I am working on automated counting and one of my solutions is the use of the template matching algorithm (specifically using Chamfer Matching Algorithm). However, granted it is a template matching approach, I need to establish my template. The objects I am trying to count are individual coconut trees. Based on the definition of coconut tree crowns, they are star-shaped with feather-like leaves. I had the idea in my mind that I could derive that given a polar equation. However, since I am not a Math major, I looked for polar equations that could describe the above mentioned description and stumbled upon the link below An equation that generates a beautiful or unique shape for motivating students in mathematics and noticed the answer given by Luscian.
My question is, is this equation
$$r(t)=|cos(nt)|^{sin(2nt)}\text{ for }2n\text{ in between 1 and 8 and }t\in(0,2\pi)$$
an already defined one? I am not familiar with this, since my background on polar equations reached only up to cardiods and rhodonea curves.
Any help is greatly appreciated. Thanks!
No, these aren't classical curves, and IMO they are uselessly contrived and asymmetric.
You could consider using a rose instead https://en.wikipedia.org/wiki/Rose_(mathematics). Considering the probable high variability/complexity of the crowns, the exact shape of the "petals" doesn't matter.
I also believe that this approach is flawed, because even when seen from above, coconut trees do not exhibit a regular pattern at all, and a regular template will overlap both leaves and voids. You will need a deformable template, for which a "nice" equation doesn't work.