Rao Transforms: A New Approach to Integral and Differential Equations

Transcription

1 Rao Tranform: A New Approach to Integral and Differential Equation Dr. Muralidhara SubbaRao (Rao) SUNY at Stony Brook, murali@ece.unyb.edu,, rao@integralreearch.net Rao Tranform (RT) provide a brand new approach to the century old problem of Integral and Differential Equation in one and multiple dimenion. The new approach i unified, localized, implified, and computationally efficient. Both ymbolic and numerical olution are provided by RT. Differential Equation are firt converted to Integral Equation by incorporating boundary condition and then olved. Many fundamental law of phyic are tated uing differential equation. Therefore RT are expected to have wide application in cience and engineering. RT are ueful in hift-variant image/ignal filtering, modeling linear/non-linear integral ytem, optic, invere optic, computer viion, mathematical oftware (e.g. Matlab, Mathematica), and medical image analyi. The RT approach i fully localized and therefore permit extremely fine-grained parallel implementation on a computer. Error are minimized at boundarie. Localized tructure of data i naturally exploited (e.g. a 2D image data i not retructured a a 1D vector for hift-variant deblurring). Patent application are pending and a book ha been elf-publihed on RT. 1

2 Integral Tranform/Equation In the definition below, x and are real variable, f i an unknown real valued function that we need to olve for, g, k, and h are known (or given) real valued function. r and may be real contant or one of them can be the real variable x. 1. Integral Tranform (IT/E) 2. Rao Localization Equation (RLT/E) 3. Rao Tranform (RT/E) α g ( x) = k( x,α ) f ( α) x: point of meaurement of effect, alpha: poition of ource x r Rao Firt Theorem (RFT): If h i defined by RLT in (2), then IT in (1) and RT in (3) above are exactly equivalent, i.e., RHS(1)=RHS(3). RT i inverted to olve for f(x). A ytem of linear algebraic equation will have to be olved. The derivative of f(x) at x are the unknown. The olution i given in term of the derivative of g(x) at x. Therefore the olution i a local olution at x. r hx (, α ) kx ( + α, x) g( x) = h( x α, α) f( x α) x 2

3 RFT: Proof of Equivalence of RT and IT when RLT i ued R.H.S. of RT = = = x r x x r x x r x hx ( αα, ) f( x α) k(( x α) + α, ( x α)) f( x α) from RLT kxx (, α) f( x α) Change the variable of integration to β where β = x α, and dβ =. The new limit of integration are α = x β =, α = x r β = r R.H.S. of RT = = = r r kx (, β ) f ( β) dβ kx (, α) f ( α) R.H.S. of IT. 3

4 Serie Expanion and Inverion of RT g( x) = h( x α, α ) f ( x α ) Derive ymbolic expreion for the k-th derivative of g(x) ( ) denoted by g k for k=1,2, N, and recurively ubtitute in the above equation. Aume x r x x r h x a f x d x ( α, α ) N n n α ( n) ( ) α (Taylor-erie expanion) n = 0 x r n ( α α α α x ) N an ( n) f ( x) h( x, ) d n = 0 N ( n ) an f x hn x n = 0 ( ) ( ) ( k f ) ( x) = 0 for k > N. (0) (0) g h00 h01 h0 N f (1) (1) h10 h11 h g 1N f = ( N) h ( N) g 00 h01 h0 N f Invere RT: 1 f x = H x g x g = H f x x x x r f () x = h'( x α, α) g( x α) x The reolvent kernel h i determined by the invere of H. 4

5 Rao Tranform: A New Approach to Integral and Differential Equation Unknown Function f ( x ) Conventional Linear Integral Equation with term g( x) = k( x,α ) f ( α) OR r OR Differential Equation ODE/PDE + Boundary Condition Apply Rao Localization Tranform h( x, α ) = k ( x + α, x) Derive Equivalent Integral Equation uing Rao Tranform x r g ( x) = h( x α, α) f ( x α). x Subtitute truncated Taylor-erie expanion for h( x α, α ) around ( x, α ) and f ( x α ) around x. ( n) Simplify. Group term baed on derivative of f ( x). f Take derivative with repect to x and derive a ( n ) ytem of at leat N algebraic eqn in N unknown f and olve them by imple back-ubtitution. f ( x ) Solution Figure 1. Solving Linear Integral and Differential Equation 5

