Tutorial on Image Reconstruction Based on Weighted Sum (WS) Filter Approach: From Single Image to Multi-Frame Image

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1 AU J.. 3(): (Oct. 009) utoral on Image Reconstructon Based on Weghted Sum (WS) Flter Approach: From Sngle Image to ult-frame Image Vorapoj Patanavjt Department of Computer and Network Engneerng, Faculty of Engneerng Assumpton Unversty, Bangkok, haland E-mal: Abstract For large magnfcaton factors, the pror classcal smoothness leads to overly smooth results wth very lttle hgh-frequency content. he classcal mage restoratons are falng to reconstruct the desred mage. Consequently, the Recognton-Based Restoraton s desred for these purposes and one of the most effectve technques s of a weghted sum (WS) flter. hs paper revews the research framework of weghted sum (WS) flter approach for mage reconstructon. hs research framework frst starts wth the Hard-Partton-based Weghted Sum (HP-WS) flter proposed n 999 and then consequently revews the Subspace HP-WS (S-HPWS) usng PCA (Prncpal Component Analyss) Flter proposed n 005, he Soft-Partton-based Weghted Sum (SP-WS) proposed n 006, and the fast Adaptve Wener Flter proposed n 007. he paper revews each flter technque n terms of ts computatonal concepts, demonstrates parameter optmzaton from the pont of vew of the mathematcal analyss, and dscusses advantages and dsadvantages. Keywords: Dgtal mage processng, dgtal mage reconstructon, weghted sum (WS) flter, super resoluton reconstructon (SRR).. Introducton ypcally, dgtal mages are nvarably degraded by many phenomena that lmt mage resoluton and utlty durng acquston (Harde 007). Consequently, there are many tradeoffs to be consdered durng desgnng an magng system for mage acquston (Narayanan et al. 007). For hgh spatal bandwdth, a focal plane array (FPA) wth suffcently small detector spacng s requred to sample the scene at the Nyqust rate. (Narayanan et al. 007). However, fabrcaton complextes and correspondng cost may be the lmtng factor (Narayanan et al. 007). For these reasons, many magng systems produce mages sufferng from alasng artfacts resultng from undersamplng (Narayanan et al. 007). he lnear flters are notorous for ntroducng blur when appled to nonstatonary sgnals (such as edge nformaton) n an attempt to reduce the nose. (Barner et al. 999) However, the nonlnear flters tradtonally offer lttle or no mprovement over lnear flters n the case of Gaussan nose (Barner et al. 999). he Weghted Sum flter approach dea s based on parttonng the observaton mage usng vector quantzaton (VQ) and then approprate weghtng coeffcents are appled to each parttoned mage. hs secton roughly revews the Weghted Sum flter approach. Barner et al. (999) proposed the sngle mage restoraton algorthm usng Hard-Partton-based Weghted Sum (HP-WS) flter. he observed mage s parttoned nto several blocks and each block s quantzed n order to select the most proper flter coeffcent for that block to acheve restoraton. he weghts are turned by tranng on a representatve mage and the two-state suboptmal tranng s proposed to determne optmzng weghts. Alam et al. (000) proposed the SRR algorthm for the nfrared mages. he gradent-based regstraton s used for each observed R mage and a weghted nearest-neghbor approach s used for placng Revew Artcle 75

