Re: Ever decreasing circles

Posted by John Brear on Dec 12, 2016; 2:37pm
URL: http://imagej.273.s1.nabble.com/Ever-decreasing-circles-tp5017755p5017763.html

Thanks, Gabriel,  for the reference - very worthwhile reading, but it needs
a bit more explicit maths to convince me fully!

Happy to accept pixels as points mathematically - ie a number-value encoding
all information of interest at a location in 2(or more)-space (and time).
But what does such a 'point sample' sample and encode?
In my optical microscopes the detector is an array of sensor elements,
regular as in a CCD or CMOS, or quasi-regular as the crystals in a
photographic emulsion or the rods and cones in my retinas.  Whatever type,
each sensor element has a finite area which can be mapped via the laws of
classical optics to a finite area on the object being viewed.  So the
mathematical-point pixel is a sample over (not from) a real-world area -
albeit encoded as a 'point' value.

We must not let our concepts of the real world constrain our mathematical
representations.
Nor must we let our mathematical representations constrain our understanding
of how the world actually works.

I hadn't intended to open a can of philosophical worms, but I suspect most
IJ users (myself included) need to be more aware of these concepts and their
implications.

Attached is a typical image - converted to 8-bit.  You can see that several
types, sizes and spatial distributions of particles are present.
Thresholding at the default level of 109 and analyzing particles with
appropriate parameters set yields a table with area, perimeter and
circularity data for 2101 particles.    The list won't accept it as an excel
or pdf file, I'm afraid, so I attach an image of the first 40 results.
I added a column which calculates circularity from the reported area and
perimeter values, using the conventional formula, then sorted these into
ascending order and plotted against cumulative fraction, yielding the
attached graph.  1061 particles have calculated  circularities greater than
1.
To be explicit, particle 37 has an area of 1 and a perimeter of 2.828,
giving a calculated circularity of 1.5713 - whereas a value of 1 is
reported.
My interest in exploring the high circularity region becomes clear.  What is
going on at 1.08?

The calculated circularities have distinct values, which naturally arise
from integer areas and perimeters of the form a + b x sqrt(2).
Whatever pixels might be, IJ is measuring my particles as if they were
continuous assemblages of little squares...
...or maybe I'm readily confused.

Whether or not these calculated circularities are strongly meaningful is
beside the point.  Practically they are calculable and show a trend with
interesting features that may betoken something of real interest, or serve
to highlight a processing artifact.
All I wish to do is select my set for analysis in a manner that allows
choice of circularity values >1.

Again, I appreciate your time and responses
Best wishes
John

-----Original Message-----
From: ImageJ Interest Group [mailto:[hidden email]] On Behalf Of
Gabriel Landini
Sent: 11 December 2016 18:13
To: [hidden email]
Subject: Re: Ever decreasing circles

On Sunday, 11 December 2016 16:41:45 GMT John M Brear wrote:
> A pixel has an area and a perimeter.

Some would disagree, see for example:
http://www.cs.princeton.edu/courses/archive/spr05/cos426/papers/smith95b.pdf

> If it did not, then two pixels would have neither area nor perimeter,
> and three pixels would have neither area nor perimeter...

Well that is exactly how Particles8 and 4 work when considering "area" and
the values they return. That being said, they also compute "pixels" but I
think that circularity and other morphometrical parameter are better
constructed using the concept of area as the space inside the polygon
defined by the perimeter. If the polygon defines no area, then some
morphometrical descriptors cannot not exist (you get division by 0 and so
on). That is useful too, e.g.
in mereotopology there are operators for detecting regions with an without
'interior', so no, not everybody believes a single pixel has an area.

Of course one can count the number of pixels a region is made of,  and
assume that 2 pixels side by side have an area of 2, but this then has other
consequences that need to be considered later (such as that no points or
lines exist in digital images).
According to Freeman's papers, two pixels side by side define a line of
length 1, while two pixels joined by their corners define a line of length
sqrt(2).

> ...and 122,880 pixels would have neither area nor perimeter.

Not necessarily. It depends on the connectivity considered between samples
and how we define a "region". If they are all disconnected pixels, that is
also correct, you got 122,880 point samples. If they are somehow connected
forming regions (defining "connected" as the property of being able to
travel between any two pixels in that region through a path that is all
contained in that region), then you can have a perimeter and area associated
to that region.

The smallest "area" you can detect with Particles8 (which uses
8-connectivity, hence its name) is defined by 3 pixels (area of 0.5) in an
"L" configuration.
Its perimeter is 1+1+sqrt(2).
A 2x2 pixel square blob as an area of 1 and a perimeter of 4.

> But induction apart, IJ demonstrably reports an area of 1 and a
> perimeter of 2.828.. for a feature occupying one pixel.

Yes, in IJ the default Particle Analyzer does things differently and it is
important to be aware of those, that is what I have been trying to explain.

> It also reports a circularity
> of 1, but the value calculated from the reported area and perimeter is
> 1.5713..

Have a look at the documentation in the IJ site. It says in page 137 that
"values may not be valid for very small particles". That is because for very
small particles IJ just outputs "1".  I do not know exactly the reason for
this,  I think this was set long ago. Wayne might clarify it.
In practical terms it is not a good idea to use such small regions for the
analysis of captured images, but sometimes we might want to analyse
synthetic images, where you know that detecting a pixel is relevant.

If you need to know exactly the value of circularity, you can either compute
it separately (from the area (which in IJ is same a number of pixels) and
perimeter columns) or you can try the Particles 8 and compute it from either
pixels or area , and the perimeter (the default circ. is computed form the
"polygonal area").  You will note that since a pixel has no area, it cannot
have a circularity (and hence it returns -1).

> I have thousands of imaged particles with IJ reported areas and
> perimeters giving calculated circularities >1 against the reported
> values of 1.  All I need to do is to discriminate in this area.

Can you post just one of those regions? I would be interested in seeing how
a value >1 is generated. How do you compute that value?

> I don't think we differ at all in our understanding of the issues of
> polygonization; the problem is that IJ prevents those of us who
> understand the issues from accessing and using the real values.

I am sure that you understand all this, but sometimes users are not aware of
how every detail is implemented and the logic behind the choices made in the
programs.
Anyway, I hope I helped a bit in explaining how things some things work in
IJ and why I implemented it differently in my plugins. A few people found
the alternative implementation useful too.

Regards

Gabriel

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