요하다.

Coelho, C., Heller, A., Mundy, J. L., Forsyth, D. A., and Zisserman, A.1992. An experimental evaluation of projective invariants. In Geometric invariance in Computer Vision, J. L. Mundy and A. Zisserman, Eds. Mit Press Series Of Artificial Intelligence Series. MIT Press, Cambridge, MA, 87-104.


> initial identification process
extracting the pattern in an image: chromakeying -> gradient filtering: a first-order derivative of Gaussian (DoG) -> line fitting: deriving a distorted line (that is actually a curve) equation -> feature point tracking (using intersection filter)


R1x = 0



http://en.wikipedia.org/wiki/Difference_of_Gaussians



real-time camera parameter extraction

이상적인 렌즈의 optical axis가 영상면에 수직이고 변하지 않는다고 할 때, 영상 중심은 카메라의 줌 동작 동안 고정된 값으로 계산된다. (그러나 실제 렌즈의 불완전한 특성 때문에 카메라의 줌 동작 동안 영상 중심 역시 변하게 되는데, 이 변화량은 적용 범위 이내에서 2픽셀 이하이다. 따라서 본 연구에서는 이러한 변화를 무시하고 이상적인 렌즈를 가정하여 줌동작에 의한 영상 중심을 구하게 된다.)

For zoom lenses, the image centers vary as the camera zooms because the zooming operation is executed by a composite combination of several lenses. However, when we examined the location of the image centers, its standard deviation was about 2 pixels; thus we ignored the effect of the image center change.


calculating lens distortion coefficient

Zoom lenses are zoomed by a complicated combination of several lenses so that the effective focal length and distortion coefficient vary during zooming operations.

When using the coplanar pattern with small depth variation, it turns out that focal length and z-translation cannot be separated exactly and reliably even with small noise.

카메라 변수 추출에 있어서 공간상의 특징점들이 모두 하나의 평면상에 존재할 때는 초점거리와 z 방향으로의 이동이 상호 연관 (coupling)되어 계산값의 안정성이 결여되기 쉽다.


collinearity

Collinearity represents a property when the line in the world coordinate is also shown as a line in the image. This property is not preserved when the lens has a distortion.


Once the lens distortion is calculated, we can execute camera calibration using linear methods.


filtering

가상 스튜디오 구현에 있어서는 시간 지연이 항상 같은 값을 가지게 하는 것이 필수적이므로, 실제 적용에서는 예측 (prediction)이 들어가는 필터링 방법(예를 들면, Kalman filter)은 사용할 수가 없었다.

averaging filter 평균 필터








Orad  http://www.orad.co.il

Evans & Sutherland http://www.es.com









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Jules Bloomenthal & Jon Rokne "Homogeneous Coordinates"

2009. 8. 20. 23:33 Computer Vision
Jules Bloomenthal and Jon Rokne (Department of Computer Science, The University of Calgary)
Homogeneous Coordinates
http://portal.acm.org/citation.cfm?id=205426
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.3319
http://www.springerlink.com/content/p356406661505622/
homog-coords.pdf


Introduction

http://en.wikipedia.org/wiki/Barycentric_coordinates_%28astronomy%29
무게중심좌표 http://100.naver.com/100.nhn?docid=64206

http://mathworld.wolfram.com/BarycentricCoordinates.html
Barycentric coordinates are homogeneous, so
(t_1,t_2,t_3)=(mut_1,mut_2,mut_3)
(1)

for mu!=0.


http://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates
(d:m) are the Plücker coordinates of L.
Although neither d nor m alone is sufficient to determine L, together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between x and y. That is, the coordinates
(d:m) = (d1:d2:d3:m1:m2:m3)
may be considered homogeneous coordinates for L, in the sense that all pairs (λd:λm), for λ ≠ 0, can be produced by points on L and only L, and any such pair determines a unique line so long as d is not zero and d•m = 0.

http://en.wikipedia.org/wiki/Ray_tracing_%28graphics%29
technique for generating an image by tracing the path of light through pixels in an image plane

http://en.wikipedia.org/wiki/Projective_space

http://mathworld.wolfram.com/GrassmannCoordinates.html

Plücker embedding = Grassmann coordinates
http://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding


Projective Plane

http://en.wikipedia.org/wiki/Point_at_infinity
http://en.wikipedia.org/wiki/Hyperplane_at_infinity
The real projective plane  By Harold Scott Macdonald Coxeter

http://en.wikipedia.org/wiki/Projective_plane
"A projectivity is any conceivable invertible linear transform of homogeneous coordinates."

A projective transformation in P2 space is an invertible mapping of points in P2 to points in P2 that maps lines to lines. A P2 projectivity has the equation

x′ = Hx
where H is an invertible 3 × 3 matrix.

http://mathworld.wolfram.com/ProjectivePlane.html

http://vision.stanford.edu/~birch/projective/

ideal line = line at infinity
http://en.wikipedia.org/wiki/Line_at_infinity

http://en.wikipedia.org/wiki/Linear_perspective
http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm

Quadrilateral Perspective, drawing in perspective, parallel, oblique and integrated perspectives
by Yvonne Tessuto Tavares

AERIAL PARALLEL PERSPECTIVE (2 VANISHING POINTS)


PARALLEL PERSPECTIVE - AERIAL VIEW GEOMETRIC STRUCTURE


PARALLEL PERSPECTIVE - AERIAL VIEW WITH A VIEW FROM BOTTOM TO TOP




The mapping from planes and lines through the center of projection to lines and points on the projective plane is the transformation of the usual Euclidean space into projective space.