2 AU J.. 3(): (Oct. 009) the frame onto a unform grd to form a hghresoluton mage. Subsequently, the Wener flter s used for deblurrng the fnal hghresoluton mage. ater, n et al. (005, 005b) proposed the sngle mage restoraton algorthm (recognton-based) usng Subspace HP-WS (S-HPWS) flter. he observaton vectors nto a subspace usng PCA (Prncpal Component Analyss) are used n order to reduce the computatonal burden, especally for large wndow szes. Due to the nondfferentable characterstcs, the global optmzaton of HP-WS s dffcult to acheve. Consequently, n et al. (006) proposed a novel radal bass functon nterpretaton of the Soft-Partton-based Weghted Sum (SP-WS) flters and presented an effcent optmzaton procedure based on the gradent method (both the quas-newton method and the steepest descent method). Next, Shao and Barner (006) proposed the sngle mage restoraton algorthm usng Soft-Partton-based Weghted Sum (SP-WS) flter. hus, an analytcal soluton for the global optmzaton s easy to obtan due to the dfferentable SP-WS flter functon. oreover, they compared the proposed SP-WS soluton and the HP-WS soluton computed by two-state suboptmal tranng and by GA algorthm. Narayanan et al. (007) proposed the fast recognton-based SRR algorthm usng Hard-Partton-based Weghted Sum (HP-WS) flter. Due to computatonal complexty, the HP-WS flter s Block Block Observaton Vector { x( n) or x( nx, ny) Blockng ncorporated n ths algorthm nstead of the SP-WS flter. Hence, HP-WS flters are employed to smultaneously perform nonunform nterpolaton and perform deconvoluton of the system s pont spread functon (PSF). Subsequently, Harde (007) proposed the fast SRR algorthm usng the fast recognton-based SRR algorthm wth Adaptve Wener Flter. he postons of the R pxels are not quantzed to a fnte grd as wth some prevous technques and the weghts for each HR pxel are desgned to mnmze the SE and they depend on the relatve postons of the surroundng R pxels. he parametrc statstcal model s used for these correlatons that ultmately defne the flter weghts.. Introducton to Hard-Partton- Based Weghted Sum (HP-WS) Flter (Barner et al. 999) hs secton wll frst revew the concept of the HP-WS flter that s desgned for the sngle mage restoraton algorthm. hs algorthm frst separates the observed mage usng VQ (Vector Quantzaton) and the parttoned mage wll be fltered. ater, the mathematcal analyss of the parameter optmzaton s presented. Fnally, the advantage and the dsadvantage are dscussed. x ( n, n ) x y 5 5 pxels excographc 5 pxels 5 x VQ,,, { Ω Ω K Ω Optmum Wndow Coeffcent ˆ d = w x Fg.. Imagng system, or observaton model of Hard-Partton-Based Weghted Sum (HP-WS) Flter. Revew Artcle 76 Ω w

3 AU J.. 3(): (Oct. 009). he Concept of Hard-Partton-Based Weghted Sum (HP-WS) Flter he man procedure of the HP-WS Flter s shown as follows and n Fg.. x n or { (, ) x y blocks ( x( n ) or ( x, he observed mage ( x n n ) wll be parttoned nto small consstng of 5x5 pxels. Each block ( x n n ) that are x n or ( x, x n n ) wll be vector quantzed n order to determne the best texture smlarty from VQ Code Book ( Ω ) wth a hard threshold. After determnng Ω, the process wll select the most matchng flter w for that block of the observed mage ( x( n ) or ( x, x n n ). he sets of VQ Code Book ( Ω ) and coeffcent flter w are calculated from the tranng set of mages.. athematcal Analyss of the Parameter Optmzaton hs secton revews the optmzaton of flter coeffcents from mathematcal pont of vew. d n s the estmated mage Assume that and { Assume that FPWS F PWS x = wp( x ) x x n s the observed mage. x s the fltered mage. where wp( x ) s the coeffcent of flter. he cost functon ( J ) of SE (ean Square Error) s defned as ( ) J = E d F PWS x, J = Ed w x I( x Ω), = d ( d) I( x Ω) = J = E, + I ( Ω) x = E( d ) E( d) I( x Ω) = J =, E I + ( Ω) x = σ d E w d x I( x Ω) = J =, + E I( x Ω) xw = σ d E( d I( Ω) ) w x x = P J =, + E( I( Ω) ) w x x x w = R J = σ d w P+ w R w, = = where R = E Ω xx x and P = E d x x Ω. One can determne the flter coeffcent that mnmzes the cost functon as n J = ( J) = 0, w w σ d w P = = 0, w + w R w = ( σ d) + w P w w = 443 = 0 = 0, + w R w w = 0 w P w = = 0, + w R w w = + P R w = 0, = = Revew Artcle 77

4 AU J.. 3(): (Oct. 009) P + R w = 0, = = R w = P, = = R w = P, w = R P or w = R P..3 he Advantage and the Dsadvantage Although ths algorthm can be effectvely appled on the mage, an analytcal soluton for the global optmzaton s dffcult to obtan due to the non-dfferentable HP-WS flter functon. 3. Introducton to Subspace HP-WS (S-HPWS) usng PCA (Prncpal Component Analyss) Flter (n et al. 005, 005b) hs secton revews the sngle mage restoraton algorthm usng Subspace HP-WS (S-HPWS) flter. hs secton wll frst revew the concept of S-HPWS Flter that separates the observed mage usng VQ (Vector Quantzaton) wth a subsequent fltraton of the parttoned mage. ater, the mathematcal analyss of the parameter optmzaton s Block Block Observaton Vector { x( n) or x( nx, ny) Blockng x ( n, n ) x y presented. Fnally, the advantage and the dsadvantage are dscussed. 3. he Concept of Subspace Hard- Partton-Based Weghted Sum (S-HPWS) Flter he man procedure of the S-HPWS Flter s shown as follows and n Fg.. x n or { (, ) x y blocks ( x( n ) or ( x, he observed mage ( x n n ) wll be parttoned nto small consstng of 5 pxels. Each block ( x n n ) that are x n or ( x, x n n ) wll be havng a reduced number of pxels from 5 pxels to 5 pxels usng PCA process. PCA nformaton (5 pxels) wll be vector quantzed n order to determne the best texture smlarty from VQ Code Book ( Ω ). After determnng Ω, the process wll select the most matchng flter w for that block of observed mage ( x( n ) or ( x, x n n ). he set of VQ Code Book ( Ω ) and excographc x PCA A VQ pxels { Ω, Ω, K, Ω ( 5 pxels) Optmum Wndow Coeffcent Ω w K pxels ( 5 pxels) dˆ = w x Fg.. Imagng system, or observaton model of Subspace Hard-Partton-Based Weghted Sum (S- HPWS) Flter. Revew Artcle 78

5 AU J.. 3(): (Oct. 009) coeffcent flter w are calculated from the tranng set of mages. 3. athematcal Analyss of the Parameter Optmzaton he parameter optmzaton s closely smlar to the HP-WS parameter optmzaton. 3.3 he Advantage and the Dsadvantage he observaton vectors nto a subspace are usng PCA (Prncpal Component Analyss) n order to reduce the computatonal burden, especally for large wndow szes. Due to the non-dfferentable characterstcs, a global optmzaton of HP-WS s dffcult to acheve. 