The Hermite transform (Martens 1990a, b) is a special case of polynomial transform, which is a technique of signal decomposition. The original signal *X*(*x*, *y*), where (*x*, *y*) are the coordinates of the pixels, can be located by multiplying the window function *V* (*x* − *p*, *y* − *q*), by the positions *p*, *q* that conform the sampling lattice *S*, Eq. 1:

$$\begin{aligned} X(x,y) = \frac{1}{W(x,y)} \sum _{p,q \in S} X(x,y)V(x-p,y-q) \end{aligned}$$

(1)

The periodic weighting function is then defined as Eq. 2:

$$\begin{aligned} W(x,y)= \sum _{p,q \in S} V(x-p,y-q) \end{aligned}$$

(2)

The unique condition that allows the polynomial transform to exist is that the weighting function must be different from zero for all coordinates (*x*, *y*).

The local information within every analysis window will then be expanded in terms of an orthogonal polynomial set. The polynomials \(G_{m,n-m} (x,y)\), used to approximate the windowed information are determined by the analysis window function and satisfy the orthogonal condition, Eq. 3:

$$\begin{aligned} \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } V^2(x,y)G_{m,n-m}(x,y)G_{j,i-j}(x,y)dxdy = \delta _{ni} \delta _{mj} \end{aligned}$$

(3)

for \(n,i=0,1,\ldots , \infty ; m=0, \ldots , n\) and \(j=0, \ldots , i.\)

The polynomial coefficients \(X_{m,n-m} (p,q)\) are calculated by convolving the original image *X*(*x*, *y*) with the filter function \(D_{m,n-m} (x,y)=G_{m,n-m}(-x,-y)V^2(-x,-y)\) followed by a sub-sampling in the positions (*p*, *q*) of the sampling lattice *S*: (Eq. 4)

$$\begin{aligned} X_{m,n-m}(p,q) = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } X(x,y) D_{m,n-m} (x-p, y-q) dxdy \end{aligned}$$

(4)

The orthogonal polynomials associated with \(V^2(x)\) are known as Hermite polynomials: (Eq. 5)

$$\begin{aligned} G_{n-m,m}(x,y) = \frac{1}{\sqrt{2^n(n-m)n!}} H_{n-m}\displaystyle {x \atopwithdelims ()\sigma } H_{m}\displaystyle {y \atopwithdelims ()\sigma } \end{aligned}$$

(5)

where \(H_n(x)\) denotes the Hermite polynomial of order *n*.

In the case of the Hermite transform, it is possible to demonstrate that the filter functions \(D_{m,n-m}(x,y)\) correspond to Gaussian derivatives of order *m* in *x* and \({n-m}\) in *y*, in agreement with the Gaussian derivative model of early vision (Young 1985). Moreover, the window function resembles the receptive field profiles of human vision, Eq. 6:

$$\begin{aligned} V(x,y)=\frac{1}{2\pi \sigma ^2} \exp \left( \displaystyle -\frac{x^2+y^2}{2\sigma ^2} \right) \end{aligned}$$

(6)

Besides constituting a good model for the overlapped receptive fields found in physiological experiments, the choice of a Gaussian window can be justified because they minimize the uncertainty principle in the spatial and frequency domains. The recovery process of the original image consists in interpolating the transform coefficients through the proper syntheses filters. This process is known as inverse polynomial transform, and is defined by Eq. 7:

$$\begin{aligned} \hat{X}(x,y)=\sum _{n=0}^{\infty }\sum _{m=0}^{n}\sum _{p,q \in S} X_{m,n-m}(p,q) P_{m,n-m}(x-p, y-q) \end{aligned}$$

(7)

The synthesis filters \(P_{m,n-m} (x,y)\) of order *m* in *x*, and \({n-m}\) in *y*, are defined by Eq. 8:

$$\begin{aligned} P_{m,n-m}(x,y)= \frac{G_{m,n-m}(x,y)V(x,y)}{W(x,y)} \end{aligned}$$

(8)

for \(m=0, \ldots , n\) and \(n=0, \ldots , \infty\).

In a discrete implementation, the Gaussian window function may be approximated by the binomial window function Eq. 9:

$$\begin{aligned} V^{2}(x) = \frac{1}{2^M}\displaystyle {M \atopwithdelims ()x} \end{aligned}$$

(9)

with \(x=0,\ldots , M\). The orthonormal polynomials discrete associated to binomial window are known as the Krawtchouck’s polynomials Eq. 10:

$$\begin{aligned} G_{n}[x]=\frac{1}{\sqrt{\displaystyle {M \atopwithdelims ()n}}} \sum _{k=0}^{n} (-1)^{n-k} \displaystyle {M-x \atopwithdelims ()n-k} \displaystyle {x \atopwithdelims ()k} \end{aligned}$$

(10)

with \(x,n=0, \ldots , M\). For long values of *M*, the binomial window approaches a Gaussian window, Eq. 11:

$$\begin{aligned} \lim _{M \rightarrow 2}\frac{1}{2^M} \left( \displaystyle {M \atopwithdelims ()x+(M/2)} \right) =\frac{1}{\sqrt{\pi }\sqrt{\frac{M}{2}}} \exp \left[ \displaystyle - \left( \frac{x}{\sqrt{\frac{M}{2}}} \right) ^2\right] \end{aligned}$$

(11)

Discrete Hermite transform of length *M* approaches to continuous Hermite transform with standard deviation \(\sigma =\frac{x}{\sqrt{M/2}}\).

Analysing the case where *M* is even, we have that filter functions and pattern functions can be centered at the origin moving the window \(\frac{M}{2}\) points. Thus the filter function are Eq. 12:

$$\begin{aligned} D_{n}(x)=G_n \displaystyle \left( {\frac{M}{2} -x }\right) V^2 \displaystyle \left( {\frac{M}{2} -x}\right) \end{aligned}$$

(12)

with \(x=-(M/2),\ldots , (M/2)\). These functions can be expressed Eq. 13:

$$\begin{aligned} D_n \displaystyle \left( {\frac{M}{2} -x }\right) =\frac{(-1)^n}{2^M\sqrt{\displaystyle {M \atopwithdelims ()n}}} \Delta^n \left[ \displaystyle {M \atopwithdelims ()x} \displaystyle {x \atopwithdelims ()n}\right] \end{aligned}$$

(13)

Calculating *Z* transform of this filter function, Eq. 14:

$$\begin{aligned} d_{n}(Z)= \sum _{x=-M/2}^{M/2} D_{n}(x)z^{-x} \sqrt{\displaystyle {M \atopwithdelims ()n}} \left( {\frac{1-z}{2}}\right) ^{n}\left( {\frac{1+z}{2}}\right) ^{M-n} \end{aligned}$$

(14)

with \(n=0,\ldots , M\). These filters have advantage that they can be performed applying successively a number of simplest filters \(z^{-1}{(1+z)}^2\), \(z^{-1}{(1-z)(1+z)}\), \(z^{-1}{(1-z)}^2\), with their respective kernels [1 2 1], [−1 0 1] and [1 −2 1].

Hermite coefficients are arranged as a set of \(N \times N\) equal-sized subbands; one coarse subband \(X_{0,0}\) representing a Gaussian-weighted image average and detail subbands \(X_{n,m}\) corresponding to higher-order Hermite coefficients, as shown in Fig. 1.