Graphics Reference
In-Depth Information
that with one capacitance per node, a multinode generalization, such as the one in
Fig.
1.8
, or the larger one in the full 2D imager, would have similar dynamics in
exiting its metastable states. Yet I had no proof.
Hopfield showed in 1984 that with symmetric interconnection weights, such a
network can be characterized by an energy function, and that any state change
reduces the energy, until it settles into a stable state, a local minimum [
17
]. Hop-
field's differential equation formulation with one state variable per node exactly
describes the network of interconnected nMOS NOR gates in the optical mouse
imager, so we can be sure, in retrospect, that the metastable states of that cir-
cuit are nonoscillatory. The cross-coupled TTL gates that Chaney and Molnar
analyzed had extra internal (not symmetrically interconnected) nodes with their
own state and delay, which is why the metastable states in that case could be
oscillatory.
1.5 Two Dimensions
The idea of a 1D line tracker such as that of Fig.
1.9
has various possible extensions
into two dimensions. Extensions that use axis-aligned patterns of lines or dots have
problems when the mouse is rotated, which is why pre-1980 attempts at PARC did
not get very far. Using the idea of a radius of inhibition in two dimensions, I set
about finding a way to track less rigid patterns of light dots in a dark field. An ideal
pattern seemed to be a close-packed hexagonal array of dots, allowing the imager
at arbitrary angles to treat the dots as being randomly arrayed, with a characteristic
distance between them.
After evaluating various radii of inhibition and imager sizes, I found that a 4
4
array with a “3.0 special” radius of inhibition would yield a set of stable images
containing either one light dot in the central 2
×
2 area, or two light dots on oppo-
site edges—a total of 30 possible state binary images, as shown in Fig.
1.14
.Here
“special” means that pixels at a distance of exactly 3.0 pixel spaces will inhibit each
other if they are corners, but not otherwise; this scheme eliminates patterns of 3
and 4 dots, but allows patterns of dots on opposite edges, so that motion can be
detected, much as with the lines on opposite ends of the linear array of 4. In the
2D case, a pattern of light dots in a hexagonal array, as shown in Fig.
1.15
, works
well.
×
When the radius of inhibition exceeds 2
√
2, a light spot in an image cannot be
adjacent or diagonally adjacent to two different spots in a previous or subsequent
image, so we will not have any ambiguity of which direction a spot moved. Further-
more, with the 3.0 special inhibition pattern, since there can only be at most two
spots (as shown in Fig.
1.14
), there will be at most two motion directions involved
in a correlation of one image with the next. Computing the average of two moves
is easy, using a half-step bit in a position counter: when an old image and a new
image each have two spots, with different apparent move directions, their average
can still be represented in terms of half steps. The logic to drive the half and full steps
