SURFACE Verification

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SURFACE Verification

Invocation of SURFACEs is largely based on summary characteristics (e.g. areas), rather than detailed shape. As surface regions are characterized by their boundaries and internal shapes, verification could then ensure that:

The observed surface has the same shape as that of the forward-facing portions of the oriented model SURFACE.
The surface image boundaries are the same as those predicted by the oriented model.

Several problems complicate this approach: unmodeled extremal boundaries on curved surfaces, inexact boundary placement at surface curvature discontinuities, and information lost because of occlusion.

As extremal boundaries are not modeled, they should not be considered, except perhaps for verifying that the surface has the appropriate curvature directions.

The second problem causes variable sized surface regions and hence makes it difficult to compare surfaces and boundaries exactly. But, some possibilities remain. In particular, all model boundaries are either orientation or curvature discontinuity boundaries. The former should remain stable and appear as either predictable shape segmentation or front-side-obscuring boundaries. Detailed shape analysis may distinguish front-side-obscuring boundaries arising from orientation discontinuities from extremal boundaries. Curvature discontinuity boundaries should probably be ignored.

Occlusion causes data loss, but is detectable as the back-side-obscuring boundaries associated with the surface indicate the initial point of occlusion. As the visible data must be a subset of the predicted data, the back-side-obscuring boundary must be internal to the predicted surface. Concave boundaries are also ambiguous regarding surface ordering, so may not be true surface boundaries.

Figure 10.2 illustrates these points, which are summarized as:

: [] All data boundaries labeled as front-side-obscuring and surface orientation discontinuity should closely correspond to portions of the boundaries predicted by the model. The back-side-obscuring and concave boundaries must lie on or be internal to the predicted region.
: [] The data surface should have the same shape as a subset of the oriented model SURFACE, except where near curvature discontinuities. This entails having similar areas, surface curvatures and axis orientations.

**Figure 10.2:** *Boundary and Surface Comparison*
$\begin{figure}\epsfysize =5in \epsfbox{FIGURES/Fig10.2.ps}\end{figure}$

Because of errors in estimating SURFACE reference frames, it was difficult to predict surface orientation and boundary locations accurately enough for direct comparison. As a result, only test was implemented:

Surface Shape Verification Test

Let:

and $\hat{S}$ be the predicted and observed surface shape class

and $\hat{M}$ be the predicted and observed major curvatures

and $\hat{m}$ be the predicted and observed minor curvatures

$\vec{a}$ and $\vec{\hat{a}}$ be the predicted and observed major curvature axes

$\tau_c$ , $\tau_a$ be thresholds

If:

is the same as $\hat{S}$ ,

$\mid M - \hat{M} \mid < \tau_c$ , ( $\tau_c$ = 0.05)

$\mid m - \hat{m} \mid < \tau_c$ , and

$\mid \vec{a} \circ \vec{\hat{a}}\mid > \tau_a$ ( $\tau_a$ = 0.80)

(planar surfaces do not use this last test)

Then: the proposed identity of the surface is accepted.

Next: Rigid ASSEMBLY Verification Up: Constraining Object Existence and Previous: Constraining Object Existence and

Bob Fisher 2004-02-26

Surface Shape Verification Test
Let:
	and $\hat{S}$ be the predicted and observed surface shape class
	and $\hat{M}$ be the predicted and observed major curvatures
	and $\hat{m}$ be the predicted and observed minor curvatures
	$\vec{a}$ and $\vec{\hat{a}}$ be the predicted and observed major curvature axes
	$\tau_c$ , $\tau_a$ be thresholds
If:
	is the same as $\hat{S}$ ,
	$\mid M - \hat{M} \mid < \tau_c$ , ( $\tau_c$ = 0.05)
	$\mid m - \hat{m} \mid < \tau_c$ , and
	$\mid \vec{a} \circ \vec{\hat{a}}\mid > \tau_a$ ( $\tau_a$ = 0.80)
	(planar surfaces do not use this last test)
Then:	the proposed identity of the surface is accepted.