A projective space is not a vector space in the same manner as the Euclidean space.

Riesenfeld, R. F. 1981. Homogeneous Coordinates and Projective Planes in Computer Graphics. IEEE Comput. Graph. Appl. 1, 1 (Jan. 1981), 50-55. DOI= http://dx.doi.org/10.1109/MCG.1981.1673814
Homogeneous Coordinates and Projective Planes in Computer G.pdf

Unification of the translation, scaling and rotation of geometric objects
: "All affine transformations are matrix multiplication."


Affine Transformations

Homogeneous Lines

Conics
"matrix of the second degree curve"
http://en.wikipedia.org/wiki/Ellipse
http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
http://en.wikipedia.org/wiki/Conic_section

Rational Curves
: extended parametric curve (control points + basis functions)


The use of homogeneous coordinates not only produces polynomials of fixed degree, it also provides a method for consistent manipulation of the Euclidean space.

Perspective Projection
perspective divide

A loss of depth information is due to the linear dependence of the third and fourth columns of the matrix.

Introducing a second non-zero term, e.g. -1, into the third column does not affect x’ and y’, but z’ becomes D-D/z. The purpose of this additional term is to compress the Euclidean space z Î [1, ¥] to z’ Î [0, D].
perspective-projection


Perspective Space

"The homogeneous perspective transformation transforms Euclidean points to new homogeneous points."

perspective space (of the transformed points) vs. object space

The perspective matrix is invertible whereas the perspective-projection matrix is singular.

http://en.wikipedia.org/wiki/Viewing_frustum
http://en.wikipedia.org/wiki/Frustum




Perspective Transformation

Homogeneous Clipping





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Richard Hartley <In defense of the eight-point algorithm>

2009. 8. 19. 00:35 Computer Vision
In defense of the eight-point algorithm
Hartley, R.I.  
Corp. Res. & Dev., Gen. Electr. Co., Schenectady, NY, USA;
This paper appears in: Pattern Analysis and Machine Intelligence, IEEE Transactions on
Publication Date: June 1997
Volume: 19 , Issue: 6
On page(s): 580 - 593
fundamental.pdf

In Defense of the Eight-Point Algorithm.pdf





Zhengyou Zhang

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Five-Point algorithm

2009. 8. 18. 21:29 Computer Vision
Nistér, D. 2004. An Efficient Solution to the Five-Point Relative Pose Problem. IEEE Trans. Pattern Anal. Mach. Intell. 26, 6 (Jun. 2004), 756-777. DOI= http://dx.doi.org/10.1109/TPAMI.2004.17
An efficient solution to the five-point relative pose problem
 Nister, D.  
Sarnoff Corp., Princeton, NJ, USA;
This paper appears in: Pattern Analysis and Machine Intelligence, IEEE Transactions on
Publication Date: June 2004
Volume: 26,  Issue: 6
On page(s): 756-770 An Efficient Solution to the Five-Point Relative Pose Problem
nisterd_efficient.pdf

David Nist´er
Sarnoff Corporation
Center for Visualization and Virtual Environments and the Computer Science Department of University of Kentucky


H. Stewenius, C. Engels, and D. Niste. Recent developments on direct relative orientation.
ISPRS Journal of Photogrammetry and Remote Sensing, 60:284-294, June 2006.
stewenius_engels_nister_5pt_isprs.pdf


Calibrated Fivepoint Solver
http://www.vis.uky.edu/~dnister/Executables/RelativeOrientation/


Dhruv Batra, Bart Nabbe, and Martial Hebert. An Alternative Formulation for the Five Point Relative Pose Problem. IEEE Workshop on Motion and Video Computing 2007 (WMVC '07).
http://www.ece.cmu.edu/~dbatra/research/fivept/fivept.html
batra-5pt.pdf


Five-Point Motion Estimation Made Easy
Hongdong Li and Richard Hartley (RSISE, The Australian National University. Canberra Research Labs, National ICT Australia.)
new5pt_cameraREady_ver_1.pdf





preview


SfM = structure from motion
http://en.wikipedia.org/wiki/Structure_from_motion

eight-point algorithm
http://en.wikipedia.org/wiki/Eight-point_algorithm
algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding image points

Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press.
http://www.robots.ox.ac.uk/~vgg/hzbook/
ch.8 - ch.10

Richard I. Hartley (June 1997). "In Defense of the Eight-Point Algorithm". IEEE Transaction on Pattern Recognition and Machine Intelligence 19 (6): 580–593. doi:10.1109/34.601246.

Richard Szeliski, Microsoft Research
Computer Vision: Algorithms and Applications
ch.7 Structure from Motion

Epipolar geometry - Fundamental matrix


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Superformula

2007. 7. 20. 12:29 Method/Nature
Superformula
: a generalization of the superellipse and was first proposed by Johan Gielis


Wolfram MathWorld: Superellipse:
A superellipse is a curve with Cartesian equation
|x/a|^r+|y/b|^r==1,
(1)

first discussed in 1818 by Lamé. A superellipse may be described parametrically by

x=acos^(2/r)t
(2)
y=bsin^(2/r)t.
(3)


Math Trek: A Geometric Superformula
Ivars Peterson, Science News Online, May 3, 2003

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