By Simon Haykin

ISBN-10: 013322760X

ISBN-13: 9780133227604

Represents the main entire therapy on hand of neural networks from an engineering point of view. Examines the entire very important points of this rising know-how. DLC: Adaptive filters.

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**Additional info for Adaptive Filter Theory (3rd Edition)**

**Example text**

X−y)∈B1 (x−z)∈B2 f (x − z) + μn (z)} ⎛ ⎞ {k ♦ ( f (x − y) + λ1 (y))} ∪ { f (x − z) + μ1 (z)}, ⎜{k ♦ ( f (x − y) + λ2 (y))} ∪ { f (x − z) + μ2 (z)},⎟ ⎟ = max(k) ⎜ ⎠ (x−y)∈B1 ⎝. . (x−z)∈B2 {k ♦ ( f (x − y) + λn (y))} ∪ { f (x − z) + μn (z)} n ⎛ = max(k) [{k ♦ ( f (x − y) + λi (y))} ∪ { f (x − z) + μi (z)}] (x−y)∈B1 (x−z)∈B2 i=1 26 M. I. VARDAVOULIA, A. GASTERATOS, AND I. ANDREADIS This equation can be expressed in terms of order statistics of the multiset as follows: f ⊕ [α, β, k](x) n = max(k) [max(N) ({k ♦ ( f (x − y) + λi (y))} ∪ { f (x − z) + μi (z)}), i=1 (x−y)∈B1 (x−z)∈B2 (N−1) max (x−y)∈B1 (x−z)∈B2 ({k ♦ ( f (x − y) + λi (y))} ∪ { f (x − z) + μi (z)}), ..

This agrees with the notion of fuzzy ﬁtting, because the structuring element ﬁts better only at these points than at the rest points of the image. Fuzzy erosion quantiﬁes the degree of structuring element ﬁtting. The larger the number of pixels of the structuring element, the more difﬁcult the ﬁtting. Furthermore, fuzzy soft erosion shrinks the image. If fuzzy image A is considered as a noisy version of a binary image (Sinha and Dougherty, 1992), then the object of interest consists of points (0, 1), (0, 2), (0, 3), (0, 4), (1, 1), (1, 2), (1, 3), (1, 4), (2, 1) and (2, 2), and the rest is the background.

D 34 x(16) ... x(1) (a) Maximum k=1 d 33 .. d1 d 69 ... ... d 23 x(28) x(27) ... x(1) d22 .. d1 19th-order statistic k=2 d 57 ... d 12 x(40) x(39) x(38) ... x(1) d 11 .. d1 d 45 ... d1 x(52) x(51) x(50) x(49) ... ... ... x(1) d 57 ... d 47 x(40) ... x(3) x(2) x(1) d 46 ... x(52) ... ... ... x(4) x(3) x(2) x(1) d 45 ... 26th-order statistic k=3 33th-order statistic k=4 (b) 45 d 81 ... d 49 x(12) ... x(1) d 48 ... ... d1 d 69 ... d 48 x(20) ... Minimum k=1 x(2) x(1) d47 ... ...

### Adaptive Filter Theory (3rd Edition) by Simon Haykin

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