Photogrammetric Computer Vision at Universität Weimar | Flashcards & Summaries

Lernmaterialien für photogrammetric computer vision an der Universität Weimar

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TESTE DEIN WISSEN

What are major tasks for a 3D object reconstruction?

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- Perspective 

-Motion

-Shading

-Silhouette and color consistency

- Texture

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What does one understand by DLT?

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Direct Linear transformation

-originally limited to transformation betweend 3D space and 2D image coordinates

- now generalized on many problems of projective geometry

---> mathematical approach to problems in photogrammetry

For/Contra

+ direct unique solution (without iteration(initial value)

-minimization of algebraic error without geometrical/statistical meaning


Use DLT as initial value for non-linear optimization with meaningful geometrical cost function

Lösung ausblenden
TESTE DEIN WISSEN

What is a homogeneous equation system and what must be considered for a stable solution?

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TESTE DEIN WISSEN








In general, the equation AX=B representing a system of equations is called homogeneous if B is the nx1 (column) vector of zeros. Otherwise, the equation is called nonhomogeneous.



Linear homogeneous equation systems

Ax = 0

A = Design matrix (wth obeservations(measurements))

x wanted solution vector

unique solution , if at least as many independent homogeneous equations are presented ass unknown parameters

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TESTE DEIN WISSEN

Compare the methods for parameter estimation 

a. linear (in-)homogeneous equation systems,

b. robust parameter estimation

c. non-linear optimization

concerning the requirements, complexity and accuracy.

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TESTE DEIN WISSEN

Linear (in-)homogeneous equation systems:

Advantages: no initialization required, 1 iteration

Disadvantages: sensitive to impercise measurements and outliers, usually no minimal parametrization, constraints enforced afterwards


Robust parameter estimation:

- M-estimators

-RANSAC --> Robust fit of model to data set S, which contains outliers


non-linear optimization:

- often slower than  DLT

- requires initialization --> typically use linear solution or sample parameter space

- no guaranteed convergence, local minima

- stopping criterion required

Many optimization algorithms exist, minimize geometric error function by adjusting the 9 matrix elements in Homography

Lösung ausblenden
TESTE DEIN WISSEN

Specify for each case a procedure for the computation of the parameter estimation. (linear (in-)homogeneous equation systems, robust parameter estimation, non-linear optimization)


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TESTE DEIN WISSEN

Linear (in-)homogeneous equation systems --> direct linear estimation

Robust parameter estimation --> RANSAC

Non-linear optimization --> Least  squares estimation

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TESTE DEIN WISSEN

How many degrees of freedom have these spaces?

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Euclidean space: 6 DoF --> rotation :3 + translation(x,y,z): 3

Metric space: 7 DoF --> rotation: 3 + translation(x,y): 3 + global scaling: 1

Affine space: 12 DoF --> Metric (rotation, translation, scaling): 7 + individual scaling: 3 + skew factor: 2

Projective space: 15 DoF --> Affine: 12 + persepctive projection --> 4x4 matrix with 15 DoF and last element is homogeneous

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TESTE DEIN WISSEN

What are homogeneous coordinates and what are the advantages?

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TESTE DEIN WISSEN

They are a system of coordinates used in  the projective  space.

Allow us to describe Eculidean objects in  the projective space

Allow us to describe points at infinity as well as calculate intersection of two parallels


- Representation of Euclidean by adding component and a free scaling to mathematical representation

- Redundant (überflüssige) representation of vectors and matrices

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TESTE DEIN WISSEN

What does incidence mean?

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Realtion between points, lines

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What is a homography and which property has it?

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TESTE DEIN WISSEN

Any two images of the same planar surface in space related by a homography:

- invertible mapping h from P^2 to P^2 exactly if three points x1, x2, x3 of a straight line are located again on a straight line after mapping h(x1), h(x2), h(x3)


h(x) = Hx


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TESTE DEIN WISSEN

How many degrees of freedom does a 

a. planar homography

b. spatial homography

c. projective transformation from space into a plane have?

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TESTE DEIN WISSEN

planar homography: number of unknown parameter: n*(n+2) --> 8 DoF for plane

spatial homography: 3*(3+2)= 15 DoF

projective transformation from space into a plane: 3x4 projection matrix P (11 DoF)

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TESTE DEIN WISSEN

How many points are needed in each case (planar, spatial, projectie transformation from space into a plane) for the computation?

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TESTE DEIN WISSEN

planar homography: n*(n+2) --> n+2=2+2=4 --> at least 4 points

spatial homography: n*(n+2) --> n+2=3+2=5 --> at least 5 points

projective transformation: we need 11 equations/information, each object contains 3 components (x,y,z) , we lose depth, so we need 5 1/2 points --> gives us 6 points

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TESTE DEIN WISSEN

What does the term photogrammetry mean?

