We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is...

θ. This definition is deficient as it depends on a choice of basis. A definition in R is that a rotation is a linear map g with an axis, a line L fixed by g, and on the orthogonal complement L⊥ of L the restriction of g is a two-dimensional rotation. For this to make sense, one must have understood that the two-dimensional definition is independent of basis, and that g does stabilize the orthog...

The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors, matrices, etc) into an orthonormal basis (a set of orthogonal, unit-length vectors, bi or tri dimensional matrices). The process consists of taking each array and then subtracting the projections in common with the previous arrays. This paper introduces an enhanced version of the Gra...

Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. Previously, we have developed a basis of functions generated from gradients of Zern...

Detecting and recognizing face images automatically is a difficult task due to the variability of illumination, presentation angle, face expression and other common problems of machine vision. In this paper, we represent face images as combinations of 2-D Gabor wavelet basis which are non-orthogonal. Genetic Algorithm (GA) is used to find an optimal basis derived from a combination of frequenci...