By Peter W. Hawkes

Advances in Imaging and Electron Physics merges long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The sequence beneficial properties prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technology and electronic photograph processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in some of these domain names.

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**Sample text**

Granulometric Spectral Theory Given a Euclidean granulometry { is deÞned by t }, the size distribution of a compact set S (t) = ν[S] − ν[ t (S)] (80) The size distribution gives the volume of the set removed by t . It is increasing and continuous from the left (Matheron, 1975). We let (0) = 0 and assume at least one generating set for { t } contains more than a single point, so that (t) = ν[S] for sufÞciently large t. The normalization, (t) = (t)/ν[S] is called the pattern spectrum of S relative to { t }.

51) for the uniform densities state Þlter error: e[r ] = E[T ] P(S) S and N yields the steady- b3 − r 3 r 3 − c3 + P(N ) 3 3 3 d −c b − a3 (58) Minimization yields the optimal solution for various conditions: min{e[ r r ]} ⎧ b3 − c3 ⎪ ⎪ E[T ]P(S) ⎪ ⎪ d 3 − c3 ⎪ ⎪ ⎨ b3 − c3 = E[T ]P(N ) ⎪ b3 − a 3 ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎩ E[T ]P(N ) b − c b3 − a 3 at r = b at r = c at r ∈ [c, b] P(S) P(N ) < 3 3 −c b − a3 P(S) P(N ) (59) if 3 > 3 3 d −c b − a3 P(S) P(N ) if 3 = 3 3 d −c b − a3 if d3 DeÞne the discriminant D= P(N ) P(S) − 3 3 −c b − a3 d3 (60) For D < 0, the optimum occurs at the upper endpoint b of the interval over which the states can vary; for D > 0, the optimum occurs at the lower endpoint c; for D = 0 , all states in [c, b] are equivalent, and hence optimal.

Then e[ N] =E = = = ∞ i=1 ∞ i=1 ∞ N i=1 k=1 ν[Nk ](T[X k ; ti+1 ] − T[X k ; ti ]) M N (tk+1 ) − M N (tk ) tk+1 H N (t) dt tk H N (t) dt (94) DESIGN OF LOGICAL GRANULOMETRIC FILTERS 45 where the Þrst equality follows from the deÞnition of the Þlter, the second from Eq. (92), and the third by the fundamental theorem of calculus. A similar representation holds for e[ S ] in terms of the fail set and the noise GSD. From Eq. (94) and the analogous expression for e[ S ], we see that e[ N ] and e[ S ] are minimized by choosing the pass set in accordance with the differential determinant of Eq.