The major criticism of the variational approach to shape-from-shading is that the quadratic regularizing term over-smoothes the recovered needle-map. In particular, the quadratic regularizing term discourages sudden changes in surface normal direction across a surface. The main consequence is to blur high-curvature surface features. Our aim in this paper is to illustrate how the smoothness constraints can be controlled more effectively using error kernels suggested by robust statistics. A review of the alternatives offered in the literature leads us to a new class of error-kernel for the shape-from-shading problem.
In this section we illustrate how the variational calculus can be
applied to a general robust regularizer to develop iterative equations
for the recovery of needle-maps. The robust error function
is defined on the data error-residual
. The quantity
controls the width of the error kernel. It is also convenient to couch
the process of robust estimation in terms of an
influence-function
which can be used to weight parameter estimates according to their
associated error-residuals. Formally, the influence function is related
to the error-function by
.
The novel contribution in this paper is to use the robust error kernel as a smoothness prior for the derivatives of the needle-map. Specifically, we adopt the following form for regularised energy-function
In other words, we apply robust-error kernels separately to the
magnitudes of the derivatives of the needle-map in the
x
and
y
directions. Applying variational calculus to the energy function in a
directly analogous fashion to Section 2, yields the Euler equation
As a result, the fixed-point iterative equation for updating the
components of the needle map is
This result is entirely general. Any robust error kernel
can be inserted into the above result to yield a shape-from-shading
scheme. However, it must be stressed that performance is critically
determined by the choice of error-kernel.
Although robust estimation is usually posed in terms of the influence
function
, it is the form of the associated error or ``energy'' function
that is of primary importance in smoothness regularization. Formally,
the energy is related to the first-moment of the influence function,
i.e.
. Moreover, the asymptotic properties of the derivative of the energy
function allows us to establish a broad-based taxonomy of the available
influence functions. Broadly speaking, there are three classes of
influence function. The first of these are referred to as re-descending.
Here the derivative of the energy function asymptotically approaches
zero. In the second case the derivative of the energy function is
referred to as sigmoidal if it becomes asymptotically constant. The
third category includes those which are monotonically increasing, into
which falls the quadratic prior.
According to this taxonomy, Tukey's bi-weight is re-descending. Huber's robust kernel [ 8 ] is sigmoidal. Both are defined in a piecewise manner, and so are not particularly amenable to variational treatment. Li's [ 11 ] adaptive potential functions are continuous, but all fall into the re-descending category.
Influence functions with monotonically increasing energy function derivatives tend to over-smooth genuine discontinuities in image brightness, since such discontinuities lead to large values of smoothness error. Conversely, re-descending influence functions do not penalise sharp changes in surface orientation. Although this leads to improved treatment of discontinuities, it can be at the expense of increased noise sensitivity. Sigmoidal influence functions, represent a compromise between the dual aims of recovering discontinuities and rejecting noise artifacts.
Unfortunately, none of the available sigmoidal error-kernels are defined in a continuous manner. In order to pursue the variational analysis of the sigmoidal case, we will present a continuous variant Huber's kernel, based on a hyperbolic tangent function.
The classical example of a sigmoidal-derivative energy function is
Huber's estimator (Figure 1), defined by the following influence
function
and error-function
Figure 1:
The Huber regularizer:
(left),
(middle) and
(right).
As pointed out earlier, the piecewise nature of Huber's estimator
renders it unsuitable for variational analysis. To provide a continuous
counterpart to Huber's error-kernel, we have investigated the following
influence and energy functions, which have the qualitative shapes shown
in Figure 2
Figure 2:
The sigmoidal-derivative regularizer:
(left),
(middle) and
(right).
Substituting the sigmoidal regularizer into the generalised needle-map update equation yields the following result
It is illuminating to consider the behaviour of this update equation for
small and large smoothness errors. Firstly, the averaging of the
neighbourhood normals is moderated by a function of the form
. This averaging effect is most pronounced when the smoothness error is
small. The remaining contribution to the smoothness process is of the
form
. This term vanishes at the origin and tends towards zero for large
values of
, only kicking-in at intermediate error conditions.
Benoit Huet