Maybe you should utilise a ven digram with shape,location size, etc. to derive a basic hypothosis. I always tuaght my students to go back to nuts and bolts before jumping in. Time is an iomportant factor here.
Think outside the square then square it.
As a practical matter, there are two basic problems involved. First is the problem of looking at a photo and interpreting what is in it - i.e. whether a blob is a leaf or a pile of leaves, tracing the outline of a leaf as a curve, etc. If you rely on human beings to do this, it would be very labor intensive. So it would be nice to have computer programs to do the job. However, the mathematics needed to program computers to recognize what is in a picture is not perfected although there are many methods that work in special situations. Efforts to solve the problem are currently topics in Computer Science rather than topics in Mathematics. The general topic is "Pattern Recognition". Specific methods would be "Neural Nets", "Fuzzy Classification", "Attention Selection".
Let's assume you have gotten data about types of leaves or curves that outline leaves. Telling whether two curves are "similar" is a special case of the first problem, which was recognizing what is in a picture. I think I can find Computer Science papers written about this problem, but it is not a separate branch of Mathematics. If you have data that does not represent a shape (such as leaf length, width, area,... etc.) then the mathematics called "Statistical Pattern Recognition" is applicable. (Don't restrict your thinking to an x-y graph in two dimensions.) Statistical Pattern Recognition may not do a job. Whether it works depends on whether your data really has enough information to solve our problem. (The mathematical field called "Statistics" is not the same as "Statistical Pattern Recognition".)
Ok, let's say you do. But are you doing conceptual daydreaming or are you actually going to do something concrete? If you are going to do something concrete then you have to be specific about the data that you have. Can you give an example of the type of data that you are imagining?
Leaves are concrete things, but your descriptions of the data you intend to collect about leaves is not specific. Give a numerical example of the type of data you are talking about if you conceptualize anything specific. Otherwise, you are just daydreaming. It's fine to daydream, but you don't need our help to do that.
We have to be clear about what "this problem" is. Does it have to do with the idea of estimating the "forrest type" from the leaf piles? We can use your model of leaf piles, if you wish. But if you want to estimate forrest type, we have to define what that means.
If you want count all possible types of leaf piles, that's a different problem. It doesn't necessarily have anything to do with estimating the properties of forrests.
I'll make up a more specific example. Let's say that I am reading some books that were written by hand. In these books, three pronouns appear a lot - "item," "she," and "he". I am only interested in pages that have a combination of 10 or more of any of these words on a page, and I'm only interested in comparing the order that the first 10 of these words appear on each page for the first 10 pages of each book on which these pronouns appear in this quantity. I am also interested in representing how the part of "e" without the "-" that looks like a "c" in each of these words changes in each word. Specifically, I'm not so much interested in the way that the handwritten curve that looks like a "c" in the letter "e" in the word "item" changes when it appears next in the word "she" or "he" on the same page. Rather, I'd be interested in the way that this curve changes subsequent times that it appears in the letter "e" in the same word - in this case the word "item" – but in other cases the word “she” when I am comparing the appearances of the word “she” and the word “he” when I am comparing the appearances of that word. I want to group books according to these patterns - the order in which the these three pronouns appear starting from the beginning of each page, how the curve that looks like a "c" but is really part of the letter "e" changes in these words as described, and how these patterns change for the first 10 pages of books. I just want to identify patterns, but maybe these patterns could also be compared to categorize how people have related to words that have anthropomorphic meanings or meanings that have to do with gender in books that were written for people of different ages during different historical periods.When you say things like "how curves that had a minimum length, were least straight, and made up the letters changed from letter to letter in the word – say, from left to right on the page", you aren't saying anything specific enough for a mathematician or programmer to use. No, I can't help you define the problem. I'm not a mind reader. And I don't think you have a clear definition of what you want to do in your mind.
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