FFTs in Grahics and Vision Rotational and Relectie Symmetry Detection
Outline Reresentation Theory Symmetry Detection Rotational Symmetry Relectie Symmetry
Reresentation Theory Recall: A rou is a set o elements G with a binary oeration oten denoted such that or all h G the ollowin roerties are satisied: Closure: h G Associatiity: h= h Identity: There exists an identity element G s.t.: = = Inerse: Eery element has an inerse - s.t.: - = - =
Reresentation Theory Obseration : Gien a rou G={ n } or any G the set-theoretic ma that multilies the elements o G on the let by is inertible. The inerse is the ma multilyin the elements o G on the let by -.
Reresentation Theory Obseration : In articular this imlies that the set { n } is just a re-orderin o the set { n }. Or more simly G=G.
Reresentation Theory Obseration : In articular this imlies that the set { n } is just a re-orderin o the set { n }. Or more simly G=G. Similarly the set { - n - } is just a reorderin o the set { n }. Or more simly G - =G.
Reresentation Theory Recall: A Hermitian inner roduct is a ma rom VxV into the comlex numbers that is:. Linear: For all uw V and any real scalar u w u w w w w. Conjuate Symmetric: For all u V w w 3. Positie Deinite: For all V:
Reresentation Theory Obseration : Gien a Hermitian inner-roduct sace V and ien a set o ectors { n } V the ector minimizin the sum o squared distances is the aerae o { n }: n n ar min n V
Reresentation Theory Recall: A unitary reresentation o a rou G on a Hermitian inner-roduct sace V is a ma sends eery element in G to an orthoonal transormation on V satisyin: or all h G. h h that
Reresentation Theory Deinition: We say that a ector V is inariant under the action o G i G sends bac to itsel: or all G.
Reresentation Theory Notation: We denote by V G the set o ectors in V that are inariant under the action o G: V G V G
Reresentation Theory Obseration 3: Note that the set V G is a ector sub-sace o V.
Reresentation Theory Obseration 3: Note that the set V G is a ector sub-sace o V. I w V G then or any G we hae: and w w
Reresentation Theory Obseration 3: Note that the set V G is a ector sub-sace o V. I w V G then or any G we hae: and w And or all scalars and we hae: w w w w
Reresentation Theory Obseration 3: Note that the set V G is a ector sub-sace o V. I w V G then or any G we hae: and w And or all scalars and we hae: So + w V G as well. w w w w
Reresentation Theory Obseration 4: Gien a inite rou G and ien ector V the ector obtained by aerain oer G: Aerae G G G is inariant under the action o G.
Reresentation Theory Obseration 4: To see this let h be any element in G. We would lie to show that h mas the aerae bac to itsel: Aerae G Aerae G h
Reresentation Theory Obseration 4: To see this let h be any element in G. We would lie to show that h mas the aerae bac to itsel: Aerae G Aerae G Exandin the riht hand side we et: h h Aerae G h G G
Reresentation Theory Obseration 4: h Aerae G h G G By the linearity o the reresentation we et: h Aerae G h G G
Reresentation Theory Obseration 4: h Aerae G h G G Since the reresentation reseres the rou structure we et: h Aerae G h G G
Reresentation Theory Obseration 4: h Aerae G h G G But this can be re-written as a summation oer the set hg: h Aerae G G hg
Reresentation Theory Obseration 4: h Aerae G G hg And since hg=g this imlies that: h Aerae G G G Aerae G
Reresentation Theory Obseration 5: Gien a inite rou G and ien a ector V the aerae o oer G is the closest G-inariant ector to : Aerae G ar min V G
Reresentation Theory Obseration 5: Aerae G ar min V G Since is inariant under the action o G we can write out the squared distances as: G G
Reresentation Theory Obseration 5: Since the reresentation is unitary we can rewrite this as: G G G G
Reresentation Theory Obseration 5: Since the reresentation reseres the rou structure we et: G G G G
Reresentation Theory Obseration 5: Re-writin this as a summation oer G - we et: G G G G
Reresentation Theory Obseration 5: And inally usin the act that the set G - is just a re-orderin o the set G we et: G G G G
Reresentation Theory Obseration 5: G G Thus is the G-inariant ector minimizin the squared distance to i and only i it minimizes the sum o squared distances to the ectors: n
Reresentation Theory Obseration 5: G G Thus is the G-inariant ector minimizin the squared distance to i and only i it minimizes the sum o squared distances to the ectors: n So must be the aerae o these ectors: Aerae G G G
Reresentation Theory Note: Since the aerae ma: Aerae G G is a linear ma returnin the closest G-inariant ector to the aerae ma is just the rojection ma rom V to V G : G G Aerae G
Outline Reresentation Theory Symmetry Detection Rotational Symmetry Relectie Symmetry
Symmetry Detection For unctions on a circle we deined measures o: Relectie Symmetry: or eery axis o relectie symmetry. Rotational Symmetry: or eery order o rotational symmetry. o 45 o Relectie 3-Fold Rotational 8-Fold Rotational
Symmetry Detection For unctions on a shere we would lie to deine a measure o: Relectie Symmetry: or eery lane o relectie symmetry.
