Archive for the ‘nucleator’ Category

A more general way to examine the nucleator and the circle

January 30, 2010

The nucleator is a local stereological probe.  Being a local probe means that the nucleator is based on the selection of a reference point. The probe, i.e. the geomtrical shape that is used to sample, is constructed relative to the reference point. A nucleator probe is a ray that starts at the reference points and projects outward. The nucleator is an unbiased probe because there is some essence of randomness to the sampling. That is accomplished by making the ray IUR. The abbreviation IUR stands for isotropic uniform random. That’s a fancy way of saying that all directions in space have the same chance of being chosen. On a flat surface a ray should have the same chance of choosing any of the directions from 0 to 360 degrees.

The measure of the nucleator is the positions along the ray where the ray crosses the boundaries of whatever is of interest. If the nucleator’s reference point is inside of the object of interest, and the ray goes out and never crosses the boundary again, then the measure is sort of like the length of the ray inside of the object.

In the case of the circle it is possible to come up with a formula that relates the circle and nucleator and the intercepts.

  1. The circle is defined by its size which is its radius. That will be called r.
  2. The nucleator is defined by its reference point and the orientation of the ray. These will be shown to be c and theta.

The first step is to work out the circle. The circle is defined to be centered at (0,0) and to have a radius r. A rather simple equation defines the point (x,y) that lie on the circle. Remember that a circle is the curve. A disc is the area inside of a circle.

The nucleator is a bit trickier. A circle is radially symmetric. No matter how it is rotated it is still the same. That allows us to simplify the choice of point inside of a circle. All points a distance c from the center of the circle are really the same. Therefore, all reference points inside of the circle a distance c from the center are essentially the same. Think about drawing a line through the reference point and the center of the circle. The line is a diameter of the circle. Use this line as the x-axis of a coordinate system.  For another point also a distance c from the center use a different coordinate system. The angle theta is the angle between the ray and the x-axis.

Knowing r, c, and theta it is possible to derive a formula that gives the intercept distance I.

The mean of this formula is found as follows:

Copyright of the author - click to view

It should be fairly obvious that this is not related to the circumference of the circle. Actually this result gives the correct value if c=0. But what if I squared was used instead of I? The mean of I squared is:

Copyright of the author - click to view

This result, r squared, is in units-squared. Length requires units. The square root of this result is proportional to the circumference of the circle.

The question is: Does this work for shapes other than the circle?

The answer is seen to be a resounding NO. All stereologists already know that perimeter is related to another measure, not the mean intercept length or the mean intercept squared, or on any variation of that except for a few trivial cases.

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The special case of the nucleator and the circle

November 24, 2009
Two plots of a circle: with 4-way nucleator and finding the center

The daisy fails, but do things work for a circle

The 4-way nucleator, when clicked inside of a convex profile creates 4 lengths – one for each ray of the nucleator. In this case we examine the circle, the simplest of shapes.

A circle is simple, because it is isotropic. Rotate the circle and nothing changes. The circle looks the same. That makes many questions about the circle easy to answer.

Here an arbitrary point is selected inside of the circle. Four rays start at the arbitrary point and intercept the circle. Because the circle is isotropic it is possible to simplify the math and use rays parallel to the coordinate axes. Also, without loss of generality let’s select the arbitrary point from the first quadrat.

These two simplifications mean that the lengths show in the drawings, a, b, c, and d, can be used to identify the 4 intercepts as (0, a), (b, 0), (0, -c), and (-d, 0). The axes cross at the arbitrarily chosen point. That point has coordinates (0,0). The intercepts on the y-axis are at a and -c. The x-axis intercepts are at -d and b.

So the question is whether or not the information given here is capable of estimating the circumference of the circle? Yes. Let’s see how that is done.

<insert math>

In general, the nucleator cannot be used to estimate perimeter. There is at least 1 case in which the nucleator does work. Are there other cases? The answer to that is unfortunately no. No shape other than a circle has a perimeter that can be estimated by the nucleator  without bias.

Can’t the nucleator work for special cases?

November 20, 2009

It’s easy to show that there are cases in which the nucleator fails to estimate perimeter. The nucleator involves measurements of radius. The mean radius is not related to perimeter. That has been known for at least 70 years.

So now I’m asked, “None of my work deals with things that look like gears. I’m not trying to estimate the perimeter of a daisy or an octopus. I’m not looking at the perimeter of a leaf. My things are all convex. They are simple blobs.  Can’t the nucleator work for those kinds of shapes?”

This is a very good question. Unfortunately, the answer is NO.

I reply, “Think about a circle. Imagine that the reference point is at the center. The measured radius is the same in all direction. That means that the mean radius is the same as any of the measured radii. The perimeter can be computed knowing the radius. Just multiply by 2 pi.”

“Think about stretching or squishing a circle and you get an ellipse. The ellipse is tricky in that the formulas for the mean length and the perimeter involve nasty math. But what you find out is that the mean radius measured from the center of the ellipse multiplied by 2 pi is the perimeter of the ellipse.”

That sure sounds encouraging doesn’t it? The only problem is that this is not true for any other point in the ellipse. It’s not true for any other point in the circle.

“So you’re saying that the perimeter can’t be figure out from the nucleator?”

“That’s the trouble with math. It tells you what works and what doesn’t work.”

Estimating perimeter with the nucleator

November 17, 2009
Gear in metal lid along with notes on a napkin

Gear inside a metal lid

A while back I was asked why the nucleator does not estimate perimeter. My answer at the time was off the cuff – the math doesn’t work out. I knew that this was correct, but had not really looked into the matter.

The person asked me again if it were possible to use the nucleator to estimate perimeter. I needed a quick demonstration that it was simply not possible to use the nucleator probe for that purpose.

I drew a circle on a piece of paper.

“See this circle”, I said. “The circle is interesting because it has the least perimeter for the area that is enclosed. A circle is easy to draw, because all you have to do is use a compass or string to keep the radius fixed and swing around a point to draw the circle.”

I saw a glimmer of understanding, a glimmer of what I was about to do.

“Think of a gear,” I began. “Let me draw a gear inside of the circle. It is pretty clear that the radius of the gear is always smaller than the radius of the circle. Despite it being smaller it should be clear that I can make the perimeter of the gear as long as I want by getting as jaggy as I want.”

“That’s really obvious”, was the response.

In a few strokes of a pen I was able to demonstrate that the radius of the object was not related to the perimeter.

“OK?”, was the next thing I heard.

Sure it was true that the radius did not seem to be related to the perimeter if the measurements were taken from the center of the figure, but I could see what was coming next.

“You measured the radius from the center, didn’t you?”

“That’s right”, I said.

“But isn’t the nucleator unbiased on average?”

Here was the problem: why is it claimed that the nucleator is unbiased?

“There are two types of sterological probes. Most people are familiar with the probes like the counting frame or points. The other type of probe is part of local stereology. Local does not mean for small things. It means that the probe is based on the use of reference points. In the case of the nucleator the reference point is the nucleolus. The nucleator must be unbiased for the estimates made at every possible point, not the estimates made across all points.”

“To be unbiased the nucleator has to be unbiased at the center, and at every other point that is chosen. So if the nucleator is not unbiased at the center, then it is not unbiased.”

I almost missed that important qualifier.

“I meant to say unbiased in estimating perimeter.”