Why Inverse F-transform? A General Definition of . . . A - PowerPoint PPT Presentation

Data Compression F-Transform Inverse F-transform At First Glance, . . . Why Inverse F-transform? A General Definition of . . . A Compression-Based A Reasonable . . . Discussion and Main . . . Explanation Proof Conclusion Vladik Kreinovich Home Page Department of Computer Science Title Page University of Texas at El Paso 500 W. University ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ vladik@utep.edu Page 1 of 13 Irina Perfilieva and Vilem Nov´ ak Centre of Excellence IT4Innovations Go Back division of the University of Ostrava Institute for Research and Applications of Fuzzy Modeling Full Screen ul. 30. dubna 22, 701 00 Ostrava 1, Czech Republic Close Irina.Perfilieva@osu.cz, Vilem.Novak@osu.cz Quit

Data Compression F-Transform 1. Data Compression Inverse F-transform • In practice, we often need to compress the data: At First Glance, . . . A General Definition of . . . – Sometimes, it takes too much space to store all this A Reasonable . . . information. Discussion and Main . . . – Sometimes, it takes too much computation time to Proof process all this information. Conclusion • In these situations, we need to compress the data, i.e.: Home Page – to replace the original values x ( t ) corresponding to Title Page different moments of time t ◭◭ ◮◮ – with a few combinations x i of these values. ◭ ◮ • Averaging close values x ( t ) decreases meas. error, so Page 2 of 13 � we take x i = a i ( t ) · x ( t ) dt , for t ≈ t i for some t i . Go Back • Optimal a i ( t ) should not change with shift, so a j ( t ) = Full Screen a i ( t − s ) for some s , hence a i ( t ) = a ( t − t i ). Close Quit

Data Compression F-Transform 2. F-Transform Inverse F-transform • We can normalize a ( t ) by taking max a ( t ) = 1. At First Glance, . . . A General Definition of . . . • It is often helpful to assume that a i ( t ) + a i +1 ( t ) = 1. A Reasonable . . . • Let t 0 and h > 0 be real numbers, let n > 0 be an Discussion and Main . . . def integer, and let t i = t 0 + i · h . Let e ( t ) be s.t.: Proof Conclusion • e ( t ) = 0 for t �∈ ( − h, h ), e (0) = 1, Home Page e ( t ) ↑ for t ∈ ( − h, 0], e ( t ) ↓ for t ∈ [0 , h ), and Title Page • e i ( t ) + e i +1 ( t ) = 1 for all t ∈ [ t i , t i +1 ], where def e i ( t ) = e ( t − t i ). ◭◭ ◮◮ ◭ ◮ • Then, for each function x ( t ), its F-transform is: � t i +1 Page 3 of 13 t i − 1 e i ( t ) · x ( t ) dt x i = i = 1 , 2 , . . . , n − 1 . , � t i +1 Go Back t i − 1 e i ( t ) dt Full Screen Close Quit

Data Compression F-Transform 3. Inverse F-transform Inverse F-transform • Problem: once we have x i , we need to reconstruct the At First Glance, . . . original signal. A General Definition of . . . A Reasonable . . . • Natural idea: we have fuzzy rules that if t ≈ t i , then Discussion and Main . . . f ( t ) ≈ x i . Proof • We can view e i ( t ) = e ( t − t i ) as membership functions Conclusion corresponding to t ≈ t i . Home Page • Then, due to � e i ( t ) = 1 for all t , the usual defuzzifi- Title Page cation leads to ◭◭ ◮◮ n � x ( t ) = ˆ x i · e i ( t ) . ◭ ◮ i =1 Page 4 of 13 • This is known as inverse F-transform. Go Back • Comment: inverse F-transform is also useful in de- Full Screen noising, smoothing, etc. Close Quit

Data Compression F-Transform 4. At First Glance, Inverse F-Transform Is Coun- Inverse F-transform terintuitive At First Glance, . . . n • We want to find ˆ x ( t ) = � c i · e i ( t ) which is the closest A General Definition of . . . i =0 A Reasonable . . . � � n � 2 Discussion and Main . . . � to x ( t ): c i · e i ( t ) − x ( t ) → min . Proof i =0 Conclusion • Differentiating w.r.t. c i and equating derivatives to 0, n � � Home Page we get: � e i ( t ) · e j ( t ) dt = x ( t ) · e j ( t ) dt. c i · Title Page i =1 • For triangular e ( t ), we get ◭◭ ◮◮ 3 · h · c j + 1 2 6 · h · c j − 1 + 1 ◭ ◮ 6 · h · c j +1 = x j . Page 5 of 13 • Clearly, the solution c i = x i corresponding to the in- Go Back verse F-transform does not satisfy these equations. Full Screen • Thus, with respect to the above optimization criterion, the inverse F-transform is not optimal. Close Quit

