Bits per Pixel Requirements for Optimum Encoder 
For the processing chain depicted in Fig 1, considering the encoder 
operation is preceded and followed by arbitrary linear filter operations 
H(f1f2) and P(f4f2) respectively, the following relations, relevant to the 
Fourier transformed signal representation, apply: 
IN(f1f2) = Fy(f1£2) + Ny(£1£2) (8) 
UN(f1f2) 7» HN(£4£2) . IN(£4£2) (9) 
CN(f1f2) 7 Un(£9£2) - Dn(£4£2) (10) 
ON(E1f2) = By(£1£2) . Oq(f1f2) (11) 
where capitals denote the symmetrical two dimensional Fourier Transform of 
size N2 and DN(f4f2) denotes the random distortion variable introduced 
by the encoder. Applying a minimum mean square quantizer [2], the 
following relations are valid (omitting the variables f1f2 for brevity): 
E[Dy] = 0 (12) 
E[DN-CN.] = 0 (13) 
|]Cy//2 = max [0, 0/1 2 — (Dy((2] (14) 
2 1 
HN OA 
E [Ny.Dy.] = RTE (15) 
_ Ent Ball 
E[FNDN] = Hz (16) 
The distortion measure dp of the processed image € (x,y) with respect to 
the reference image f(x,y) is defined here by the squared error criterium. 
Hence: 
dr = E[£ € (xy) - f(x,y)) ?] (17) 
or equivalenty, 
dp- 1 C(t £9) = F(f1f 2 18 
T weld 1£2) - F(£1£2)l1 (18) 
Applying a minimum mean square encoder for each of the Fourier 
coefficients it follows after some manipulation (omitting var. f 1£2): 
^J 2 2. à 2 t Fu -— = + 
IIc El TRITT [P) [H[ I| ll eu (Pu ePi )-Iib [P] 
IU, à à 
~ LN . 1 (iJ ll (| D I! 
JIC-FI} = Il FIL ce lor < Uo wl SUD, 
The upperbound of the minimum average number of bits per pixel R(dg) as 
function of the encoder induced distortion dg is given by the rate 
distortion function of a corresponding Gaussian process and is expressed 
parametrically by [3]: 
2, 
R(de) = a [max [0,104 ze Ja (20) 
Sa iT 
de . Jf "^ [eur 5 In ace [ 4. a f, (21) 
448 
  
La ax a RÉ A Se JP PER S de 
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