4. Introducton to Soft-Partton- Based Weghted Sum (SP-WS) Flter (Shao and Barner 006) hs secton revews the Soft-Parttonbased Weghted Sum (SP-WS) flter that s desgned for sngle mage restoraton. hs secton wll frst revew the concept of SP-WS Flter that separates the observed mage usng VQ (Vector Quantzaton) wth a subsequent fltraton of the parttoned mage. ater, the mathematcal analyss of the parameter optmzaton s presented. Fnally, the advantage and the dsadvantage are dscussed. 4. he Concept of Soft-Partton-Based Weghted Sum (SP-WS) Flter he man procedure of the SP-WS Flter s shown as follows and n Fg. 3. x n or { (, ) x y blocks ( x( n ) or ( x, he observed mage ( x n n ) wll be parttoned nto small consstng of 5x5 pxels. Each block ( x n n ) that are x n or ( x, x n n ) wll be vector quantzed n order to determne the best texture smlarty from VQ Code Book ( Ω ) wth soft threshold. After determnng Ω, the process wll select the most matchng flter w for that block of the observed mage ( x( n ) or ( x, x n n ). he sets of VQ Code Book ( Ω ) and coeffcent flter w are calculated from the tranng set of mages. 4. athematcal Analyss of the Parameter Optmzaton (n et al. 006) hs secton presents the optmzaton of flter coeffcents from mathematcal pont of vew: Block Block Observaton Vector { x( n) or x( nx, ny) x ( n, n ) x y excographc Blockng 5 Soft-VQ { Ω Ω K Ω,,, 5 pxels Optmum Wndow 5 pxels Coeffcent x Ω w 5 dˆ = w x Fg. 3. Imagng system, or observaton model of Soft-Partton-Based Weghted Sum (SP-WS) Flter. Revew Artcle 79

6 AU J.. 3(): (Oct. 009) ( wcs,, ) = ( ) Soft PWS, (,, ) = { Soft PWS + Soft PWS E{ d E{ df Soft PWS ( wcs,, ) =, + E{ FSoft PWS E{ d E{ df Soft PWS ( wcs,, ) =, + E{ FSoft PWS σ E{ df Soft PWS ( wcs,, ) =, + E{ FSoft PWS ( wcs,, ) = σ { % + {( % ), ( wcs,, ) = σ w { x + { xw ( wcs,, ) = σ w { x + w { xx w J E d F J wcs E d df F, J J J J E d E J E d% E %%, J E d% E %%, J σ wcs,, = wp % + wrw %. 4.. athematcal Analyss of the Flter Coeffcent Optmzaton: One can determne the flter coeffcent that mnmzes the cost functon as { J ( wcs) = { σ wp+ wrw % n,, n %, w w J (,, ) = { σ + wcs wp % w w wrw %, { σ { wp% w w J ( wcs,, ) =, w + { wrw % w J ( wcs,, ) = 0 p% + Rw %, w 0= p% + Rw %, Rw % = p%, Rw % = p%, w = R% p%. 4.. athematcal Analyss of the VQ Code Book Optmzaton: One can determne the VQ Code Book ( c ) that mnmzes the cost functon as show schematcally n Fg. 4 (due to the complexty of equatons). ( wcs,, ) = σ { Soft PWS + { Soft PWS J E df E F { J( wcs) = σ E{ df + E{ F ax,, ax Soft PWS Soft PWS c c { J( wcs,, ) = σ E{ dfsoft PWS + E{ FSoft PWS { { J( wcs,, ) = + E{ dfsoft PWS + E FSoft PWS σ { { { J( wcs,, ) = 0 E{ dfsoft PWS + E FSoft PWS { J(,, ) E d { F { wcs = Soft PWS + E FSoft PWS { J(,, ) E d { FSoft PWS E F wcs = + Soft PWS { FSoft PWS { J(,, ) E d { FSoft PWS E F wcs = + Soft PWS { FSoft PWS Fg. 4. athematcal analyss of VQ Code Book optmzaton. Revew Artcle 80

