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An object which corresponds to an image detail is modeled as a two- 
dimensional reflectance function R (x,y). Some point (oo) is designated 
as the position of the object. Let Rj (x,y) denote the object positioned 
at the:origin,.so that 
R (xy) = R (x —.x 
à Xsosyo) (2-1) 
o? 0 
Let P, 4 be the value of a pixel in a digital image of the object, and let 
(x, 4755) be the position associated with the center of the pixel. We will 
assume that Py may be expressed as the quantization of a convolution; 
Py 5 = LO, * R) (X145 3: 
= (CT, à * Ra) (x4 5 EnXreïis 7 Yoldr 
= EC zx Yi Yol. 
f Ta Gu Dakgredi;iT 5d (272) 
rr 
for some transformation function Ti 4 where | denotes quantization, * 
denotes two-dimensional convolution and H,, is as discussed below. In 
; : il: : ; 
order to represent P.. in this way, the imaging system must be spatially 
invariant and linear [4]. While these are not generally true assumptions, 
they may be reasonably good approximations locally for a limited range of 
reflectance. The transformation T is subscripted with the pixel index 
(1,3) to reflect the local nature of the representation. 
The functions H;, and their quantized counterparts B, which are 
defined in terms'of T: and R over the (continuous) cóordinate variables 
(xy), will be referred to, collectively in 1 and, ji, âs. au “entity. , or 
"quantized entity” respectively. The convolutions of R with Ti. results in 
Hi being generally quite smooth and well behaved. We will preSume that 
H,, is non negative and is greater than zero only on a region of finite 
area. 
  
2 
+ 
+ 
The pixel value P,, is just a sampling (i.e., point evaluation) of the 
quantized entity. “A shift in the object's position corresponds to a shift 
in the sampling of the entity. It will be convenient to assume that all 
the Hs are identical, that is, Ti, - T. In this case we will drop the 
subscripts and say that the entity H is "spatially invariant”. The pixel 
values are then obtained by sampling a single function. If, in addition, 
the pixel grid is uniformly spaced with unit spacings, then 
  
Yes (2723 
Po. =F (1 ~ 3 = % 
13 (i E? J “0 
By a ‘locale’ of an entity, or quantized entity, we will mean an area, À, 
consisting of all points (x,y) for which the quantized entity functions Hs 
do not change. More precisely, (x',v') iie in A, if and only if 
HG rnxdrmsisdug a. ics (2-4) 
forall (x,y) in.A.and albi, Jd. The 
of equivalenced positions. Each locale 
0 
he plane into sets 
3 a { 
uncertainty for