December 27, 1997
Return to SITE CONTENTS


How Much A Signal Is Like A Standard

2-D Edge Detection: A Feature Extraction
The Work of Reiner Lenz In Alternative Notation

SYNOPSIS OF THE THESIS

Task: "To find out whether the incoming grey pattern is edgelike."
Lenz, J. Opt. Soc. Am. A 6, 827 (1989)

The property "edgelike" means "any state |>". If there exists one value of (say ) where the patterns match, then |u> is edgelike. A formal statement of the task may be written

This is a recipe impossible to implement!
It consists in matching an infinite number of pattern displays (the for each of the -values) to any given unknown display, pu(r,), and seeing whether they are equal.

Lenz shows how this impossible task may be made possible. He recasts the task into an algorithm suitable for practical computation. His statement is

In words: The fourier transform ('weighted spatial average' in Lenz) of the unknown is simply related to the same weighted spatial average of a single prototype pattern. We need not match all patterns; just one.

This appears in Lenz, Lctr Nts, p.3, as .

In the language of Dirac bra-kets this same result is

where is the 1-D irreducible representation with symmetry label k for the group member . And

where <k|> is a unitary transformation matrix element connecting the -basis (the scrutiny basis) to the k-basis (irreducible representation basis). In the present case of the group SO(2)

end of synopsis

ANALYSIS

1. The character 'edgelike' has no r dependence. The True/False condition must be true for every r. So we suppress the r.

What follows is true for every r. The distance, r, is a parameter; present but not notated.

2. Idea: = the observer rotation angle can affect (the perceived edge angle) and can affect , the position location angle coordinate. Let mean "rotate the observer by the angle ". The are the observer's group of altered scrutinies. The observer sees basis states |> in the basis <|. What state he perceives depends upon his orientation. As is evident from the figure:


Figure 3. Every member of the pattern set results merely from altered scrutiny.

Thus any basis state |> is generated by the rotation group SO(2) from a single prototype |=0>.

(Aside note: This is the rule for generating the regular representation of a group.)

Together all the |> make up an invariant space with respect to the .

3. How the operator affects the position basis is governed by intrinsic sameness against rotation. What the observer measures at the location is exactly the same as what the rotated observer measures at - so


Figure 4. Illustration of the Principle of Intrinsic Sameness: What is there is independent of the observer. This is a particular case of an induced transformation: <x|G = <G-1x| defines the effect of G in some basis x, of position.

Because (3.1) is valid for any state |> we write

4. The two bases and are both in spaces invariant to the group .


But and are in different Hilbert Spaces!

5. In the Hilbert Space |=> we may build matrix representations of . These consist of all matrices with elements <||>.

These matrix representations of SO(2) may be block diagonalized into irreducible representations labeled by k. An irrep basis state is one that produces block diagonalization so, where is one of the k,

The irreducible representations of SO(2) are known. They are 1-D and given by

The unitary transformation matrix elements relating states |> to states in the basis |k> are derived by applying (3.2) to a state, |k>, and using (5.1) and (5.2).


so

6. To describe the state in its symmetry basis is to compute <k|>. Having <|k> gives us the means to do it.


This expansion is valid because and k mark bases in the same Hilbert Space.

7. In Dirac bra-kets the task of finding "whether the grey pattern is edgelike" can be succinctly put as

This asks whether the unknown state is one of the edgelike ones - in any basis. The original statement, (P), is restricted to the -basis. But if the states are the same in the basis- they must be so in any basis of the same Hilbert space. So we may apply (B) in the symmetry basis <k|.

But, using 2.2 and 5.1,

which, with (K), results in (D) of the Synopsis.


December 1997
Marvin Chester
email: chester@physics.ucla.edu

Return to SITE CONTENTS
9'05-

© m chester 1997 Occidental CA