ROTATION MATRIX SAMPLING SCHEME FOR MULTIDIMENSIONAL PROBABILITY DISTRIBUTION TRANSFER

This paper address the problem of rotation matrix sampling used for multidimensional probability distribution transfer. The distribution transfer has many applications in remote sensing and image processing such as color adjustment for image mosaicing, image classification, and change detection. The sampling begins with generating a set of random orthogonal matrix samples by Householder transformation technique. The advantage of using the Householder transformation for generating the set of orthogonal matrices is the uniform distribution of the orthogonal matrix samples. The obtained orthogonal matrices are then converted to proper rotation matrices. The performance of using the proposed rotation matrix sampling scheme was tested against the uniform rotation angle sampling. The applications of the proposed method were also demonstrated using two applications i.e., image to image probability distribution transfer and data Gaussianization.

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Orthogonal Matrices in Four-Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1949-006-3 ◽

1949 ◽

Vol 1 (1) ◽

pp. 69-72

Author(s):

C. C. MacDuffee

Keyword(s):

Orthogonal Transformation ◽

Orthogonal Matrix ◽

Orthogonal Matrices ◽

Image Position ◽

Proper Orthogonal ◽

The Relationship

Every proper orthogonal matrix A can be writtenwhere Q is a skew matrix [6], and conversely every such matrix A is orthogonal. It is also known that every proper orthogonal transformation in real Euclidean four-space may be characterized in term of quaternions [1, 3] by the equationdetermines with the origin a vector having the coordinates (XQ, XI, x2, x3). The relationship between these two representations was clearly shown by Murnaghan [5].

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Statistical Signal Processing by Using the Higher-Order Correlation between Sound and Vibration and Its Application to Fault Detection of Rotational Machine

Advances in Acoustics and Vibration ◽

10.1155/2008/828562 ◽

2008 ◽

Vol 2008 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Hisako Masuike ◽

Akira Ikuta

Keyword(s):

Signal Processing ◽

Probability Distribution ◽

Time Domain ◽

Statistical Signal Processing ◽

Higher Order ◽

Orthogonal Expansion ◽

Mutual Relationship ◽

Diagnosis Method ◽

Correlation Information ◽

Multidimensional Probability

In this study, a stochastic diagnosis method based on the changing information of not only a linear correlation but also a higher-order nonlinear correlation is proposed in a form suitable for online signal processing in time domain by using a personal computer, especially in order to find minutely the mutual relationship between sound and vibration emitted from rotational machines. More specifically, a conditional probability hierarchically reflecting various types of correlation information is theoretically derived by introducing an expression on the multidimensional probability distribution in orthogonal expansion series form. The effectiveness of the proposed theory is experimentally confirmed by applying it to the observed data emitted from a rotational machine driven by an electric motor.

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Probabilistic Nodes Combination

International Journal of Organizational and Collective Intelligence ◽

10.4018/ijoci.2012070102 ◽

2012 ◽

Vol 3 (3) ◽

pp. 22-35

Author(s):

Dariusz Jakóbczak

Keyword(s):

Probability Distribution ◽

Orthogonal Matrix ◽

Distribution Functions ◽

Two Dimensional ◽

Curve Reconstruction ◽

Probability Distribution Functions ◽

Key Points ◽

The Family ◽

Curve Interpolation ◽

Inverse Functions

Mathematics and computer science are interested in methods of 2D curve interpolation and extrapolation using the set of key points (knots or nodes). Proposed method, called by author Probabilistic Nodes Combination (PNC), is such a method. This novel PNC method is introduced in the case of Hurwitz- Radon Matrices (MHR). MHR method is based on the family of Hurwitz-Radon (HR) matrices which possess columns composed of orthogonal vectors. Two-dimensional curve is modeled and interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions. It is shown how to build the orthogonal matrix operator and how to use it in a process of curve reconstruction.

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Sign patterns of J-orthogonal matrices

Special Matrices ◽

10.1515/spma-2017-0016 ◽

2017 ◽

Vol 5 (1) ◽

pp. 225-241

Author(s):

Frank J. Hall ◽

Zhongshan Li ◽

Caroline T. Parnass ◽

Miroslav Rozložník

Keyword(s):

Orthogonal Matrix ◽

Orthogonal Matrices ◽

Sign Patterns ◽

Exchange Operator ◽

Czechoslovak Math ◽

Siam Review ◽

G Matrices ◽

Block Diagonal

Abstract This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.

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A generating function for averages over the orthogonal group

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1955.0092 ◽

1955 ◽

Vol 229 (1178) ◽

pp. 367-375 ◽

Cited By ~ 10

Keyword(s):

Invariant Measure ◽

Generating Function ◽

Orthogonal Group ◽

Wishart Distribution ◽

Orthogonal Matrix ◽

Orthogonal Matrices

The integral, ∫ H exp ( tr H ′ X ) d V ( H ) , over the group H of orthogonal matrices with respect to the invariant measure V ( H ), is obtained. X is an n × n matrix, and H an n × n orthogonal matrix. Various applications of it are discussed, namely, to the non-central Wishart distribution and as a generating function for the integrals and averages of polynomials over the orthogonal group.

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Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2018-0037 ◽

2019 ◽

Vol 10 (4) ◽

pp. 467-492 ◽

Cited By ~ 1

Author(s):

Mario Kieburg ◽

Peter J. Forrester ◽

Jesper R. Ipsen

Keyword(s):

Probability Distribution ◽

Random Matrices ◽

Singular Values ◽

Symmetric Matrices ◽

Orthogonal Matrices ◽

Unitary Matrices ◽

Correlation Kernel ◽

Multiplicative Convolution ◽

Special Case ◽

Double Contour

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

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Cospectral Graphs and Regular Orthogonal Matrices of Level 2

The Electronic Journal of Combinatorics ◽

10.37236/2383 ◽

2012 ◽

Vol 19 (3) ◽

Cited By ~ 4

Author(s):

Aida Abiad ◽

Willem H Haemers

Keyword(s):

Adjacency Matrix ◽

Discrete Math ◽

Orthogonal Matrix ◽

Orthogonal Matrices ◽

Switching Operation ◽

Cospectral Graphs ◽

New Methods ◽

Almost All ◽

Isomorphic Graphs ◽

Level 2

For a graph $\Gamma$ with adjacency matrix $A$, we consider a switching operation that takes $\Gamma$ into a graph $\Gamma'$ with adjacency matrix $A'$, defined by $A'=Q^\top A Q$, where $Q$ is a regular orthogonal matrix of level $2$ (that is, $Q^\top Q=I$, $Q$1 $=$ 1, $2Q$ is integral, and $Q$ is not a permutation matrix). If such an operation exists, and $\Gamma$ is nonisomorphic with $\Gamma'$, then we say that $\Gamma'$ is semi-isomorphic with $\Gamma$. Semi-isomorphic graphs are $\mathbb {R}$-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [On the asymptotic behavior of graphs determined by their generalized spectra, Discrete Math. 310 (2010)] expect that almost all pairs of nonisomorphic $\mathbb {R}$-cospectral graphs are semi-isomorphic.Regular orthogonal matrices of level $2$ have been classified. By use of this classification we work out the requirements for this switching operation to work in case $Q$ has one nontrivial indecomposable block of size $4$, $6$, $7$ or $8$. Size $4$ corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of $\mathbb {R}$-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are $\mathbb {R}$-cospectral to another graph, only 44 are not semi-isomorphic to another graph.

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