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model in an incorrect pose.
or nonshiny surface. Lambert's law states that the visible intensity of a surface patch is
related to the dot product between the surface normal and the lighting. For a Lambertian
object the imaging equation is:
X~
vT x =
1.3
i li ux ;
i where the model value ux is the normal vector of a surface patch on the object, li is a
vector pointing toward light source i, and i is proportional to the intensity of that light
source Horn, 1986 contains an excellent review of imaging and its relationship to vision.
13 Paul A. Viola CHAPTER 1. INTRODUCTION Drawing the explicit parallel between 1.1 and 1.3 we can see that the imaging function is
X~
F ux; q =
1.4
i li ux ;
i where q = f i; ~ig. As the illumination changes the functional relationship between the model
l
and image will change.
Since we can not know beforehand what the imaging function will be, aligning a model and
image can be quite di cult. These di culties are only compounded if the surface properties of
the object are not well understood. For example, many objects can not be modeled as having
a Lambertian surface. Di erent surface nishes will have di erent re ectance functions. In
general re ectance is a function of lighting direction, surface normal and viewing direction.
The intensity of an observed patch is then:
X
vT x = R i; ~i;~; ux ;
lo
1.5
i where ~ is a vector pointing toward the observer from the patch and R is the re ectance
o
function of the surface. For an unknown material a great deal of experimentation is necessary to completely categorize the re ectance function. Since a general vision system should
work with a variety of objects and under general illumination conditions, overly constraining
assumptions about re ectance or illumination should be avoided.
Let us examine the relationship between a real image and model. This will allow us to
build intuition about the alignment process. Data from the real re ectance function can
be obtained by aligning a model to a real image. An alignment associates points from the
image with points from the model. If the alignment is correct, each pixel of the image can
be interpreted as a sample of the imaging function R. The imaging function could be displayed by plotting intensity against lighting direction, viewing direction and surface normal.
Unfortunately, because intensity is a function of so many di erent parameters the resulting
plot can be prohibitively complex and impossible to visualize. Signi cant simpli cation will
be necessary if we are to detect any structure in this data.
In a wide variety of real images we can assume that the light sources are far from the
object at least in terms of the dimensions of the object. When this is true and there are no
shadows, each patch of the object will be illuminated in the same way. Furthermore, we will
14 1.1. AN INTRODUCTION TO ALIGNMENT AITR 1548 assume that the observer is far from the object, and that the viewing direction is therefore
constant throughout the image. The resulting relationship between normal and intensity is
three dimensional: the normal vector has unit length and is determined by two parameters,
its x and y components; the intensity is a third parameter. A three dimensional scatter plot
of normal versus intensity is really a slice through the high dimensional space in which R
is de ned. Though this graph is much simpler than the original, three dimensional plots are
still quite di cult to interpret. We will slice the data once again so that all of the points have
a single value for the y component of the normal.
Figure 1.3 contains a graph of the intensities along a single scanline of the image of Ron.
Figure 1.4 shows similar data for the correctly aligned model of Ron. Model normals from
this scanline are displayed in two graphs: the rst shows the x component of the normal
while the second shows the y component. Notice that we have chosen this portion of the
model so that the y component of the normal is almost constant. As a result the relationship
between normal and intensity can be visualized in only two dimensions. Figure 1.5 shows the
intensities in the image plotted against the x component of the normal in the model. Notice
that this relationship appears both consistent and functional. Points from the model with
similar surface normals have very similar intensities. The data in this graph could be well
approximated by a smooth curve. We will call an imaging function like this one consistent.
Interestingly, we did not need any information about the illumination or surface properties
of the object to determine that there is a consistent relationship between model normal and
image intensity.
Figure 1.6 shows the relationship between normal and intensity when the model and
image are no longer aligned. The only di erence between this graph and the rst is that the
intensities come from a scanline 3 centimeters below the correct alignment i.e. the model is
no longer aligned with the image, it is 3...
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