Symmetry Detection For unctions on a shere we would lie to deine a measure o: Relectie Symmetry: or eery lane o relectie symmetry. Rotational Symmetry: or eery axis assin throuh the oriin and eery order o rotational symmetry.
Symmetry Detection Goal: Relectie Symmetry: Comute the sherical unction iin the measure o relectie symmetry about the lane erendicular to eery axis throuh the oriin. Rotational Symmetry: For eery order o rotational symmetry :» Comute the sherical unction iin the measure o -old symmetry about eery axis throuh the oriin.
Symmetry Detection Goal: Relectie Symmetries -Fold Rotational Symmetries Model 3-Fold Rotational Symmetries 4-Fold Rotational Symmetries
Symmetry Detection Aroach: As in the D case we will comute the symmetries o a shae by reresentin the shae by a sherical unction. Model Sherical Extent Function
Symmetry Detection Recall: To comute the measure o symmetry o a unction we: Associated a rou G o transormations to each tye o symmetry
Symmetry Detection Recall: To comute the measure o symmetry o a unction we: Associated a rou G o transormations to each tye o symmetry Deined the measure o symmetry as the size o the closest G-inariant unction: Sym G G
Symmetry Detection Recall: Usin the act that nearest symmetric unction was the aerae o the unction under the imae o the rou we ot: Sym G G G
Outline Reresentation Theory Symmetry Detection Rotational Symmetry Relectie Symmetry
Rotational Symmetry Gien a unction on the shere and ien a ixed order o rotational symmetry we would lie to deine a unction whose alue at eery oint oint is the measure o -old rotational symmetry about the associated axis. = =3 =5
Rotational Symmetry To do this we need to associate a rou to eery axis assin throuh the oriin.
Rotational Symmetry To do this we need to associate a rou to eery axis assin throuh the oriin. In articular i we denote by G the rou o - old rotations about the axis throuh : / y z x
Rotational Symmetry To do this we need to associate a rou to eery axis assin throuh the oriin. In articular i we denote by G the rou o - old rotations about the axis throuh the elements o the rou are the y rotations: j R corresondin to rotations about by the anle j /. x j z /
Rotational Symmetry Usin this notation the equation or the measure o -old symmetry o a unction about the axis becomes: Sym G j j
Rotational Symmetry Usin this notation the equation or the measure o -old symmetry o a unction about the axis becomes: Exandin this exression in terms o dotroducts we et: Sym j G j Sym j i G j i
Rotational Symmetry Usin this notation the equation or the measure o -old symmetry o a unction about the axis becomes: Usin the linearity o the inner-roduct we et: Sym j i G j i Sym j i G j i
Rotational Symmetry Usin this notation the equation or the measure o -old symmetry o a unction about the axis becomes: And usin the act that the reresentation is unitary we et: Sym j i G j i Sym j i G i j
Rotational Symmetry We can urther simliy this exression by obserin that the rotation j-i only deends on the dierence between the index j and i. Sym j j i G j i j
Rotational Symmetry We can urther simliy this exression by obserin that the rotation j-i only deends on the dierence between the index j and i. In articular usin the act that eery index in the rane [ can be exressed in exactly dierent ways as the dierence j-i with ji [ we et : Sym j j i G j i j
Rotational Symmetry Thus the measure o -old rotational symmetry about the axis can be comuted by tain the aerae o the dot-roducts o the unction with its rotations about the axis. Sym G j j
Rotational Symmetry Sym G j j So comutin the measures o rotational symmetry reduces to the roblem o comutin the correlation o with itsel: Dot R R
Rotational Symmetry Sym G j j So comutin the measures o rotational symmetry reduces to the roblem o comutin the correlation o with itsel: Dot R And this is somethin that we can do usin the Winer D-transorm rom last lecture. R
Rotational Symmetry Sym G j j Alorithm: Gien a unction : Comute the auto-correlation o. For each order o symmetry :» Comute the sherical unction whose alue at is the aerae o the correlation alues at rotations R j/ j.
Rotational Symmetry Sym G j j Comlexity: Comute the auto-correlation: On 3 lo n For each order o symmetry : Comute the sherical unction: On For a total comutational comlexity o On 4 to comute all rotational symmetries.
Rotational Symmetry Sym G Comlexity: Note: There is a lot o redundancy Comute the auto-correlation: On 3 lo n in this comutation. For each order o symmetry : Comute the sherical unction: On j j For a total comutational comlexity o On 4 to comute all rotational symmetries.