Data Compression F-Transform 5. A General Definition of an Inverse Transform Inverse F-transform n • We want a reconstruction of the type ˆ x ( t ) = � c i · e i ( t ). At First Glance, . . . i =0 A General Definition of . . . • In inverse F-transform, we take c i = x i . A Reasonable . . . Discussion and Main . . . • For the optimal least squares approximation, we have Proof a linear system Conclusion n � � �� � � Home Page c i · e i ( t ) · e j ( t ) dt = x ( t ) · e j ( t ) dt = e ( t ) dt · x i . i =1 Title Page • The resulting c i are linear comb. of x j : c i = � k i,j · x j . ◭◭ ◮◮ j ◭ ◮ • By an inverse transform , we mean a matrix K with Page 6 of 13 elements k i,j , 0 ≤ i, j ≤ n . Go Back • For each matrix K , by a K -inverse transform , we mean n n Full Screen a function ˆ x K ( t ) = � c i · e i ( t ) , where c i = � k i,j · x j . i =0 j =0 Close Quit

Data Compression F-Transform 6. A Reasonable Property: Local Consistency Inverse F-transform • The main purpose of the F-transform compression is At First Glance, . . . that x i describe the signal’s behavior signal on [ t i , t i +1 ]. A General Definition of . . . A Reasonable . . . • It is reasonable to require that the inverse transform Discussion and Main . . . follows the same idea; e.g.: Proof – if the original x ( t ) signal was constant in a neigh- Conclusion borhood of this interval, Home Page – then ˆ x K ( t ) should also be equal to the same con- Title Page stant for all t from this interval. ◭◭ ◮◮ • We say that a matrix K is locally consistent if ◭ ◮ – for every function x ( t ) which is equal to a constant Page 7 of 13 c on an interval [ t i − h, t i +1 + h ], Go Back – the reconstructed function ˆ x K ( t ) is equal to the same constant c for all t ∈ [ t i , t i +1 ]. Full Screen Close Quit

Data Compression F-Transform 7. Discussion and Main Result Inverse F-transform • One can easily check that the inverse F-transform sat- At First Glance, . . . isfies this property. A General Definition of . . . A Reasonable . . . • The F-transform example explains why ˆ x K ( t ) = c only Discussion and Main . . . on a subinterval [ t i , t i +1 ] ⊂ [ t i − h, t i +1 + h ]: Proof – if x ( t ) = 1 for all t ∈ [ t i − h, t i +1 + h ] and x ( t ) = 0 Conclusion for all other t , Home Page – then inverse F-transform ˆ x ( t ) is only equal to 1 for Title Page t ∈ [ t i , t i +1 ]; for all other t , we have ˆ x K ( t ) < 1. ◭◭ ◮◮ • We show that the inverse F-transform is the only lo- ◭ ◮ cally consistent K -inverse transform. Page 8 of 13 • Proposition. A matrix K is locally consistent if and only if it coincides with the unit matrix. Go Back • So, the above result provides the desired justification of Full Screen the inverse F-transform . Close Quit

Data Compression F-Transform 8. Proof Inverse F-transform • Let x ( t ) = 1 for all t ∈ [ t i − h, t i +1 + h ]. At First Glance, . . . A General Definition of . . . • Local consistency implies ˆ x K ( t ) = 1 for all t ∈ [ t i , t i +1 ]. A Reasonable . . . • For t ∈ [ t i , t i +1 ], only e i ( t ) and e i +1 ( t ) are non-zero, so Discussion and Main . . . c i · e i ( t ) + c i +1 · e i +1 ( t ) = 1 for all such t . Proof • For t = t i , we have e i ( t i ) = 1, e i +1 ( t i ) = 0, so c i = 1. Conclusion Home Page • Similarly, for t = t i +1 , we get c i +1 = 1. Title Page n n � � • Thus, we must have k i,j · x j = 1 and k i +1 ,j · x j = 1 . ◭◭ ◮◮ j =0 j =0 ◭ ◮ n � • By definition of x i , we get � a j ( t ) · x ( t ) dt = 1 , k i,j · Page 9 of 13 j =0 n def � Go Back i.e., w ( t ) · x ( t ) dt = 1 , where w ( t ) = � k i,j · a j ( t ). j =0 Full Screen • For t �∈ [ t i − h, t i +1 + h ], x ( t ) can be arbitrary. Close Quit

Data Compression F-Transform 9. Proof (cont-d) Inverse F-transform � • w ( t ) · x ( t ) dt = 1 no matter what the values x ( t ) for At First Glance, . . . t �∈ [ t i − h, t i +1 + h ]. A General Definition of . . . A Reasonable . . . • Thus, for for t �∈ [ t i − h, t i +1 + h ], we have Discussion and Main . . . n � w ( t ) = k i,j · a j ( t ) = 0 . Proof Conclusion j =1 Home Page • For each integer ℓ ≤ i − 2, values t from the interval [ t ℓ , t ℓ +1 ] are outside the interval [ t i − h, t i +1 + h ]. Title Page n ◭◭ ◮◮ • Hence we have � k i,j · a j ( t ) = 0 for all such t . j =1 ◭ ◮ • On this interval, only two functions a j ( t ) are different Page 10 of 13 from 0: a ℓ ( t ) and a ℓ +1 ( t ), so we get Go Back k iℓ · a ℓ ( t ) + k i,ℓ +1 · a ℓ +1 ( t ) = 0 . Full Screen • In particular, for t = t ℓ , we have a ℓ ( t ℓ ) � = 0 and a ℓ +1 ( t ℓ ) = Close 0, so k i,ℓ · a ℓ ( t ℓ ) = 0 and thus, k i,ℓ = 0. Quit