![](https://csdnimg.cn/release/download_crawler_static/16300135/bge.jpg)
patch deformation models, 3D model based intensity predictions, coupled affine or spline patches for
surface coverage, etc., [1, 52, 55, 9, 110, 94, 53, 97, 76, 104, 102]. The structure of intensity based
bundle problems is very similar to that of feature based ones, so all of the techniques studied below
can be applied.
We will not go into more detail on intensity matching, except to note that it is the real basis
of feature based methods. Feature detectors are optimized for detection not localization. To lo-
calize a detected feature accurately we need to match (some function of) the image intensities in
its region against either an idealized template or another image of the feature, using an appropri-
ate geometric deformation model, etc. For example, suppose that the intensity matching model is
f(u) =
1
2
RR
ρ(kδI(u)k
2
) where the integration is over some image patch, δI is the current intensity
prediction error, u parametrizes the local geometry (patch translation & warping), and ρ(·) is some
intensity error robustifier. Then the cost gradient in terms of u is g
>
u
=
df
du
=
RR
ρ
0
δI
>
dI
du
. Similarly,
the cost Hessian in u in a Gauss-Newton approximation is H
u
=
d
2
f
du
2
≈
RR
ρ
00
(
dI
du
)
>
dI
du
. In a feature
based model, we express u = u(x) as a function of the bundle parameters, so if J
u
=
du
dx
we have
a corresponding cost gradient and Hessian contribution g
>
x
= g
>
u
J
u
and H
x
= J
>
u
H
u
J
u
. In other
words, the intensity matching model is locally equivalent to a quadratic feature matching one on the
‘features’ u(x), with effective weight (inverse covariance) matrix W
u
= H
u
. All image feature error
models in vision are ultimately based on such an underlying intensity matching model. As feature
covariances are a function of intensity gradients
RR
ρ
00
(
dI
du
)
>
dI
du
, they can be both highly variable be-
tween features (depending on how much local gradient there is), and highly anisotropic (depending
on how directional the gradients are). E.g., for points along a 1D intensity edge, the uncertainty is
large in the along edge direction and small in the across edge one.
3.5 Implicit models
Sometimes observations are most naturally expressed in terms of an implicit observation-constraining
model h(x, z) = 0, rather than an explicit observation-predicting one z = z(x). (The associated
image error still has the form f(z
− z)). For example, if the model is a 3D curve and we observe
points on it (the noisy images of 3D points that may lie anywhere along the 3D curve), we can
predict the whole image curve, but not the exact position of each observation along it. We only have
the constraint that the noiseless image of the observed point would lie on the noiseless image of the
curve, if we knew these. There are basically two ways to handle implicit models: nuisance parameters
and reduction.
Nuisance parameters: In this approach, the model is made explicit by adding additional ‘nuisance’
parameters representing something equivalent to model-consistent estimates of the unknown noise
free observations, i.e. to z with h(x, z) = 0. The most direct way to do this is to include the entire
parameter vector z as nuisance parameters, so that we have to solve a constrained optimization prob-
lem on the extended parameter space (x, z), minimizing f(z − z) over (x, z) subject to h(x, z) = 0.
This is a sparse constrained problem, which can be solved efficiently using sparse matrix techniques
(§6.3). In fact, for image observations, the subproblems in z (optimizing f(z − z) over z for fixed z
and x) are small and for typical f rather simple. So in spite of the extra parameters z, optimizing this
model is not significantly more expensive than optimizing an explicit one z = z(x) [14, 13, 105, 106].
For example, when estimating matching constraints between image pairs or triplets [60, 62], instead
of using an explicit 3D representation, pairs or triplets of corresponding image points can be used as
features z
i
, subject to the epipolar or trifocal geometry contained in x [105, 106].
However, if a smaller nuisance parameter vector than z can be found, it is wise to use it. In the
case of a curve, it suffices to include just one nuisance parameter per observation, saying where along
13