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I have a set of 1000 images of the dorsal side of the hand. I need to identify the different joints in the fingers and measure the distance between them. I have already cropped the images to same dimensions, centered the hand, and labeled the required points.

Can someone point me to the next steps and a general direction I should be looking at for training the machine I need? Which algorithms would be the closest for my need?

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Convolution Neural Network (CNN) is the state of art algorithm for your problem. The standard terminology for this kind of problem is "feature detection".

Here is a good resource for the basics. http://cs231n.stanford.edu/

Your biggest problem probably is going to be the size of your training images. 1,000 is most likely not going to be enough. You should try to augment your data by rotating and translation etc.

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