7 AU J.. 3(): (Oct. 009) { J( wcs,, ) = E ( d FSoft PWS ) { FSoft PWS { J( wcs,, ) = E ( d F Soft PWS ) exp( β) = γ γ exp( β) exp( β) { γ = = { J(,, ) E ( d F c c wcs = Soft PWS ) γ exp( β) exp( β ) exp( β) = = γ k= { J(,, ) E ( d F c wcs = Soft PWS ) γ ( A8) ( β ) exp { exp( β) exp( β) = γ { J(,, ) E ( d F c wcs = Soft PWS ) γ ( β ) exp exp( β) { β exp( β) { β = γ { J(,, ) E ( d F c wcs = Soft PWS ) γ exp β x c x c exp( β) exp( β) = γ s s { J( wcs,, ) = E ( d F Soft PWS ) γ ( β ) x c exp w x x c exp( β) exp( β) = γ { J(,, ) E ( d F s = s wcs Soft PWS ) γ { J( wcs,, ) = E ( d F Soft PWS ) ( ) ( ) exp( β) ( ) exp( β) s s = γ ( β ) x c x c exp w x γ ( ) ( β ) ( β ) exp exp { J( wcs,, ) = E ( d FSoft PWS ) x c w x c s γ γ = ( x c ) { J( wcs,, ) = E ( d F ) ( ˆ Soft PWS p ( x) )( FSoft PWS ) s k { J(,, ) = 4E ( d FSoft PWS ) x c ( FSoft PWS ) exp x c exp x c wcs s k = s sk Fg. 4. athematcal analyss of VQ Code Book optmzaton (contnued). Revew Artcle 8

8 AU J.. 3(): (Oct. 009) 4..3 athematcal Analyss of the Radal Bass Parameter Optmzaton: One can determne the Radal Bass Parameter ( s ) that mnmzes the followng cost functon: ( wcs,, ) de ( d FSoft PWS ) dj =. ds ds he parameter optmzaton s closely smlar to the VQ Code Book optmzaton n Secton 4.. and consequently the optmzed radal bass parameter (s ) s { J ( wcs,, ) x c ( d FSoft PWS ) s = E ( FSoft PWS) x c exp s x ck exp k = sk 4.3 he Advantage and the Dsadvantage he Soft-Partton-Based Weghted Sum (SP-WS) flters represent a mult-kernel flterng scheme that combnes lnear flterng theory wth data adaptve observaton space parttonng. hs method has proven effectve n treatng both unform and detaled components of non-statonary mages n the case of Gaussan nose. (Barner et al. 999) 5. Introducton to SRR Algorthm usng HP-WS Flter (Narayanan et al. 007) hs secton revews the SRR (Super Resoluton Reconstructon) algorthm to produce hgh-resoluton vdeo from lowresoluton (R) vdeo usng partton-based weghted sum (HP-WS) flters. Due to computatonal complexty, the HP-WS flter s ncorporated n ths algorthm nstead of the SP-WS flter. Hence, HP-WS flters are employed to smultaneously perform nonunform nterpolaton and perform deconvoluton of the system s PSF. he PWS approach uses the R frames to partally populate an HR grd based on regstraton of the R frames. However, unlke more tradtonal methods, PWS flters are then employed to perform non-unform nterpolaton and mage restoraton smultaneously. hs secton wll frst revew the concept of SRR algorthm (usng HP-WS flters) that separates the observed mage usng VQ wth a subsequent fltraton of the parttoned mage. ater, the mathematcal analyss of the parameter optmzaton s presented. Fnally, the advantage and the dsadvantage are dscussed. 5. he Concept of SRR algorthm usng Hard-Partton-Based Weghted Sum (HP- WS) Flter he man procedure of the SP-WS Flter s shown as follows and n Fg. 5. S n or { he observed mage ( { (, ) x y S n n ) wll be parttoned nto small overlapped blocks ( S n or S( nx ny), ) that are consstng of 5x5 pxels. Each estmated block or reconstructed block ( x( n ) or x( nx, n ) of 3x3 pxels wll be vector quantzed n order to determne the best texture smlarty from VQ Code Book ( Ω ) wth soft threshold. After determnng Ω, the process wll select the most matchng flter w for that block of observed mage ( x( n ) or ( x, x n n ). he sets of VQ Code Book ( Ω ) and coeffcent flter w are calculated from the tranng set of mages. Revew Artcle 8

