When the computer industry
was beginning to form in the 1940s, many people saw the significance of
a raster to vector converter just as we have outlined above. One of the
earliest solutions was an entirely mathematical approach known as wavelets.
This solution is still considered state of the art today, but it has been
joined by various other solutions.
By now, a number of people claim to
have raster to vector converters with wavelets, fractals, and Bezier curves
counted among the most popular. Technically, these claims are correct (raster
data is converted to vector), but practically the proposed solutions do
not yield mathematical formulas that allow real world solutions.
In going back to the heart of the problem,
we can understand why these current solutions are inadequate. In other
words, what makes the problem of finding a real world solution so difficult?
The larger part of the problem is that
there are too many dots, and the smaller part of the problem is that the
algorithm needs to deal with the way that the dots were collected.
Let’s start with the small part of
the problem first. If our sensory system had one sensor, we could tell
whether something was there or not. We could not tell whether it was a
fly or an elephant, but we could make a good guess from the color of the
object. If the sensor was only one-third covered with the fly and the background
was white, the color mix would be a light gray (assuming it is a black
fly). So color alone is not reliable.
If we had 16 sensors we could do better
and if we had 512 sensors we could tell the fly from the elephant. With
512 sensors we would not have a very smooth picture of the elephant and
we would need to fill between the data points. The fill should take into
account the color from the sensors that are completely immersed in the
image and it should extrapolate the points between the data points from
the partially covered sensors. This is how scanners yield more data points
with greater accuracy than what the scanner actually senses.
In other words, the sensors work in
a specific and unvarying way and by knowing this we can take advantage
of the partially sensed objects to accurately project the actual image.
Therefore, by using color and integrating shapes, we can solve the smaller
part of the problem.
Before we can solve the bigger part
of the problem we have to be able to write formulas across a large number
of dots. Mathematics is only able to write equations over a few dots. For
example, with 1 dot there is one shape. With 9 dots arranged into a 3 by
3 array, there are about 21,000 possible shapes. The inclusion of each
additional dot takes the number of possibilities up exponentially. Each
case has to be considered and it must have a solution. There are many problems
in mathematics that don’t have solutions. For example, only about 33% of
second order partial differential equations have solutions. Therefore,
mathematics alone will not solve the problem.
Purely mathematical approaches, such
as wavelets and fractals, are able to convert raster to vector by limiting
the size of the converted picture to about 5 by 5 dots (this is actually
generous). Large pictures are converted by converting thousands of independent
and disconnected small pictures. All other methods do something similar.
The very idea of converting a 10 by 10 dot picture with all the attendant
possibilities seems ludicrous. Real world pictures are often 2,000 by 2,000
dots. Therefore, a real world solution seems hopeless.
On the other hand, equations running
over 5 dots or so (in some cases) do not make the solution impractical
for computer algorithms. The problem comes when adjacent equations are
independent. If an interrelationship between equations can be found, a
number of equations (related together) are a practical solution. Furthermore,
systematic philosophy predicts that such relationships must exist in every
case. In this philosophical system (from Aristotle and Saint Thomas of
Aquinas, among others), everything is made up of parts. For example, skin
is made of cells. Cells are a composition of molecules. Molecules are the
products of atoms, and there is no smallest part. The process goes on and
on.
For our part, we need to manipulate
data. Manipulation of nature can be undertaken by understanding the lower
levels of recursion. For example, we can improve crops by working with
the genetic material that the crops come from. Therefore, if we could find
the parts that made up a picture, we could build a practical raster to
vector converter. From a theoretical perspective, we completed this task
in the six years from 1986 to 1992. Since then, we have been
implementing the solution.
Even this philosophical approach is
not unique to us. Other people have tried it, but finding all of the data
possibilities is a daunting task. Finding one or two component parts will
not yield a functional product. Every case has to be discovered, analyzed
and related to every other case. It is not a certainty that the project
would be completed in even 500 years.
The basic nature of the work deals
with complex digital logic. Humans, yes even engineering nerds, are not
well suited for this and the task is unpleasant. Since the logic is extremely
complex, a high caliber digital designer would be required. These types
of people are in high demand and they don’t need to do things like this
to make a living.
Therefore, a capable engineer who has
other options, has to enter an unpleasant task of indeterminable length
to even try to find the solution. How would such a project be funded with
no one to do it and not enough time to devote to it?
For these reasons, we do not expect
and have not found another group working on this approach with a long-term
effort. |