Size Functions are a new kind of mathematical transform. More precisely each Size Function is the output of a mathematical transform that takes a topological space X and a function from it to the real numbers as an input. Each Size Function is a function from the real plane to the extended natural numbers (extended means that the infinite value is allowed). An example can make all this clearer:

Usually the topological space X is an image and the function from X to the real numbers (called

How much time does it take to compute a Size Function?

What is the complexity of computation of Size Functions?

You can also download SizeTool 1.0 , a tool for the programmers all written in standard C.

You can use Size Functions to study any topological space. In practice, it means that you can compute Size Functions of B/W pictures, color pictures, sound waves, multidimensional medical plots, subsets in the Euclidean Space and anything that can be seen as a topological space. We give a simple example:

We point out that different measuring functions generate quite different Size Functions.

Are Size Functions resistant to occlusions?

Size Functions have been defined in such a way that they can potentially resist to non-destructive noise and occlusions. The important fact is that the information in Size Functions is

In the following example you can see three images containing a wrench and a partially occluded wrench, and a fourth image where only a nutcracker appears. If you consider the topological space of all pairs of black points in each image, and measure each pair

A simple way for comparing Size Functions is to compute the distance in the space

Another way for comparing Size Functions can be obtained by observing that the restriction of a Size Function to a compact set K coincides (almost everywhere) with the restriction to

You can find some information about this subject by downloading the gzipped PostScript file standard.ps.gz.

There have been concrete applications to the recognition of hand-drawn sketches (the already mentioned program SketchUp), of tree leaves, of handwritten numerals, of letters in the sign language, of toy cars, of white blood cells, of characters and writers in on-line handwriting, and to the measure of coronary tortuosity. Signature validation and ECG interpretation are our next targets.

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You can download free software about Size Functions at the URL: http://vis.dm.unibo.it/sizefcts/files.

Before computing Size Functions you could use some reduction processes. You can download a gzipped PostScript preprint describing two of such techniques, capable of simplifying Size Graphs without changing the corresponding Size Functions. See also the paper

SketchUp is a demonstrative program for Windows 3.1 and Windows 95/98. It shows the capability of Size Functions in describing shapes. Draw the profile of an object chosen in this set: nut, screwdriver, scissors, drill, compass, wrench, hammer: SketchUp will try to recognize your choice. You can freely download SketchUp 2.0 (file sketchup.tar - approximately 4Mb) at the URL: http://vis.dm.unibo.it/sizefcts/files/

SizeTool is a tool for the programmers all written in standard C (a part from the use of dirent.h libraries); you can find the Size Functions engine, some measuring functions implemented and the main code to use and correlate them.

You can freely download SizeTool 1.0 (file SizeTool.1.0.tar.gz - approximately 120Kb) at the URL: http://vis.dm.unibo.it/sizefcts/files/

Please send all questions to frosini@dm.unibo.it