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Analysis of Neural Networks for Edge Detection
This paper illustrates a novel method to analyze artificial neural networks so as to gain insight into their internal functionality. To this purpose, the elements of a feedforward-backpropagation neural network, that has been trained to detect edges in images, are described in terms of differential operators of various orders and with various angles of operation
Detection of Edges in Spectral Data II. Nonlinear Enhancement
We discuss a general framework for recovering edges in piecewise smooth
functions with finitely many jump discontinuities, where . Our approach is based on two main aspects--localization using
appropriate concentration kernels and separation of scales by nonlinear
enhancement.
To detect such edges, one employs concentration kernels, ,
depending on the small scale . It is shown that odd kernels, properly
scaled, and admissible (in the sense of having small -moments of
order ) satisfy , thus recovering both the location and amplitudes of all edges.As
an example we consider general concentration kernels of the form
to detect edges from the first
spectral modes of piecewise smooth f's. Here we improve in
generality and simplicity over our previous study in [A. Gelb and E. Tadmor,
Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and
nonperiodic spectral projections are considered. We identify, in particular, a
new family of exponential factors, , with superior
localization properties.
The other aspect of our edge detection involves a nonlinear enhancement
procedure which is based on separation of scales between the edges, where
, and the smooth regions where . Numerical examples demonstrate that by coupling
concentration kernels with nonlinear enhancement one arrives at effective edge
detectors
Recovery of edges from spectral data with noise -- a new perspective
We consider the problem of detecting edges in piecewise smooth functions from
their N-degree spectral content, which is assumed to be corrupted by noise.
There are three scales involved: the "smoothness" scale of order 1/N, the noise
scale of order and the O(1) scale of the jump discontinuities. We use
concentration factors which are adjusted to the noise variance, >> 1/N,
in order to detect the underlying O(1)-edges, which are separated from the
noise scale, << 1
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