Continuous Domain Analysis of Graph Laplacian Regularization for Image Denoisng. Presenter: Gene Cheung National Institute of Informatics

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1 Cotiuous Domai Aalysis of Graph Laplacia Regularizatio for Image Deoisg Preseter: Gee Cheug atioal Istitute of Iformatics [1] Jiahao Pag, Gee Cheug, Wei Hu, Oscar C. Au, "Redefiig Self-Similarity i atural Images for Deoisig Usig Graph Sigal Gradiet," i APSIPA ASC, Siem Reap, Cambodia, December, GSP Workshop, 31 Oct 2014

2 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 2

3 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 3

4 Motivatio (I) Image deoisig a basic restoratio problem: observatio y x e oise desired sigal It is uder-determied, eeds image priors for regularizatio fidelity term 2 mi y x prior( x) x 2 prior term Graph Laplacia regularizer: should be small for target patch x S ( x) T x Lx G L D A graph Laplacia matrix May works use Gaussia kerel to compute graph weights [2]: w ij dist( i, j) is some distace metric betwee pixels i ad j dist i, j exp 2 2 [2] D. Shuma, S. arag, P. Frossard, A. Ortega, ad P. Vadergheyst, The emergig field of sigal processig o graphs: extedig highdimesioal data aalysis to etworks ad other irregular domais, IEEE Sigal Processig Magazie, vol. 30, o. 3, pp ,

5 Motivatio (II) However S T x Lx a. Why is a good prior? G ( x) b. Why usig Gaussia kerel for edge weights? T x Lx c. How to desig a discrimiat for restoratio? We aswer these by viewig discrete graph as samples of high-dimesioal maifold. approximate discrete graph cotiuous maifold 5

6 Our Cotributios 1. Usig Gaussia kerel to compute graph weights, SG( x) coverges to a cotiuous fuctioal S, which ca be iterpreted as regularizer i cotiuous domai. T x Lx Graph Laplacia regularizer S G coverge A cotiuous fuctioal for regularizatio S S 2. Aalysis of fuctioal provides uderstadig of what sigals are beig discrimiated ad to what extet, o a poit-by-poit basis i the cotiuous domai. S 3. We desig a discrimiat for regularizatio i cotiuous domai, the obtai the graph Laplacia regularizer S G Desig discrimiat S obtai The correspodig S G 6

7 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 7

8 Road Map Cotiuous Domai Choose the cotiuous feature fuctios{ f } 1 SAMPLE Discrete Domai Sample { f } 1 to obtai the discrete { f D } 1 Get metric space G R o poit-by-poit basis 2 2 Compute the weights ad Laplacia L R M M Obtai cotiuous fuctioal S ( h) COVERGE Graph Laplacia D Regularizer S ( h ) G Differet { f } leads to differet regularizatio behavior! 1 8

9 Graph Costructio (I) First, defie: 2 2D domai R the shape of a image T s [ x y ] s,1 i M i i i i a set of M radom samples uiformly distributed o, costrued as pixel locatios Roadmap Features Samples { f } { D } Matrix G R Fuctioal S ( h) coverge f 1 Graph weights, M M ad L R Regularizer D S ( h ) G (Freely) choose cotiuous fuctios f (, ) :, 1 x y R f (x, y) y called feature fuctios, ca be itesity for gray-scale image ( 1) R, G, B chaels for color image ( 3) O x Ω 9

10 Graph Costructio (II) Samplig f at positios i gives discretized feature fuctios f D [ f ( x, y ) f ( x, y ) f ( x, y )] M M y f (x, y) f (x i, y i ) T Roadmap Features Samples { f } { D } 1 Matrix G R 2 2 f 1 Graph weights, M M ad L R O s i Ω Fuctioal S ( h) coverge Regularizer D S ( h ) G For each pixel locatio s i, defie a legth 2 vector v [ x y f ( i) f ( i) f ( i)] is a tuable costat D D D i i i 1 2 Build a graph G with M vertices, each pixel locatio have a vertex V i x T s i 10

