# Finding similarity between the objects ( Actually how to justify the similarity of objects in mathematics form )

I have developed a system to trace the outlines of (images of) objects. Now I want to test whether two independent traces represent a common feature.

Imagine two people (or machines) tracing the outline of a feature in an image, recording it as sequence of vertices. Inaccuracies in recognizing the feature boundaries and in specifying the vertices can be viewed as random errors in vertex positions. The problem is the two traces might use completely different (Cartesian) coordinate systems (set up on two digitizing tablets, for instance). The null hypothesis to test is that they represent a common feature.

This is illustrated below. I drew a figure and recorded the $x$ and $y$ co-ordinates of its vertices in its coordinate system. Let's call this figure $M$. It is represented as a sequence $(x_i,y_i), i=1, 2, \ldots, m$.

Then I drew the same figure in a bigger size in another coordinate system (with no known relationship to the first coordinates system) and recorded the $x$ and $y$ co-ordinates. Let's call this $N$, represented as a sequence $(x_i^\prime, y_i^\prime), i=1, 2, \ldots, n$.

Question: Having these data, how can I test whether the figures $M$ and $N$ represent the same image features even though they have different sizes and co-ordinates?

If the conclusion is that yes, they do represent a common object, then how can I estimate a similarity transformation between $M$ and $N$ so that I can work on formulas or equations to check the results with different figures?

• This question might be on topic, but it is unclear what is going on. Your sketches won't be random, to begin with. More importantly, it is not evident what you are really doing when you "capture the coordinates," nor is it apparent how the scales and coordinates differ between sketches. If you find a way to communicate what you are doing, please edit your question accordingly so that the community can vote to re-open it.
– whuber
Apr 29, 2014 at 15:51
• We still need a more precise sense of "similarity." Do you mean similarity in the Euclidean sense that there exists some combination of a translation, rotation, and uniform scaling that sends $M$ to $N$? If not, exactly what does "M ≈ N" mean? Are you perhaps asking how to measure a degree of similarity when the vertices of both $M$ and $N$ are traced with random-looking errors or how to test whether $M$ and $N$ might be representations of the same underlying geometric object?
– whuber
Apr 29, 2014 at 16:36
• You really need to address @whuber's specific questions. If you can't understand them, then so be it, but you're not making this question statistical in any sense so far as I can see. It's a question on (computational) geometry, it seems. Apr 29, 2014 at 16:40
• @Nick Actually I see a potentially very interesting statistical question--provided I am correctly interpreting it. Imagine two people (or machines) tracing the outline of a feature in an image, recording it as sequence of vertices. Inaccuracies in recognizing the feature boundaries and in specifying the vertices can be viewed as random errors. The problem is the two traces might use completely different (Cartesian) coordinate systems (set up on two digitizing tablets, for instance). The null hypothesis to test is that they represent a common feature. Shakthydoss, am I close?
– whuber
Apr 29, 2014 at 16:47
• I have incorporated your comments and my preceding comment into a major edit to the question. Please read it over and correct any misunderstandings I may inadvertently have introduced.
– whuber
Apr 29, 2014 at 17:19