Rotational Symmetry Sym G Comlexity: Note: There is a lot o redundancy Comute the auto-correlation: On 3 lo n in this comutation. For Examle: each order Comutin o symmetry the -old : symmetries Comute the sherical unction: On we re-comute the -old symmetry allowin or a -old imroement in eiciency For a total comutational comlexity o On 4 to comute all rotational symmetries. j j
Outline Reresentation Theory Symmetry Detection Rotational Symmetry Relectie Symmetry
Relectie Symmetry Gien a sherical unction we would lie to comute a unction whose alue at a oint is the measure o relectie symmetry with resect to the lane erendicular to.
Relectie Symmetry Relections throuh the lane erendicular to corresond to a rou with two elements: G Id Re
Relectie Symmetry Relections throuh the lane erendicular to corresond to a rou with two elements: G Id Re So the measure o relectie symmetry becomes: Sym G Re Re
Relectie Symmetry Relections throuh the lane erendicular to corresond to a rou with two elements: G Id Re So the measure o relectie symmetry becomes: Sym G Re Re
Relectie Symmetry How do we comute the dot-roduct o the unction with the relection o throuh the lane erendicular to?
Relectie Symmetry How do we comute the dot-roduct o the unction with the relection o throuh the lane erendicular to? Since relections are not rotations they hae determinant - we cannot use the alues o the autocorrelation.
Relectie Symmetry General Aroach: I we hae two relections S and T we can set R to be the transormation: R TS
Relectie Symmetry General Aroach: I we hae two relections S and T we can set R to be the transormation: R TS Since S and T are both orthoonal the roduct R must also be orthoonal.
Relectie Symmetry General Aroach: I we hae two relections S and T we can set R to be the transormation: R TS Since S and T are both orthoonal the roduct R must also be orthoonal. Since both S and T hae determinant - R must hae determinant.
Relectie Symmetry General Aroach: I we hae two relections S and T we can set R to be the transormation: R Thus R must be a rotation and we hae: RS TS so T is just the relection S ollowed by some rotation. T
Relectie Symmetry General Aroach: Thus i we comute the correlation o with some relection S : Dot R S We can obtain the dot roduct o with its relection throuh the lane erendicular to by ealuatin: Re Dot S R Re S Rotation S
Relectie Symmetry Sym Alorithm: Gien a unction : G Dot Re Comute the correlation o with the relection S Comute the sherical unction whose alue at is the aerae o the size o and the dot-roduct o with the rotation o S by Re S -. S S
Relectie Symmetry Sym Comlexity: G Dot Re Comute the correlation: On 3 lo n Comute the sherical unction: On S S For a total comutational comlexity o On 3 lo n to comute all relectie symmetries.
Relectie Symmetry Sym Comlexity: G Dot Re Comute the correlation: On 3 lo n For comutin relectie symmetries the comutation o the correlation is oerill as we don t use most o the correlation alues. Comute the sherical unction: On For a total comutational comlexity o On 3 lo n to comute all relectie symmetries. S S
Relectie Symmetry There are many dierent choices or the relection S we use to comute: Dot S T
Relectie Symmetry There are many dierent choices or the relection S we use to comute: Dot T The simlest relection we can use is the antiodal ma: S S
Relectie Symmetry The adantae o usin the antiodal ma is that it maes it easy to exress Re S -.
Relectie Symmetry The adantae o usin the antiodal ma is that it maes it easy to exress Re S -. In articular ixin a oint we can thin o S as the combination o two mas: A relection throuh the lane erendicular to and A rotation by 8 o about the axis throuh.
Relectie Symmetry Thus a relection throuh the lane erendicular to can be exressed as the roduct o the antiodal ma and a rotation by 8 o around the axis throuh : Re R S
Relectie Symmetry Thus settin S to be the antiodal ma we et: Sym G R S
Relectie Symmetry Usin the sherical harmonics we et a simle exression or S :
Relectie Symmetry Y Im Y Y Re Y Im Y Im Y Y Re Y Re Y Im Y 3 3 Im Y 3 Im Y 3 Y 3 Re Y Re 3 Y 3 Re 3 Y 3
Relectie Symmetry Usin the sherical harmonics we et a simle exression or S : S l l ˆ l m l m l Y m l
Relectie Symmetry Additionally i the unction is antiodally symmetric: the equation or the relectie symmetries becomes: S Sym Sym R S R G G
Relectie Symmetry Additionally i the unction is antiodally symmetric: the equation or the relectie symmetries becomes: S Sym Sym R S R G G
Relectie Symmetry Additionally i the unction is antiodally symmetric: the equation or the relectie symmetries becomes: S Sym Sym R S R G G
Relectie Symmetry That is in the case that is antiodally symmetric the relectie symmetries o and the -old rotational symmetries o are equal. -Fold Symmetry Benzene Relectie Symmetry