11 Graph Costructio (III) Weight betwee vertices degree before ormalizatio i ( d ) j 1 ij w m ormalizatio factor clipped Gaussia kerel Vi ad ( ) ( d ) ij i j ij 2 d exp d r, 2 ( d) 2 0 otherwise where r C r ad is a costat C r V j distace Roadmap Features Samples { f } { D } Matrix G R Fuctioal S ( h) d v v 2 ij i j coverge 2 2 D D 2 i j ( ( ) ( )) 2 1 i j s s f f Graph weights, M M ad L R 2 2 f 1 Regularizer D S ( h ) G G is a r-eighborhood graph, i.e., o edge coectig two vertices with distace greater tha r 11

12 Graph Costructio (IV) Our graph G is very geeral e.g., choose a small with proper, obtai the 2D grid graph r Roadmap Features Samples { f } { D } 1 Matrix G R 2 2 f 1 Graph weights, M M ad L R wij A its ( i, j) etry is uormalized Graph m D its ( i, j) etry is w Laplacia 1 ij L D A j Fuctioal S ( h) coverge Regularizer D S ( h ) G h( x, y) : R is a cotiuous cadidate fuctio h S D [ h( x, y ) h( x, y ) h( x, y )] ( h ) ( h ) Lh D D T D M M samples of graph Laplacia regularizer, a fuctioal o G T h( x, y) M R 12

13 Covergece of the (I) S G The cotiuous couterpart of is a fuctioal o domai S Roadmap Features Samples { f } { D } 1 f G G T 1 S( h) ( h) ( h) det dxdy T h [ h h] is the gradiet of h x G is a 2-by-2 matrix: y x f 1 x f y f 2 G I f 2 f 1 1 x f y f 1 y f I 2x2 idetity matrix Matrix G R Fuctioal S ( h) is computed from { f } o a poit-by-poit basis G coverge Graph weights, M M ad L R Regularizer D S ( h ) Structure tesor [3] of the gradiets { f (, )} xy 1 T G [3] H. Kutsso, C.-F. Westi, ad M. Adersso, Represetig local structure usig tesors ii, i Image Aalysis. Spriger, 2011, vol. 6688, pp

14 Covergece of the (II) Theorem : covergece of to M lim 4 1 M M 1 umber of samples eighborhood M S G icreases shriks ~ meas there exist a costat such that equality holds. S G S Roadmap Features Samples { f } { D } Matrix G R Fuctioal S ( h) coverge With results of [4], we proved it by viewig a graph as proxy of a 2 -dimesioal Riemaia maifold Vertex Coordiate o Ω Coordiate o (+2)-D maifold Vi si xi, yi D h ~ S h r C r D D D i i i 1 2 f 1 Graph weights, M M ad L R Regularizer D S ( h ) v [ x y f ( i) f ( i) f ( i)] G T [4] M. Hei, Uiform covergece of adaptive graph-based regularizatio, i Learig Theory. Spriger, 2006, pp

15 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 15

16 Justificatio of (I) 2 1 G G T 1 S( h) ( h) ( h) det dxdy 2 f 1 f G I S G ( h ) ( h ) Lh D D T D S S G coverges to, With S, ay ew isights we ca gai o?? S G T Roadmap Features Samples { f } { D } Matrix G R Fuctioal S ( h) coverge f 1 Graph weights, M M ad L R Regularizer D S ( h ) G The eige-space of reflects statistics of T 1 ( h) G ( h) measures legth of h i a metric space established by G! S itegrates the gradiet orm G { f } 1 16

17 Justificatio of (II) Metric space defied by G y Ellipses are orm-balls, reflects how cocetratio of { f } 1 O l x Gree dots are { f (, )} xy 1 l: Eigevector correspods to the largest eigevalue of G, goes through the cluster of { f } G G T 1 S( h) ( h) ( h) det dxdy 2 f 1 f G I T 17

