Finite frames theory Scalable frames References Scalable frames Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Department of Mathematics & Norbert Wiener Center University of Maryland, College Park The Math/Stat Department Colloquium American University, Washington, DC Tuesday November 18, 2014 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Scalable frames References Outline Finite frames theory 1 Motivations and definition Tight frames Frame potential Scalable frames 2 Transforming a frame into a tight frame Some generic properties of scalable frames Characterization of scalable frames Fritz John’s ellipsoid theorem and scalable frames References 3 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential A standard problem Question i =1 ⊂ R N be a complete set. Recover x from ˆ Let Φ = { ϕ i } M y given by ˆ y given by y = Φ ∗ x + η, ˆ where Φ is the N × M matrix whose k th column is ϕ k , and η is an error (noise). Solution Need to design “good” measurement matrix Φ , e.g., Φ should lead to reconstruction methods that are robust to erasures and noise. Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential A standard problem Question i =1 ⊂ R N be a complete set. Recover x from ˆ Let Φ = { ϕ i } M y given by ˆ y given by y = Φ ∗ x + η, ˆ where Φ is the N × M matrix whose k th column is ϕ k , and η is an error (noise). Solution Need to design “good” measurement matrix Φ , e.g., Φ should lead to reconstruction methods that are robust to erasures and noise. Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Minimal requirements on the measurement matrix Fact i =1 ⊂ K N is complete ⇐ Φ = { ϕ i } M ⇒ ∃ A > 0 : M � A � x � 2 ≤ |� x, ϕ i �| 2 for all x ∈ K N i =1 Clearly, there exists B > 0 , e.g., B = � M i =1 � ϕ i � 2 such that M � |� x, ϕ i �| 2 ≤ B � x � 2 for all x ∈ K N . i =1 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Definition of finite frames Definition i =1 ⊂ K N is called a finite frame Let K = R or K = C . { ϕ i } M for K N if ∃ 0 < A ≤ B : M � A � x � 2 ≤ |� x, ϕ i �| 2 ≤ B � x � 2 , for all x ∈ K N . (1) i =1 i =1 ⊂ K N is called a finite tight frame for If A = B , then { ϕ i } M K N . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Frame operator & Reconstruction formulas � � k =1 ⊂ K N let Φ = For Φ = { ϕ k } M ϕ 1 ϕ 2 . . . ϕ M . ⇒ S = ΦΦ ∗ is positive definite. Φ is a frame ⇐ M M � � x = S ( S − 1 x ) = � x, S − 1 ϕ i � ϕ i = � x, ϕ i � S − 1 ϕ i i =1 i =1 � ϕ i } M i =1 = { S − 1 ϕ i } M Φ = { ˜ i =1 is the canonical dual frame . A opt = λ min ( S ) and B opt = λ max ( S ) . The condition number of the frame is κ (Φ) = λ max ( S ) /λ min ( S ) ≥ 1 . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Frame operator & Reconstruction formulas � � k =1 ⊂ K N let Φ = For Φ = { ϕ k } M ϕ 1 ϕ 2 . . . ϕ M . ⇒ S = ΦΦ ∗ is positive definite. Φ is a frame ⇐ M M � � x = S ( S − 1 x ) = � x, S − 1 ϕ i � ϕ i = � x, ϕ i � S − 1 ϕ i i =1 i =1 � ϕ i } M i =1 = { S − 1 ϕ i } M Φ = { ˜ i =1 is the canonical dual frame . A opt = λ min ( S ) and B opt = λ max ( S ) . The condition number of the frame is κ (Φ) = λ max ( S ) /λ min ( S ) ≥ 1 . