Interative approximation environments for modeling the evolution of an image propagating through a physical medium in restoration and other applications

A method for approximating the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said method comprising:

• (a) representing an image using a data array;
• (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P−1) of said ordered similarity transformation operator (P);
• (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator;
• (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P−1) to produce a first partial result;
• (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array;
• (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator;
• and
• (g) repeating operations (c) and (d) for each of said plurality of values of said power (α).

A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α), said controlling provided by said computer system being accomplished according to operations comprising:

• (a) representing a plurality of images using an individual data array for each of said plurality of images;
• (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P−1) of said ordered similarity transformation operator (P);
• (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to said power (α) to produce a fractional power diagonal operator;
• (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P−1) to produce a first partial result;
• (e) multiplying a data array of one of said plurality of images by said ordered similarity transformation operator (P) to produce a modified data array;
• (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator;
• and
• (g) repeating operations (e) and (f) for each of said plurality of images.

A computer-readable medium containing instructions for controlling a computer system to approximate the evolution of images propagating through a physical medium by calculating a fractional power of a numerical operator, said numerical operator being defined by said physical medium and comprising a diagonalizable numerical linear operator raised to a power (α) having any one of a plurality of values, said controlling provided by said computer system being accomplished according to operations comprising:

• (a) representing an image using a data array;
• (b) representing said numerical operator with a linear operator formed by multiplying an ordered similarity transformation operator (P) by a correspondingly-ordered diagonal operator (Λ), the result of which is multiplied by an approximate inverse (P−1) of said ordered similarity transformation operator (P);
• (c) raising diagonal elements of said correspondingly-ordered diagonal operator (Λ) to one of said plurality of values of said power (α) to produce a fractional power diagonal operator;
• (d) multiplying said fractional power diagonal operator by an approximate inverse of said ordered similarity transformation operator (P−1) to produce a first partial result;
• (e) multiplying said data array by said ordered similarity transformation operator (P) to produce a modified data array;
• (f) multiplying said modified data array by said first partial result to produce said fractional power of said numerical operator;
• and
• (g) repeating operations (c) and (d) for each of said plurality of values of said power (α).