Circle Tool - Cookie Cutter Button.
11/5/2003
All of the cookie cutter commands work on the image selected by the Image! button.
Overlap, No overlap
For cutting maximum inscribed circles out of an arbitrary shape. Repeatedly cuts circles out of a shape by finding the maximum inscribed circle, removing it, then working on the remaining shape.First, erases the circles list. The slow mode recalculates the distance mapevery time so that the circles do not overlap. This is the correct way to do things, but takes much longer. The fast mode just cuts a circle out of the distance map, then finds the global max of what is left. This results in many circles being centered on the edge of a previous cut out, thus many circles overlap. On cursory examination, it seems that the size distributions of the circles are about the same, although the overlapping set has more of them.
The largest circle is the same in each set.
Radius images
These images have pixel values that are calculated in some way from the Euclidian distance map. The pixel value is that of the radius of the circle that contains the pixel. The pixel will likely not be at the center of the circle. The cookie cutter radius image is the overlapping algorithm from the above menu item. The max disk radius image is the radius image calculated by accumulating disks, in order of radius, for every object pixel (every pixel with a value for the distance map.
Background
The radius images are for calculating an average width of an arbitrarily shaped region, after Hildebrand:
Here is Figure 1 from this paper:
and here is some of the text about the figure.
The radius images are two-dimensional adaptations of this idea. They have been tested using images similar
to those in Fig. 2 of the article:
The figure below shows a mask of two overlapped disks, the distance map, the max disk image and the
Note that the max disk image tapers between the two radii values in the region corresponding to the hashed region of Fig. 2 from the article. The cookie cutter image does not, but otherwise it is the same.
The next example is one focal plane of a confocal microscope image series of a collagen fiber network. We wish to determine an average measurement of the ‘width’ of the black areas, as the average of the max disk image. This series shows the original image, the max disk image, the cookie cutter disk image, and the difference of the two disk images. The disk images were made from the original by thresholding it at value 150 (thresholded image not shown).
Original
Max width disk image
Cookie cutter disk image
Difference of max width image and cookie cutter image. Light gray = 0. Note that the differences of the cookie cutter approximation with respect to the max width image are largely around the edges of the disks.
Here are some comparative statistics for the two images.
| statistics for |
maxw |
maxw |
cc |
cc |
| n pixels |
262144 |
262144 |
||
| total n pixels |
262144 |
262144 |
||
| pixel value mean |
42.53457 |
41.14106 |
||
| pixel value sum |
11150182 |
10784882 |
||
| pixel value min |
0 |
0 |
||
| min value location |
0 |
0 |
0 |
0 |
| pixel value max |
136 |
140 |
||
| max value location |
0 |
374 |
0 |
371 |
| min positive pixel value |
1 |
1 |
||
| min pos val location |
142 |
0 |
33 |
0 |
| pixel val wtd centroid |
180.7061 |
288.1245 |
179.7647 |
290.8076 |
| sample std. dev. |
46.3427 |
45.27816 |
||
| std. error of mean |
0.090513 |
0.088434 |
||
| variance / mean |
50.49177 |
49.83128 |