6 General Integral Tranform/Equation (Non-linear Integral Equation) General IT (GIT/E) General RLE (GRLT/E) gx ( ) = kx (,α, f( α)) r hx (, α, f( x)) kx ( + α, x, f( x)) General RT (GRT/E) x r gx ( ) = hx ( α, α, f( x α)) x Theorem: If h i defined a in GRLT, then GIT and GRT are exactly equivalent. GRT i inverted to olve for f(x). A ytem of non-linear algebraic equation will have to be olved. The derivative of f(x) at x are the unknown. The olution i given in term of the derivative of g(x) at x. Therefore the olution i a local olution at x. 6

7 Application Example and Advantage Many pecific example have been olved, including the Fredholm and Volterra integral equation of both the Firt and the Second kind. A general differential equation correponding to an n-th order initial value problem ha been olved. RT have been applied to the problem of retoring image defocued by a hift-variant point pread function with ignificant computational aving. Additional problem are being invetigated for olving them uing RT and comparing computational / theoretical advantage relative to exiting method. In comparion with exiting method, RT have the unique feature of (1) completely localizing the problem, (2) olving the local problem accurately and efficiently, and then (3) yntheizing the local olution eamlely to obtain a complete or global olution. RT are epecially uited for practical application due to the locality property of phyical ytem, i.e. the influence or effect of an agent or ource i higher in it vicinity and generally decreae with increaing ditance. Numerou future reearch topic are open for invetigation. The approach of RT eem to hold great promie. See. 7

8 Application: Lit of Ueful Practical Problem Mot of the ueful practical problem are covered by the following lit where the variable may be one, two, or three dimenional vector. 1. Fredholm Integral Equation of the Firt Kind b g( x) = k( x,α ) f ( α) a 2. Fredholm Integral Equation of the Second Kind b g( x) = f ( x) + k( x,α ) f( α) 3. Volterra Integral Equation of the Firt Kind x g( x) = k( x,α ) f ( α) a 4. Volterra Integral Equation of the Second Kind a x g( x) = f ( x) + k( x,α ) f ( α) a Solution method for all of the above problem type are the ame in the new RT approach hown in Fig. 1 earlier. 8

9 Application: RT Formulation and Solution All the four linear integral equation (Fredholm/Volterra Firt/Second Kind) which account for mot of the practical application can be reformulated and olved a a ingle integral equation uing RT a follow: x r gx () = cf() x+ hx ( α, α) f( x α) x In the above equation, etting c=0/1 reult in Firt/Second Kind equation and etting =contant/x reult in Fredholm/Volterra integral. Uing the ame notation a before, and denoting an identity matrix by I,we obtain the olution in all cae to be: g = ( ci+ H ) f x x x f = ( ci+ H ) g 1 x x x 9

10 Exiting Approache (current tate of the art) Approache are different for different problem type (e.g. Firt/Second kind) Computational requirement are too high. Accuracy of olution and numerical tability are eriou problem. Some method are iterative. Computation are global and o not amenable to fine-grain parallel computation. When approximate localization are employed by olving the problem eparately in mall interval, the olution at the boundarie will not be fully compatible. The olution will have noticeable eam at the boundarie of interval. So yntheizing approximate local olution into a ingle eamle global olution i difficult. The current tate of the art on olving integral equation i very unatifactory in practical application. Thi wa ummarized a follow by an expert Program Director of a US Government Reearch Funding agency who reviewed a whitepaper on RT: [My agency] eem to motly have problem in the category for which thee [currently exiting] method are uele." 10

11 Example of current approache Current method can be broadly claified a Matrix Inverion and Matrix multiplication method. Matrix inverion method correpond to inverting a matrix of ize NxN which i proportional to the ize of input data g ampled at a et of point. Example of thi method are Fredholm method, Eigen vector/value method uch a Singular Value Decompoition, Collocation, Galerkin, and Nytrom. Matrix multiplication method correpond to iteratively multiplying an NxN matrix with an Nx1 vector where N i the ize of ampled input data g. Example of thi method are Volterra iterated kernel, Neumann erie, and Born approximation. Thee are mainly applicable to Second Kind equation. Typical computational complexity i O(N^3) which i too much. In comparion, the typical computational complexity of the new RT approach i around O(N^2) reulting in a peed up of O(N). Thi can be very ignificant in the cae of 2D and 3D problem. 11