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Frame operator & Reconstruction formulas � � k =1 ⊂ K N let Φ = For Φ = { ϕ k } M ϕ 1 ϕ 2 . . . ϕ M . ⇒ S = ΦΦ ∗ is positive definite. Φ is a frame ⇐ M M � � x = S ( S − 1 x ) = � x, S − 1 ϕ i � ϕ i = � x, ϕ i � S − 1 ϕ i i =1 i =1 � ϕ i } M i =1 = { S − 1 ϕ i } M Φ = { ˜ i =1 is the canonical dual frame . A opt = λ min ( S ) and B opt = λ max ( S ) . The condition number of the frame is κ (Φ) = λ max ( S ) /λ min ( S ) ≥ 1 . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Frame operator & Reconstruction formulas � � k =1 ⊂ K N let Φ = For Φ = { ϕ k } M ϕ 1 ϕ 2 . . . ϕ M . ⇒ S = ΦΦ ∗ is positive definite. Φ is a frame ⇐ M M � � x = S ( S − 1 x ) = � x, S − 1 ϕ i � ϕ i = � x, ϕ i � S − 1 ϕ i i =1 i =1 � ϕ i } M i =1 = { S − 1 ϕ i } M Φ = { ˜ i =1 is the canonical dual frame . A opt = λ min ( S ) and B opt = λ max ( S ) . The condition number of the frame is κ (Φ) = λ max ( S ) /λ min ( S ) ≥ 1 . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Frame operator & Reconstruction formulas � � k =1 ⊂ K N let Φ = For Φ = { ϕ k } M ϕ 1 ϕ 2 . . . ϕ M . ⇒ S = ΦΦ ∗ is positive definite. Φ is a frame ⇐ M M � � x = S ( S − 1 x ) = � x, S − 1 ϕ i � ϕ i = � x, ϕ i � S − 1 ϕ i i =1 i =1 � ϕ i } M i =1 = { S − 1 ϕ i } M Φ = { ˜ i =1 is the canonical dual frame . A opt = λ min ( S ) and B opt = λ max ( S ) . The condition number of the frame is κ (Φ) = λ max ( S ) /λ min ( S ) ≥ 1 . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential The canonical dual frame Lemma i =1 ⊂ K N is a frame, and that Assume that Φ = { ϕ i } M i =1 ⊂ K N is the canonical dual frame. For each x ∈ K N , ϕ i } M { ˜ � M ϕ i �| 2 minimizes � M i =1 | c i | 2 for all { c i } M i =1 |� x, ˜ i =1 such that x = � M i =1 c i ϕ i . Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Why frames? Question i =1 ⊂ R N be a unit norm frame, and assume we Let Φ = { ϕ i } M wish to recover x where we have access to ˆ y given by y = Φ ∗ x + η. ˆ Solution If no assumption is made about η we can just minimize � Φ ∗ x − ˆ y � 2 . This leads to M � x = (Φ † ) ∗ ˆ ˆ y = ( � x, ϕ i � + η i ) ˜ ϕ i . i =1 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Why frames? Question i =1 ⊂ R N be a unit norm frame, and assume we Let Φ = { ϕ i } M wish to recover x where we have access to ˆ y given by y = Φ ∗ x + η. ˆ Solution If no assumption is made about η we can just minimize � Φ ∗ x − ˆ y � 2 . This leads to M � x = (Φ † ) ∗ ˆ ˆ y = ( � x, ϕ i � + η i ) ˜ ϕ i . i =1 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Finite unit norm tight frames Definition i =1 ⊂ K N with � ϕ k � = 1 for each k is A tight frame { ϕ i } M called a finite unit norm tight frame (FUNTF) for K N . In this case, the frame bound is A = M/N . Remark Tight frames and FUNTFs can be considered optimally conditioned frames since the condition number of their frame operator is unity. Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
Finite frames theory Motivations and definition Scalable frames Tight frames References Frame potential Finite unit norm tight frames Definition i =1 ⊂ K N with � ϕ k � = 1 for each k is A tight frame { ϕ i } M called a finite unit norm tight frame (FUNTF) for K N . In this case, the frame bound is A = M/N . Remark Tight frames and FUNTFs can be considered optimally conditioned frames since the condition number of their frame operator is unity. Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Scalable frames
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