470 BELL SYSTEM TECHNICAL JOURNAL 



the +m, -w with the -m, -\-n terms (giving the single (m, -n)th com- 

 ponent). This brings equation (7) back to a cosine series, 



nix ny . 



(8) 



00 +00 



E{x,y) = Y. H CLrnn COS 

 m = n= — 00 



when 



Amn = (l/2)a;„„exp (^V^n) 

 and 



A-m-n = (l/2)a„„exp {-icpmn) 



and where a„in is always a real quantity. Each term of this series 

 represents a real, two-dimensional, sinusoidal variation in brightness 

 extending across the image field. The image is built up of a superposi- 

 tion of a series of such waves extending across the field in various 

 directions and having various wave lengths. 



Imagining brightness as a third dimension, we may, as an aid in 

 visualizing the components of an image field, draw separate examples 

 of various components as shown in Fig. 3. It will be noted that any 

 given component (m, n) passes through m periods along any horizontal 

 line in the image field, and through n periods along any vertical line. 

 The slope of the striations with respect to the x-axis is therefore 

 — mb/na (the negative reciprocal of the slope of the line of fastest 

 variation in brightness). For the same values of m and of n, the m, -\-n 

 component and the m, —n component have equal wave lengths but 

 are sloped in opposite directions to the x-axis. If m is zero the crests 

 are parallel to the x-axis; if n is zero they are parallel to the 3'-axis. 

 The component with both m and n zero is a uniform distribution of 

 brightness covering the entire image field. The wave length of a 

 component is 



A complete array of the components, up to m and n equal to 4, is 

 illustrated in Fig. 4, 



As of course is characteristic of the harmonic analysis, the wave 

 lengths and orientations of the components are seen to vary only with 

 the shape and size of the rectangular field, and to be independent of 

 the particular subject in the field. A change of subject, or motion of 

 the subject, merely alters the amplitudes of the components and shifts 

 their phase; but their wave length and inclination with respect to the 

 X-axis remain unchanged. Consequently, for the same rectangular 

 field all subjects appearing in it may be considered as built up from the 

 same set of components. For a "still" subject, the amplitudes and 



