First, [[define precisely what the computer is looking for|Rigorous definitions]].\n\n\nNext, start simple: What's the simplest method that you can imagine for finding symmetries?\n\n\nFinally, think about computation: how much time will your naive method take? [[How could we speed that up?|Speeding up symmetry detection]]
\nA few examples:\n\nShape editing:\nhttp://www.youtube.com/watch?feature=player_embedded&v=5jIuaMP7Nao#t=187s\n\nShape compression:\n<html><a href="compression.png">compression.png</a></html>\n\nShape search:\n<html><a href="shapesearch.png"/>shapesearch.png</a></html>\n\nFilling in missing details of scanned shapes:\n<html><a href="scan_shape_completion.png">scan_shape_completion.png</a></html>\n\nExtending a shape?\n<html><a href="extendshape.png">extendshape.png</a></html>\n\nLet an artist increase or decrease symmetry?\nhttp://www.youtube.com/watch?v=ZNwr9TfRKkM&feature=player_detailpage&list=UUtDgwcfWbowb3fvfcJ5JAFQ#t=177s\n\nAutomatic viewpoint selection for objects in a database?\n<html><a href="viewpoints.png">viewpoints.png</a></html>\n\n
<html><img src="basic_distance_idea.png"/></html>\n\n- How about: Distance from points on M to closest points on T(M)? (Or vice versa?)\n\n- Do we care about the average distance? Average squared distance? The 'worst case' deviation?
Some ideas: \n\nRandom sampling?\n\nIdentify 'features' that reduce the number of possible symmetry transformations you need to consider:\n<html><a href="butterfly_ex.png">clustering.png</a></html>\n\nMake rough, approximate guesses, then refine them:\n<html><a href="http://groups.csail.mit.edu/graphics/classes/6.838/F01/lectures/IterativeAlgs/ICP/stages.html">iterative closest point</a></html>\n\nAsk a user? \nhttp://www.youtube.com/watch?feature=player_detailpage&v=vda2RAEuW_g#t=2s\n
Creatures can be highly symmetric, but only in a symmetric pose.\n\nGiven a creature in general pose, how can we compute its symmetries?\n\n<html><img src="intrinsic.png"/></html>\n\nTwo ideas:\n\n1. Change the pose to be symmetric? <html><a href="http://www.youtube.com/watch?feature=player_detailpage&v=ZNwr9TfRKkM&list=UUtDgwcfWbowb3fvfcJ5JAFQ#t=86s">Symmetrization</a></html>.\n\n2. Change our notion of symmetry to include more general transformations, so we can handle arbitrary poses directly. Which 'more general transformations' should we allow?\n\nTypes of symmetry: \nhttp://www.youtube.com/watch?feature=player_detailpage&v=eO_ZACTJtL0#t=12s
For noisy scan data, we can establish a probability of a given symmetry being correct.\n\nAn idea due to Thrun: \n<html><img src="scan_symmetry_eval.png"/></html>\n\nMatching points => Evidence in favor of symmetry\n\nPoints that 'conflict' with other points => Evidence against symmetry\n
How we can use computers to find symmetry in 2D images and in 3D models?\n\n<html><img src="palace.jpg" width=70%/><img src="alligatorfullcolumn.jpg"/></html>\n\n\nWarm up:\n\n[[Why are we interested in having the computer find symmetry?|Applications]]\n\n----\n\n[[How can a computer find symmetry?|How to detect symmetry]]\n\n----\n\nLet's look at an overview of a complete solution: \n\nhttp://www.youtube.com/watch?list=UUtDgwcfWbowb3fvfcJ5JAFQ&feature=player_detailpage&v=ZNwr9TfRKkM#t=4s\n\n----
First, decide what kind of symmetry we care about:\n-- Global symmetry?\n-- Partial symmetry?\n-- [[Pose-invariant symmetry|Pose invariant]]?\nWhat are these, and when would you look for each?\n\n\nNow, decide [[how to measure that symmetry|How to measure]].
<html><img src="symmetrydist_ex.png"/></html>\nKahzdan's idea: The distance between the image and the nearest symmetric image.\n\nWould this also work for shapes?
Using computers to find symmetry in 2D images and in 3D models
Symmetry in real data is rarely exact. The computer needs a low-level way of quantitatively evaluating "how symmetric" a shape is.\n\nFor example, if we transform a shape:\n<html><img src="basic_distance_idea.png"/></html>\n[["How symmetric" is M with respect to T?|Symmetry measure on shapes]]\n\n\nWhat about the symmetry of an image? [[How symmetric is this image with respect to a mirroring across the grey line?|Image symmetry]]\n<html><img src="symmetrydist_image.png"/></html>\n\n[[What about the symmetry of noisy data collected by sensors?|Scan symmetry]]\n
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