The purpose is to identify the computational requirements for accurate
characterization of luminance
transitions caused by defocused edges, shadow penumbrae and
high-curvature shading.
Results have shown that attempts to understand human sensitivity to
spatial luminance modulations in terms of
scene structure have typically distinguished
the problems of edge detection and shading interpretation. We show
that this distinction is artificial, in that the luminance variations
produced by defocused step edges, the penumbrae of cast shadows, and
shading over high-curvature surfaces are mathematically equivalent.
We therefore propose that the goal of early visual processing of luminance
transitions is not to * distinguish* but rather to
reliably * detect, localize and characterize* these stimuli.
This observation motivates a computational theory
which rests on two key findings. First,
we show that while 1st derivative (2-lobe, odd-phase) filters are
by themselves inadequate, the
conjunction of first- and second-order (3-lobe, even-phase)
derivative estimates are
sufficient to accurately localize and estimate the
blur and contrast of luminance edges. Second, while
previous considerations of the problem of * scale* in edge
detection have led to global solutions over a complete scale space,
we show that in fact this problem has a local solution. This solution
relies on the notion of a * minimum reliable scale* which can be
related to the parameters of the luminance transition, blurring processes, and
receptor uncertainty. This result is relevant to the puzzle of how
responses of filters of different sizes are arbitrated to produce a
unique characterization of the local luminance pattern.
We demonstrate the effectiveness of this theory by experiments on
images with small depth of field and shadows cast by extended light
sources. We show that edges over a wide range of blur and contrasts
can be reliably recovered, and that the accurate estimation of blur
allows the inference of complete space curves from the image. These
space curves provide continuous estimates of depth from the
focal plane in the case of defocus and of the distance between objects in
the scene in the case of cast shadows.
The conclusions show that

- The characterization of generalized luminance transitions is a second order differential problem.
- The problem of scale in edge detection has a local solution
in the form of a
*minimum reliable scale*defined by the contrast and blur of the luminance transition and the statistics of the receptor noise.

J. Elder, S.W. Zucker

Mon Nov 13 10:43:02 EST 1995