Generalized Shearlets and Representation Theory Emily J. King - PowerPoint PPT Presentation

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Generalized Shearlets and Representation Theory Emily J. King Laboratory of Integrative and Medical Biophysics National Institute of Child Health and Human Development National Institutes of Health Norbert Wiener Center University of Maryland February Fourier Talks February 18, 2011 E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Outline Overview 1 Shearlets and reproducing groups 2 Shearlets in L 2 ( R k ) 3 Reducibility and coorbit spaces 4 E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al. ) For f : R k → C , y ∈ R k , and A ∈ GL( R , d ) define the following (unitary) operators T y f ( x ) = f ( x − y ) and D A f ( x ) = | det A | − 1 / 2 f ( A − 1 x ) . Let H be a locally compact Hausdorff topological group, and let π : H → GL ( R , k ) be a continuous homomorphism. Define G = R k ⋊ π H which has product ( y, a ) · ( z, b ) = ( y + π ( a ) z, ab ) . One unitary representation ν of G (the wavelet representation ) is ν ( y, a ) = T y D π ( a ) . We consider systems of the form { T y D A ψ ( x ) : A ∈ M ≤ GL( R , d ) , y ∈ R k } . E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al. ) For f : R k → C , y ∈ R k , and A ∈ GL( R , d ) define the following (unitary) operators T y f ( x ) = f ( x − y ) and D A f ( x ) = | det A | − 1 / 2 f ( A − 1 x ) . Let H be a locally compact Hausdorff topological group, and let π : H → GL ( R , k ) be a continuous homomorphism. Define G = R k ⋊ π H which has product ( y, a ) · ( z, b ) = ( y + π ( a ) z, ab ) . One unitary representation ν of G (the wavelet representation ) is ν ( y, a ) = T y D π ( a ) . We consider systems of the form { T y D A ψ ( x ) : A ∈ M ≤ GL( R , d ) , y ∈ R k } . E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al. ) For f : R k → C , y ∈ R k , and A ∈ GL( R , d ) define the following (unitary) operators T y f ( x ) = f ( x − y ) and D A f ( x ) = | det A | − 1 / 2 f ( A − 1 x ) . Let H be a locally compact Hausdorff topological group, and let π : H → GL ( R , k ) be a continuous homomorphism. Define G = R k ⋊ π H which has product ( y, a ) · ( z, b ) = ( y + π ( a ) z, ab ) . One unitary representation ν of G (the wavelet representation ) is ν ( y, a ) = T y D π ( a ) . We consider systems of the form { T y D A ψ ( x ) : A ∈ M ≤ GL( R , d ) , y ∈ R k } . E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al. ) For f : R k → C , y ∈ R k , and A ∈ GL( R , d ) define the following (unitary) operators T y f ( x ) = f ( x − y ) and D A f ( x ) = | det A | − 1 / 2 f ( A − 1 x ) . Let H be a locally compact Hausdorff topological group, and let π : H → GL ( R , k ) be a continuous homomorphism. Define G = R k ⋊ π H which has product ( y, a ) · ( z, b ) = ( y + π ( a ) z, ab ) . One unitary representation ν of G (the wavelet representation ) is ν ( y, a ) = T y D π ( a ) . We consider systems of the form { T y D A ψ ( x ) : A ∈ M ≤ GL( R , d ) , y ∈ R k } . E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Motivation Commonly multidimensional data was analyzed using tensor products of 1 -dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R 2 . Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory. E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Motivation Commonly multidimensional data was analyzed using tensor products of 1 -dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R 2 . Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory. E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Motivation Commonly multidimensional data was analyzed using tensor products of 1 -dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R 2 . Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory. E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Motivation Commonly multidimensional data was analyzed using tensor products of 1 -dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R 2 . Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory. E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Shearlets Definition (Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005) Given ψ ∈ L 2 ( R 2 ) , the continuous shearlet system is { T y D ( S ℓ A a ) − 1 ψ = a − 3 / 4 ψ ( A − 1 a S − 1 ℓ ( · − y )) : a > 0 , ℓ ∈ R , y ∈ R 2 } , � a � � 1 � 0 ℓ √ a where A a = and S ℓ = . 0 0 1 E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Sheared . . . Figure: Linear Algebra and its Applications , 3rd ed., David C. Lay E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Sheared . . . sheep Figure: Linear Algebra and its Applications , 3rd ed., David C. Lay E. J. King Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L 2( R k ) Reducibility and coorbit spaces Shearlet tiling Figure: http://www.shearlet.org/theory.html E. J. King Generalized Shearlets