9 AU J.. 3(): (Oct. 009) 5. athematcal Analyss of the Parameter Optmzaton he optmzaton of flter coeffcents from mathematcal pont of vew s smlar to the parameter optmzaton n Secton.. combned wth a low computatonally complexty and hghly parallel structure, makes t attractve for real-tme SR enhancement of vdeo. For arbtrary moton, the complexty of the PWS method ncreases. 5.3 he Advantage and the Dsadvantage he SRR algorthm usng HP-WS flters can be appled to vdeo applcatons. Narayanan et al. (007) beleve that the good performance of the proposed PWS SR method, S S Observaton Image Observaton Image Observaton Image 3 Observaton Image 4 V - V -3 V -4 Regstraton Regstraton Regstraton 7 pxels Nonunform Interpolaton 7 pxels N = 5 x ( n, n ) x y x 5 N = Block 3 3 Block Blockng ( Overlappng) 5 pxels excographc 5 pxels VQ { Ω Ω K Ω,,, Optmum Wndow Coeffcent Ω w Hgh Resoluton Image { x( n) or x( nx, ny) 5 pxels = % p( ), j xn n y n w x n 3 pxels 3 pxels Fg. 5. Imagng system, or observaton model of SRR algorthm usng Hard-Partton-Based Weghted Sum (S-HPWS) Flter. Revew Artcle 83

10 AU J.. 3(): (Oct. 009) 6. Introducton to Fast SRR Algorthm usng HP-WS Flter (Harde 008) hs secton revews the Fast SRR Algorthm usng HP-WS Flter that s desgned for mult-frame mage restoraton. hs secton wll frst revew the concept of fast SRR algorthm usng HP-WS flters that separates the observed mage usng VQ wth a subsequent fltraton of the parttoned mage. ater, the mathematcal analyss of the parameter optmzaton s presented. Fnally, the advantage and the dsadvantage are dscussed. 6. he Concept of fast SRR algorthm usng Hard-Partton-Based Weghted Sum (HP- WS) Flter he man procedure of the SP-WS Flter s shown as follows and n Fg. 6. S n or { he observed mage ( { (, ) x y S n n ) wll be parttoned nto small overlapped blocks ( S n or S( nx ny) that are consstng of 5x5 pxels., ) 5 pxels 5 pxels K Observaton Image θ V - Observaton Image θ 3 V -3 Observaton Image 3 θ P ( V -P ) Observaton Image P Parameter urnng Regstraton Regstraton K Regstraton Nonunform Interpolaton K ( 75 pxels) pxels (, ) r x y = σ ρ dd d x + y g g (, ) r x y ff (, ) r x y df σ n Block Block g g Hgh Resoluton Image { x( n) or x( nx, ny) Block g Blockng ( Overlappng) 30 pxels ( 6mes) g, g, g, g, g = = g, K g,75 excographc 30 pxels 900 pxels dˆ = W g Wndow Statstcs R Wndow Wegths w = R P P w Combne Outputs Fg. 6. Imagng system, or observaton model of fast SRR algorthm usng Hard-Partton-Based Weghted Sum (S-HPWS) Flter. ẑ Revew Artcle 84

11 AU J.. 3(): (Oct. 009) Each estmated block or reconstructed block ( x( n ) or x( nx, n ) of 3x3 pxels wll be vector quantzed n order to determne the best texture smlarty from VQ Code Book ( Ω ) wth soft threshold. After determnng Ω, the process wll select the most matchng flter w for that block of observed mage ( x( n ) or ( x, x n n ). he sets of VQ Code Book ( Ω ) and coeffcent flter w are calculated from the tranng set of mages. 6. athematcal Analyss of the Parameter Optmzaton he P and R optmzaton of flter coeffcents from mathematcal pont of vew s shown schematcally n Fg. 7 (due to the complexty of equatons). 6.3 he Advantage and the Dsadvantage he algorthm has a low computatonal R = E ff { gg { R = E f + n f + n { R = E ff + nn + fn + nf R = E ff + E nn + E fn + E nf σ R = E ff + I n (, ) = (, ) (, ) (, ) r x y r x y h x y h x y dd = 0 = 0 complexty and may be readly sutable for real-tme and/or near real-tme processng applcatons for translatonal moton, but the computatonal complexty goes up sgnfcantly wth other moton models. However, regardless of the moton model, the algorthm lends tself to parallel mplementaton (Harde 007). 