12 Application Example 1: Image Retoration Thi example i relevant to deblurring an image of a 1D barcode printed on a lanted plane or a curved urface and canned by a laer canner that ue a len for imaging. Lene have limited depth-of-field, and o the image will be blurred by a hift-variant point pread function (PSF). Approximation: hx ( α, α) 0for α > forallx. (0) Computing the blurred image g(x) or g : Forward RT: g 1 h (1/ 2) h f (0) (1) (0) (0) 2 2 (1) (1) g 0 1 (3 2) h (1) = / 2 f g(2) f (2) m ( m) ( m) n hx (, α) hn = hn ( x) = α m x (0) Computing the focued image f(x) or f : Invere RT: f 1 h (1/ 2)( h 3 h h ) g (0) (1) (0) (1) (1) (0) (1) (1) f 0 1 (3 2) h (1) = / 2 g f (2) g(2) 12

13 Intrepretation of condition on the kernel h In order to interpret the meaning of the condition on the kernel h for convergence, conider the example of a Gauian blurring kernel in image defocuing of a lanted plane or a curved urface. In thi cae we have the following ituation. Conventional (global) kernel of an IT: 2 1 ( x α ) k( x, α ) = exp 2 2 πσ( α) 2 σ ( α ) Standard aumption for conventional kernel in olving an IT: k ( x, α ) 0 for x > T for all x. RT (local) kernel 1 hx (, α) = exp 2 πσ( x) 2 α 2 2 σ ( x) hx (, α ) = kx ( + α, x) Aumption for RT kernel: hx ( α, α) 0for α > forall x. Note that the two aumption are imilar, but the new aumption may be a little weaker and cleaner.. 13

14 New Concept: Local Eigen Value and Local Eigen Function gx ( ) = λφ( x) Set in RT a: (0) (0) g h00 h01 h0 N φ (1) (1) h10 h11 h g 1N φ = ( N) h ( N) g 00 h01 h0 N φ g = H φ x x x λφ = H φ ( H Iλ) φ = 0 x x x x x Local Eigen Value are the root of Local Eigen Function are determined by the olution of ( H Iλ ) φ = 0, λ = 0,1,2,, Actual Local Eigen Function (N-th order) are: Det ( H Iλ) = 0 x i xi i N 2 N (0) (1) α (2) α ( N ) φi( x+ α) = φi ( x) + αφi ( x) + φi ( x) + + φi ( x), i= 0,1,2,, N. 2 N! They approximate the Global Eigen Function in a mall interval around x. A N tend to infinity, Local Eigen Function tend to the Global Eigen Function. x 14

15 Application Example 2: N-th Order Initial Value Problem Ordinary Differential Equation: d y n ( n k) Ak ( x) = F( x), ( n k ) ( n k) () k= 0 dx y r = q n k k = 0,1,, n 1 Solution i obtained by inverting the following RT: x r g( x) = f( x) h( x α, α) f( x α) 0 n k 1 α hx (, α) = Ak ( x+ α) ( k 1)! k= 1 g( x) = F( x) q A ( x) [( x r) q + q ] A ( x) n 1 1 n 1 n 2 2 n 1 { n 1 1 0} [( x r) /( n 1)!] q + + ( x r) q + q A ( x) n 1 1 k x n ( x α) ( x r) y( x) = f ( α) + q r ( n 1)! k! k= 0 k n 15

16 Application Example 3: Direct Viion Sening by Invere Optic 1. Model the camera a a linear hift-variant ytem where the captured image g(x,y) i a function of camera parameter e=(,f,d,λ), focued image f(x,y), hape z(x,y) with parameter ditance Z0, lope Zx, Zy, and curvature if needed. g ( x, y) = k( x, y, α, β,, ) f( α, β) dβ p e p z 2. Localize the hift-variant point pread function (SV-PSF) uing RLT: h( x, y, α, β, e, z) = k( x+ α, y + β, x, y, e, z) p 3. Apply RFT to get a localized model of image formation g ( x, y) = h( x α, y βαβ,,,, ) f( x α, y β) dβ p e p z p 4. Capture P image with different camera parameter etting for p=1,2,,p. e = ( D,, f, λ ) p p p p p 16