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- is the acquirement of geometric object information using photographic images 

- the science and tecnology of deducing reliable spatial information from measurements on photographs

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Beispielhafte Karteikarten für deinen photogrammetric computer vision Kurs an der Universität Weimar - von Kommilitonen auf StudySmarter erstellt!

Q:

What are major tasks for a 3D object reconstruction?

A:

- Perspective 

-Motion

-Shading

-Silhouette and color consistency

- Texture

Q:

What does one understand by DLT?

A:

Direct Linear transformation

-originally limited to transformation betweend 3D space and 2D image coordinates

- now generalized on many problems of projective geometry

---> mathematical approach to problems in photogrammetry

For/Contra

+ direct unique solution (without iteration(initial value)

-minimization of algebraic error without geometrical/statistical meaning


Use DLT as initial value for non-linear optimization with meaningful geometrical cost function

Q:

What is a homogeneous equation system and what must be considered for a stable solution?

A:








In general, the equation AX=B representing a system of equations is called homogeneous if B is the nx1 (column) vector of zeros. Otherwise, the equation is called nonhomogeneous.



Linear homogeneous equation systems

Ax = 0

A = Design matrix (wth obeservations(measurements))

x wanted solution vector

unique solution , if at least as many independent homogeneous equations are presented ass unknown parameters

Q:

Compare the methods for parameter estimation 

a. linear (in-)homogeneous equation systems,

b. robust parameter estimation

c. non-linear optimization

concerning the requirements, complexity and accuracy.

A:

Linear (in-)homogeneous equation systems:

Advantages: no initialization required, 1 iteration

Disadvantages: sensitive to impercise measurements and outliers, usually no minimal parametrization, constraints enforced afterwards


Robust parameter estimation:

- M-estimators

-RANSAC --> Robust fit of model to data set S, which contains outliers


non-linear optimization:

- often slower than  DLT

- requires initialization --> typically use linear solution or sample parameter space

- no guaranteed convergence, local minima

- stopping criterion required

Many optimization algorithms exist, minimize geometric error function by adjusting the 9 matrix elements in Homography

Q:

Specify for each case a procedure for the computation of the parameter estimation. (linear (in-)homogeneous equation systems, robust parameter estimation, non-linear optimization)


A:

Linear (in-)homogeneous equation systems --> direct linear estimation

Robust parameter estimation --> RANSAC

Non-linear optimization --> Least  squares estimation

Mehr Karteikarten anzeigen
Q:

How many degrees of freedom have these spaces?

A:

Euclidean space: 6 DoF --> rotation :3 + translation(x,y,z): 3

Metric space: 7 DoF --> rotation: 3 + translation(x,y): 3 + global scaling: 1

Affine space: 12 DoF --> Metric (rotation, translation, scaling): 7 + individual scaling: 3 + skew factor: 2

Projective space: 15 DoF --> Affine: 12 + persepctive projection --> 4x4 matrix with 15 DoF and last element is homogeneous

Q:

What are homogeneous coordinates and what are the advantages?

A:

They are a system of coordinates used in  the projective  space.

Allow us to describe Eculidean objects in  the projective space

Allow us to describe points at infinity as well as calculate intersection of two parallels


- Representation of Euclidean by adding component and a free scaling to mathematical representation

- Redundant (überflüssige) representation of vectors and matrices

Q:

What does incidence mean?

A:

Realtion between points, lines

Q:

What is a homography and which property has it?

A:

Any two images of the same planar surface in space related by a homography:

- invertible mapping h from P^2 to P^2 exactly if three points x1, x2, x3 of a straight line are located again on a straight line after mapping h(x1), h(x2), h(x3)


h(x) = Hx


Q:

How many degrees of freedom does a 

a. planar homography

b. spatial homography

c. projective transformation from space into a plane have?

A:

planar homography: number of unknown parameter: n*(n+2) --> 8 DoF for plane

spatial homography: 3*(3+2)= 15 DoF

projective transformation from space into a plane: 3x4 projection matrix P (11 DoF)

Q:

How many points are needed in each case (planar, spatial, projectie transformation from space into a plane) for the computation?

A:

planar homography: n*(n+2) --> n+2=2+2=4 --> at least 4 points

spatial homography: n*(n+2) --> n+2=3+2=5 --> at least 5 points

projective transformation: we need 11 equations/information, each object contains 3 components (x,y,z) , we lose depth, so we need 5 1/2 points --> gives us 6 points

Q:

What does the term photogrammetry mean?

A:

- is the acquirement of geometric object information using photographic images 

- the science and tecnology of deducing reliable spatial information from measurements on photographs

photogrammetric computer vision

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