7. Concludng Remarks hs artcle surveys the theory of mage reconstructon based on Weghted Sum (WS) flter approach, an alternatve reconstructon paradgm that goes aganst the common wsdom n tradtonal reconstructon. Weghted Sum (WS) flter theory asserts that one can recover certan sgnals and mages from far fewer samples or measurements than tradtonal methods use (Candès and Wakn 008). he author s ntent n ths artcle s to provde an overvew of the basc Weghted Sum (WS) flter that emerged n recent research works, present the key mathematcal deas underlyng ts theory, and survey a couple P = E { gd df { P = E f + n d P = E fd + nd E P = E fd + nd 443 P = E { fd (, ) = (, ) (, ) r x y r x y h x y dd = 0 x + y, = = ( 0.75) r x y σ ρ σ dd d d x + y x + y, = = ( 0.75) r x y σ ρ σ dd d d x + y R = σ E ff + σ I d n P = σ E { fd d W = R P n W = E W = σd E{ ff + σ ni σd E fd { ff + I E { fd σ d σ Fg. 7. athematcal analyss of parameter optmzaton. Revew Artcle 85

12 AU J.. 3(): (Oct. 009) of mportant results n the feld. he author s goal s to explan the Weghted Sum (WS) flter as planly as possble, and so ths artcle s manly of a tutoral nature. Consequently, the comprehensve coverage gven n ths artcle should lead to further research n reconstructon algorthms, as well as new applcatons. he author hopes that the readers wll enjoy ther journey through the theory and applcatons of mage reconstructon based on the Weghted Sum (WS) flter approach. Acknowledgement hs paper partally descrbes the foundaton of a research project afflated wth Assumpton Unversty of haland enttled, Vdeo enhancement usng an teratve SRR based on a robust stochastc estmaton wth an mproved observaton model, that has been supported by Research Grant for New Scholar (RG58063) from RF (ha Research Fund) and CHE (Commsson on Hgher Educaton). References Alam,.S.; Bognar, J.G.; Harde, R.C.; and Yasuda, B.J Infrared mage regstraton and hgh-resoluton reconstructon usng multple translatonally shfted alased vdeo frames. IEEE rans. on Instrumentaton and easurement 49(5): Barner, K.E.; Sarhan, A..; and Harde, R.C Partton-based weghted sum flters for mage restoraton. IEEE rans. on Image ng 8(5): Candès, E.J.; and Wakn,.B An ntroducton to compressve samplng. IEEE Sgnal ng ag. 5(): -30, arch. Harde, R A fast mage super-resoluton algorthm usng an adaptve Wener flter. IEEE rans. on Image ng 6():, n, Y.; Harde, R.C.; and Barner, K.E Subspace partton weghted sum flters for mage deconvoluton. Proc. IEEE Int. Conf. on Acoustcs, Speech, and Sgnal ng (ICASSP'05), IEEE Sgnal ng Socety, Pscataway, NJ, USA. n, Y.; Harde, R.C.; and Barner, K.E. 005b. Subspace partton weghted sum flters for mage restoraton. IEEE Sgnal ng etters (9): n, Y.; Harde, R.C.; Sheng, Q.; Shao,.; and Barner, K.E Improved optmzaton of soft partton weghted sum flters and ther applcaton to mage restoraton. Appled Optcs 45():, Narayanan, B.; Harde, R.C.; Barner, K.E.; and Shao, A computatonally effcent super-resoluton algorthm for vdeo processng usng partton flters. IEEE rans. on Crcuts and Systems for Vdeo echnology 7(5): Shao,.; and Barner, K.E Optmzaton of partton-based weghted sum flters and ther applcaton to mage denosng. IEEE rans. on Image ng 5(7):, Revew Artcle 86