17 Len Image Detector λ D F O Image/video Data gxy (, ) 3D object f ( xy, ) zxy (, ) Camera controller Digital Image Digital Video Focued image f(x,y) 3D hape z(x,y) IMAGE PROCESSING COMPUTER Image normalization and moothing oftware Image derivative computation oftware Invere Rao Tranform coefficient computation oftware Algebraic equation olving oftware Virtual image/video camera oftware 3D animation oftware Animation Video e = ( D', F', ', λ ') f(x,y) : Focued image (input from cene to camera) z(x,y): 3D hape of object urface (input from cene to camera) g(x,y): blurred image recorded by camera (output from camera) New Camera Parameter 17

18 φ ref ( x1, y1) h'( x, y, α, β, z, e) z ENP O EXP +φ ( x len, y ) 1 1 PS( α, β ) PM( x, y) 3D object f ( α, β ) z( α, β ) φ ( x in 1, y 1 1 1) fout x1 y1 f ( x, y ) in 1 1 f ( len x, ) 1 y1 ( x1, y1) ( x1, y1) φ out ( x, y ) (, ) wx y φ φ ( 1, 1) = out ref ( xy, ) ( α, β ) 18

19 Application Example 3: Direct Viion Sening by Invere Optic 5. Subtitute truncated Taylor erie expanion of f, h, and z, to obtain the invere RT: In one pecific cae, we get: f = g g h g h (0, 0) (0, 0) (1, 0) (1, 0) (0, 1) (0, 1) p p p,2, 0 p p,0, g h + h h h (2, 0) (1, 0) 2 (0, 1) (0, 1) (0, 0) p ( p,2, 0 ) p,0, 2 p,2, 0 p,2, g ( h ) + h h h (0, 2) p N n (0,0) ( n i, i) S pni,, p n= 0 i= 0 f ( x, y) = f = g (01), 2 (10), (10), (00), p,0, 2 p,2, 0 p,0, 2 p,0, 2 6. Solve the ytem of P equation to obtain the P unknown f(x,y), ditance Z0, lope Zx, Zy, and curvature if needed. 7. Syntheize the olution at all point (x,y) to get complete 3D hape and focued image f(x,y). 19

20 Generalization of RT: From Kernel localization to Kernel tranformation In the conventional integral equation g( x) c f ( x) k( x, ) f ( ) d we can replace the integration variable α with a one-to-one and onto function β=β(x,α) which i bijective with repect to α. For example β=x-α reult in localization of the kernel een earlier. Scaling and tranlation can be incorporated uing β=ax+bα+d for contant a,b,d. In thi cae we obtain ax+ b+ d gx ( ) = cf( x) + kxax (, + bα + d) fax ( + bα + d) b ax+ br+ d n= 0 ( m) Now, auming f( x) = 0 for m> N we can derive expreion for the variou derivative of g(x) a before and olve the integral equation a g = ( ci+ K ) f x x x = + and r α α α n n ax+ b+ d N ( bα ) d f( ax+ d) cf( x) + kxax (, + bα + d) (Taylor erie expanion) ax br d n b + + n= 0 n! dx N n n d f( ax+ d) ax+ b+ d ( bα ) cf( x) + (, ) n dx kxax+ bα + d b ax+ br+ d n! N cf( x) + f k( x) n= 0 ( n) ( x) n f = ( ci+ K ) g 1 x x x 20

21 Future Reearch and Concluion Invetigate the application of RT to other practical problem uch a fluid mechanic and computational phyic. Compare the relative performance of RT with current tate of the art technique for practical problem in term of accuracy, tability, and computational efficiency. Invetigate the theoretical and computational apect of RT further becaue RT provide an elegant way to convert an integral equation to a ingle localized differential equation that incorporate all the boundary condition which i eaily olved both ymbolically and numerically. It may be poible to rederive previouly known theoretical reult on integral equation uch a uniquene and exitence uing RT in a impler and more unifying way that provide additional inight into problem of practical importance, The baic idea of localization may be extenible to other invere problem of practical importance. Thi hould be invetigated. Application of RT to olve medical image analyi problem uch a computer tomography (x-ray, optical, MRI, etc.) hould be invetigated. 21