02-06-2012, 11:18 AM
A Nonlinear Derivative Scheme Applied to Edge Detection
[attachment=23948]
INTRODUCTION
EDGE detection is one of the oldest topics in image
processing and has been widely studied. Methods of
edge detection have involved derivative masks, primarily
developed in the discrete case, and have been confined to
slightly noisy images [1], [2], [3]. The regularization or
smoothing [4] and optimal approaches of Canny [5] have led
to several efficient continuous operators for noisy and
blurred images [6], [7], [8], [9], [10]. Other advanced methods
that consider the Canny criterion have been developed to
deal with noise, uneven illumination, and image contrast
[11]. In a marginal way, discrete approaches for regularization
have been developed and have improved results by
considering the discrete nature of the images [12], [13].
NONLINEAR FILTERING SCHEME FOR EDGE LOCALIZATION
We first introduce the 1D principle before proposing an
extension to 2D.Acomment on the ability of the new scheme,
in terms of localization, will conclude this section. We will
denote the proposed nonlinear filtering scheme by NLFS.
Principle of NLFS
presents the configuration of two opposite profiles.
We propose to localize the edge point according to the sign
of the slope of the transition. If this slope is positive, the
point will be validated after the transition; if the slope is
negative, it will be validated before.
Rotational Invariance of Localization
It is known that an antisymmetric linear (derivative) filter
(no 0 value at the center) gives a shifted pixel localization
and this localization changes depending on edge orientation.
Since the NLFS considers the orientation of the edge
profile (the sign of the variation), it provides univocal
localization; this localization only depends on the profile
(variation). We thus have:
Edge Pixel Localization on Synthetic Images
exemplifies the difficulties of edge localization for a
synthetic image. Notably, the suggested method localizes the
edges inside the frontier of the form. The traditional method,
without regularization (CFS), gives a representation of edges
that are shifted in a particular direction. The regularized
version (with a 0: CFS0) gives thick edges; however, any
presence of noise makes this representation very chaotic.
Changing the localization of edges (to outside the frontier
of the form) is possible with a simple shift of the detectors.
Depending on the selected localization and the object
luminance (and background), the edges will always be
detected inside or outside the form.
Nondetection of Thin Lines
While the NLFS scheme has high localization and good
noise cancellation performance, it suffers from its 1D
definition in the 2D extension. Indeed, positive impulses
are interpreted as noise-inducing. For example, thin, lightgray
edge lines (thickness ΒΌ 1 pixel) are seen as noise. Fig. 7
shows this phenomenon on a noisy synthetic image.
[attachment=23948]
INTRODUCTION
EDGE detection is one of the oldest topics in image
processing and has been widely studied. Methods of
edge detection have involved derivative masks, primarily
developed in the discrete case, and have been confined to
slightly noisy images [1], [2], [3]. The regularization or
smoothing [4] and optimal approaches of Canny [5] have led
to several efficient continuous operators for noisy and
blurred images [6], [7], [8], [9], [10]. Other advanced methods
that consider the Canny criterion have been developed to
deal with noise, uneven illumination, and image contrast
[11]. In a marginal way, discrete approaches for regularization
have been developed and have improved results by
considering the discrete nature of the images [12], [13].
NONLINEAR FILTERING SCHEME FOR EDGE LOCALIZATION
We first introduce the 1D principle before proposing an
extension to 2D.Acomment on the ability of the new scheme,
in terms of localization, will conclude this section. We will
denote the proposed nonlinear filtering scheme by NLFS.
Principle of NLFS
presents the configuration of two opposite profiles.
We propose to localize the edge point according to the sign
of the slope of the transition. If this slope is positive, the
point will be validated after the transition; if the slope is
negative, it will be validated before.
Rotational Invariance of Localization
It is known that an antisymmetric linear (derivative) filter
(no 0 value at the center) gives a shifted pixel localization
and this localization changes depending on edge orientation.
Since the NLFS considers the orientation of the edge
profile (the sign of the variation), it provides univocal
localization; this localization only depends on the profile
(variation). We thus have:
Edge Pixel Localization on Synthetic Images
exemplifies the difficulties of edge localization for a
synthetic image. Notably, the suggested method localizes the
edges inside the frontier of the form. The traditional method,
without regularization (CFS), gives a representation of edges
that are shifted in a particular direction. The regularized
version (with a 0: CFS0) gives thick edges; however, any
presence of noise makes this representation very chaotic.
Changing the localization of edges (to outside the frontier
of the form) is possible with a simple shift of the detectors.
Depending on the selected localization and the object
luminance (and background), the edges will always be
detected inside or outside the form.
Nondetection of Thin Lines
While the NLFS scheme has high localization and good
noise cancellation performance, it suffers from its 1D
definition in the 2D extension. Indeed, positive impulses
are interpreted as noise-inducing. For example, thin, lightgray
edge lines (thickness ΒΌ 1 pixel) are seen as noise. Fig. 7
shows this phenomenon on a noisy synthetic image.