By André Marion (auth.)
I. The prior. the current . . . and the longer term it really is attainable to take the view that ever because it started, the "ancient" department of physics referred to as Optics has been involved in procedure ing pictures. yet because the Nineteen-Thirties more and more shut ties were forming among Optics, which until eventually then have been principally in accordance with tools, and the sciences of communique and infor mation coming up out of arithmetic and electronics. Such advancements stick to evidently, considering communique platforms and image-forming structures are all designed to obtain or transmit info. additional extra an analogous mathematical varieties are used for describing the beha viour of electric and optical platforms. it's a query of platforms idea, relatively linear structures, and of Fourier's research equipment, which jointly represent an immense a part of sign idea. when it comes to verbal exchange platforms wearing indications of an electric nature, details is time-related or temporal. Transmitted signs are one-dimensional and capabilities of a unmarried variable, time t. in relation to optical platforms info is spatial in nature. signs are distributions of sunshine depth in house. ordinarily they're handled as two-dimensional indications, being features of 2 spatia! variables written as x and y. within the early Fifties the way in which ahead turned clearer nonetheless whilst a few scientists on the Institut d'Optique in Paris all started utilizing optical filtering thoughts in coherent mild in an effort to improve the standard of photographs.
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R .......... : :. I I ;: , I : I : : I o 8 2 8 : ~ 28 t "'-_ ... :' \ f............ 28\ ,( ~ : a : '\ . , . ,. 9 - a) Square-wave signals with a height of unity and increasing widths a, 2a, 4a, b) the corresponding Fourier transforms, each of unit area. We can show this in the case of a "gate" signal. a) let us assume that the gate width increases from a to 2a, then to 4a, in general terms to 2ka. b) becomes narrower and higher. In other respects, the surface area of this spectrum remains constant because it is equal to: 26 An Introduction to Image Processing This represents the value assumed by the integral (II), when we choose t = 0, or in other words s(O) = I.
We then have s(x') = ~. To obtain the expression for s(x) we simply reverse the roles of the variables x and x' in equation (I). This gives 53 Scanning of an image by an aperture J +00 s(x) = (2) s(x) = e(x') o(x' -x) dx' -00 @ " Ix) 1 ____ J. 2 - Convolution e(x) ... o(-x); a) signal e(x), b) signal o(x), c) shifted signal o(x-x') for x' = { d) product e(x) o(x-x'): the surface area of this product J +OO represents sex') = -00 e(x)o(x-x')dx which is the value of the output signal for x = x'(= ~), e) output signals(x) = e(x) ...
I6 - Periodic square-wave signal; a) time structure, b) Fourier transform: apart from the zero frequency line representing mean signal level ~,the spectrum consists of the fundamental and odd harmonics only. s (x, Y) =8 (x, y) s (x, y) =0 (x - xo, y- Yo) Yo o )o~-+-,-:o-·- - y y Xo ... ___ ..... 17 - Two-dimensional Dirac signal c5(x,y) representing a point of light at the origin. 19 - "Comb" signal on Ox axis. 20. - "Blade" signal on Ox axis: 6(y) representing a line of light. An 8 ~- i) and, according to (9): n=-oo An = 1 T I tO+T .