Posted by
John Brear on
Dec 11, 2016; 4:41pm
URL: http://imagej.273.s1.nabble.com/Ever-decreasing-circles-tp5017755p5017757.html
Hi Gabriel,
I appreciate your reply. We certainly agree that discrete space needs
careful handling!
The essence of the problem is that circles do not exist in discrete space,
so limits on theoretical circularity are irrelevant to discrete shapes.
A pixel has an area and a perimeter.
If it did not, then two pixels would have neither area nor perimeter, and
three pixels would have neither area nor perimeter...
...and 122,880 pixels would have neither area nor perimeter. So my image
would have neither area nor perimeter!
But induction apart, IJ demonstrably reports an area of 1 and a perimeter of
2.828.. for a feature occupying one pixel. It also reports a circularity of
1, but the value calculated from the reported area and perimeter is 1.5713..
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.
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.
On the practical question of magnification, we always need to compromise
between the resolution and characterisation of small features and the
analysis of a truly representative area - at proportionate effort! Our
characterising features span over 2 orders of magnitude in linear dimension.
I shall look at your plug-in and refer to the earlier discussions - thanks
for the link!
Best wishes
John
PS ...and apologies that my results table appeared as a paperchain!
-----Original Message-----
From: ImageJ Interest Group [mailto:
[hidden email]] On Behalf Of
Gabriel Landini
Sent: 11 December 2016 15:15
To:
[hidden email]
Subject: Re: Ever decreasing circles
Hi,
On Sunday, 11 December 2016 13:38:23 GMT John M Brear wrote:
> Calculating the circularities of our observed precipitates shows that
> we would obtain much better discrimination by selecting on circularity
> values greater than 1. At present ImageJ does not allow this.
Not sure I understand your question, but that seems impossible from the
definition of circularity. A circle has the maximum theoretical circularity
of 1.
Regions with a circularity >1, would mean that a region has a shorter
perimeter than a circle enclosing the same area.
To avoid large errors generated on small regions, you could take images with
higher magnification, so the pixellation error are relatively less
prominent.
However, the discretisation problem, will not go away. You cannot get
circularity of 1. While you get closer and closer to the true area of your
region by increasing magnification, the perimeter is always overestimated
due to the polygonation of the discrete representation of your objects.
> it appears that perimeters in ImageJ are of the form a + b x sqrt(2),
> where a, b are integers. This gives a limiting circularity (for a
> single pixel) of Pi / 2.
A single pixel has no circularity because it is a sample (with neither area
nor perimeter), so I would not make assumptions as what is the limiting
circularity in that case.
IJ uses the number of pixels as area and the length of the polygon as the
perimeter. There are alternatives to this. For example my plugin
(Particles8, downloadable from my page) uses the area inside the perimeter
polygon as the estimate of "area". The circularity values are slightly
different from those in IJ The problem of the length of the perimeter,
however, remains, but this is well understood and there have been many
papers on this and the representation of "digital lines".
Freeman's papers in the 1960s are the classic reference to this. This was
discussed a bit in the ImageJ Forum some time ago and I posted some more
references
http://forum.imagej.net/t/polygon-mesh-boundary/3153Strange things happen in discrete space! :-)
Cheers
Gabriel
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