A new method for edge detection from N-dimensional digital image
Hypersurface fitting to picutre function and operators
Let R be a hyperrectangular region and x=(x
1,…..x
n) be a point in n-dimensional space, We can generally choose the coordinate system by putting the center point p0 of the region R at the origin. Let f(x) be the digital picture function over R, and f' (x) be an estimation of f(x). The f'(x) can be defined as:
Where {s
i (x) 0 < i < N} are a set of n-dimensional basis functions defined over the R and {a} are a set of coefficients. The total squared estimation error e
2 is:
Using the orthogonal properties of the basis functions, the coefficients {a} that minimize the e
2 can be obtained as:
Let the picture function f(x) can be written:
where n(x) is noise term and is assumed to be independent Gaucian from pixel with zero mean and variance
s2 . From (II-3) and (II-4) we have
and the conclusions are:
- the estimated ai are unbaised , normally distributive and independent.
Let the set of n-dimensional basis functions {Si(X) 0 < i < N} can be constructed using one dimensional discrete 2nd-class Legendre orthogonal polynomials Pil(x) as follows:
P10 (xi) = 1,
P11 (xi) = xi,
P12= (x1) = x2i - m12 / m10
P13 (xi) = x3i - xim14 / m12 ..............(II-8)
where m12 = Sxki is the kth moment of xi over the domain Xi.
From the equations (II-3) and (II-8) , the coefficients a 's, that minimize the sum of squares for errors, are given in the following:
If the size of R is defined as 3x3x3, the (II-9) can be written as the format of convolutions of the f(x1, x2, x3) with following operators:
From (II-10) the operators can be divided into two groups which are named directional gradient opertators (Prewitt) and directional Laplacian operators (Pratt and Hall). If the size of R is changed, the relationship between the coefficients and the operators can be generalized similarly.
When the coefficients are calculated the e2 and F can be
derived . If F is large in a given confidence, then a hypotheses,
H: {ai}=0, is rejected. It means that the coefficients which represent the gradients and Laplacians are significant. In other words, the significance of the edge information at the currently being studied point can be detected. Practically the R is a sliding window. So the convolution acts on all of the pixels.
