Patent ID: 8504329

Claim:
A method of determining a set of orthonormal polynomials F over a domain Σ corresponding to a noncircular pupil of an eye of a patient for use in determining a vision treatment for the patient, the method using a processor configured to perform the steps of: selecting a complete set of polynomials P as a basis for the orthonormal polynomials F; determining a set of sub-areas of the noncircular pupil; formulating a relationship between x and y within each of the sub-areas such that the set of polynomials P is integratable within each sub-area; calculating a conversion matrix M comprising an inner product of the complete set of polynomials P over the domain Σ; obtaining the set of orthonormal polynomials F based on the conversion matrix M and the complete set of polynomials P, wherein the set of orthonormal polynomials characterize ocular aberrations of the eye of the patient; and using a processing hardware having a non-transitory tangible medium embodying machine-readable code associated therewith to determine the vision treatment for the patient based on the set of orthonormal polynomials F, the vision treatment comprising a spectacle lens shape, an intraocular lens shape, a contact lens shape, a corneal ring implant shape, a corneal inlay shape, a corneal onlay shape, a corneal implant shape, or a laser treatment shape, wherein the domain Σ has a shape selected from the group consisting of a hexagon, an ellipse, an annulus, a rectangle, and a square, such that: when the domain Σ has the hexagon shape, the set of orthonormal polynomials F includes a polynomial term of 1, a polynomial term of 2 ⁢ 6 5 ⁢ ρcosθ , a polynomial term of 2 ⁢ 6 5 ⁢ ρsinθ , and polynomial term of 5 43 ⁢ ( 12 ⁢ ρ 2 - 5 ) , when the domain Σ has the ellipse shape, the set of orthonormal polynomials F includes a polynomial term of 1, a polynomial term of 2 ρ cos θ, a polynomial term of 2 e ⁢ ρsinθ , and polynomial term of 3 3 - 2 ⁢ ⅇ 2 + 3 ⁢ ⅇ 4 ⁡ [ 4 ⁢ ρ 2 - ( 1 + ⅇ 2 ) ] , when the domain Σ has the annulus shape, the set of orthonormal polynomials F includes a polynomial term of 1, a polynomial term of 2 1 + ɛ 2 ⁢ 2 ⁢ ρcosθ , a polynomial term of 2 1 + ɛ 2 ⁢ 2 ⁢ ρsinθ , and polynomial term of 3 1 - ɛ 2 ⁡ [ 2 ⁢ ρ 2 - ( 1 + e 2 ) ] , when the domain Σ has the rectangle shape, the set of orthonormal polynomials F includes a polynomial term of 1, a polynomial term of 3 a ⁢ ρcosθ , a polynomial term of 3 1 - a 2 ⁢ ρsinθ , and polynomial term of 1 2 ⁢ 5 1 - 2 ⁢ a 2 + 2 ⁢ a 4 ⁢ ( 3 ⁢ ρ 2 - 1 ) , and when the domain Σ has the square shape, the set of orthonormal polynomials F includes a polynomial term of 1, a polynomial term of √{square root over (6)}ρ cos θ, a polynomial term of √{square root over (6)}ρ sin θ, and polynomial term of 5 2 ⁢ ( 3 ⁢ ρ 2 - 1 ) , wherein ρ and Θ are polar coordinates, wherein e and a are aspect ratios, and wherein ε is an obscuration ratio.