18 Justificatio of (III) The 2D metric space provides a clear picture of what sigals are beig discrimiated ad to what extet, o a poit-by-poit basis i the cotiuous domai! y y l O x l O x (a) (b) (a) is more skewed, or discrimiat, tha (b) I (a), a small distace away from the directio orthogoal to l brigs large metric distace 18

19 Justificatio of (IV) Lesso: Select feature fuctios properly! Suppose A is the truth gradiet, choose { f } 1 such that (i) l goes through A; (ii) Ellipses stretched flat alog l. y A y A l x x l O O (a) A good scheme, { f } 1 are similar to the groud-truth A (b) A bad scheme For the case of discrete images, oe ca seek for similar patches i terms of gradiet! 19

20 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 20

21 Problem Formulatio ad Algorithm Developmet Adopt a patch-based recovery framework to deoise the image For a oisy patch p 0 o the image 1. Assume a self-similar-i-gradiet image model, search for K 1 patches similar to i terms of gradiet i pre-filtered image. p 0 2. Compute graph Laplacia from the similar patches. 3. Solve the ucostraied quadratic optimizatio iteratively: to obtai the deoised patch 0 2 Aggregate deoised patches to form a updated image. Deoise the give image iteratively to gradually ehace its quality. Our deoisig method is amed Graph-based Deoisig usig Gradiet-based Self-similarity (GDGS) q 2 T q arg mi p q q Lq q 21

22 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 22

23 Experimetal Results (I) Test images: Lea, Barbara, Boats ad Peppers i.i.d. Additive White Gaussia oise (AWG) o-local GBT (LGBT) a existig graph-based deoisig method [5] Compared to BF, LM ad LGBT 1.4 db better tha LM! Performace comparisos i PSR (db) Image Lea Barbara Boats Peppers Method Stadard Deviatio GDGS LM BF LGBT GDGS LM BF LGBT GDGS LM BF LGBT GDGS LM BF LGBT [5] W. Hu, X. Li, G. Cheug, ad O. Au, Depth map deoisig usig graph-based trasform ad group sparsity, i IEEE It l Workshop o Multimedia Sigal Processig,

24 Experimetal Results (II) GDGS vs LGBT GDGS vs LM GDGS LGBT GDGS (31.39 db) LM (30.38 db) GDGS LGBT GDGS (29.34 db) LM (28.62 db) oise stadard deviatio 25 24

25 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 25

26 Towards Optimal Graph Laplacia Regularizatio Our latest work [6] derives the optimal metric space G, leadig to optimal graph Laplacia regularizatio for deoisig. y y y l O x l O x l O x Metric space should be discrimiat to the extet that estimates of groud-truth gradiet are reliable. posterior prob. of groud truth 2 K 1 G arg mi G G0( g) Pr g g d F k g k 0 G whole gradiet domai ideal metric space give groud truth g [6] Jiahao Pag, Gee Cheug, Atoio Ortega, Oscar C. Au, "Optimal Graph Laplacia Regularizatio for atural Image Deoisig," submitted to IEEE ICASSP, Brisbae, Australia, April,

27 Outlie Itroductio Covergece of the Justificatio of the Formulatio ad Algorithm Experimetal Results Towards the Optimal Coclusio Before Lea, 30 After 27

28 Coclusio Image deoisig is a ill-posed problem ad requires good priors for regularizatio. graph Laplacia regularizer with Gaussia kerel weights coverges to a cotiuous fuctioal. Aalysis of the cotiuous fuctioal provides theoretical justificatio of why ad uder what coditios the graph Laplacia regularizer ca be discrimiat. Our deoisig algorithm with graph Laplacia regularizer ad gradiet-based similarity out-performs LM by up to 1.4 db. Our latest work obtais the optimal graph Laplacia, which is discrimiat whe the estimates are accurate, ad robust whe the estimates are ot. Before After 28

29 Thak You! Cotact: Gee Cheug Jiahao Pag GSP Workshop, 31 Oct 2014

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