| US 7,454,084 B2 | ||
| Method of generating matrix factors for a finite-dimensional linear transform | ||
| Vance Faber, Carnation, Wash. (US); and Randall L. Dougherty, Seattle, Wash. (US) | ||
| Assigned to LizardTech, Inc., Seattle, Wash. (US) | ||
| Filed on May 14, 2007, as Appl. No. 11/748,464. | ||
| Application 11/748464 is a division of application No. 10/006999, filed on Dec. 03, 2001, granted, now 7,218,789. | ||
| Claims priority of provisional application 60/250829, filed on Dec. 01, 2000. | ||
| Claims priority of provisional application 60/250850, filed on Dec. 01, 2000. | ||
| Prior Publication US 2007/0211952 A1, Sep. 13, 2007 | ||
| Int. Cl. G06K 9/36 (2006.01); G06K 9/46 (2006.01) | ||
| U.S. Cl. 382—276 [382/244] | 14 Claims |
| 1. A method of transforming a digital image signal using a factored linear transformation matrix, wherein the method includes
generating matrix factors for a finite-dimensional linear transform using a computer, each matrix factor represented by a
symbol, the finite-dimensional linear transform represented by data values stored in a linear transformation matrix A having a nonzero determinant, the method comprising:
applying a first LU-decomposition to the linear transformation matrix A;
generating four matrices from the LU-decomposition of the linear transformation matrix A, the four matrices represented by the symbols Π, Π2, L, and DŨ and satisfying the relationship ΠA Π2=LDŨ, the symbol Π representing a first permutation matrix, the symbol Π2 representing a second permutation matrix, the symbol L representing a lower triangular matrix having a unit diagonal, and
the symbol DŨ representing a first upper triangular matrix;
generating a third matrix represented by the symbol  from the linear transformation matrix A, the third matrix having a plurality
of rows and a determinant of 1;
computing a signed permutation matrix Π from the linear transformation matrix A and the third matrix  such that A=ΠÂ;
generating a permuted linear transformation matrix represented by the symbol A′ from the linear transformation matrix, the
permuted linear transformation A′ having a determinant of 1;
computing a second upper triangular matrix represented by the symbol U1 from the permuted linear transformation matrix A′ and the third matrix Â, the second upper triangular matrix U1 having a plurality of rows, all diagonal entries equal to 1, and all entries below a first row equal to 0, the second upper
triangular matrix U1 satisfying the relationship Â̂=U1A′;
factoring the permuted linear transformation A′ matrix into a product including a lower triangular matrix and an upper triangular matrix, the lower triangular matrix and
the upper triangular matrix each having a unit diagonal, the lower triangular matrix represented by the symbol L and the upper
triangular matrix represented by the symbol U;
generating the matrix factors for the linear transformation matrix A, the matrix factors including at least the lower triangular
matrix L, the upper triangular matrix U, the second upper triangular matrix U1, and the signed permutation matrix Π, the linear transformation matrix A expressed as a product of the matrix factors; and
transforming the digital image signal by applying the linear transform to the digital signal using the matrix factors generated
by the computer.
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