ADAPTIVE IMAGE FILTERING

Transcription

1 Why adaptve? ADAPTIVE IMAGE FILTERING average detals and contours are aected Averagng should not be appled n contour / detals regons. Adaptaton Adaptaton = modyng the parameters o a prrocessng block accordng to local condtons (around the current processed pxel), or each locaton wthn the ntal mage. There s teh potental o obtanng by adaptaton a derent lter or each locaton, although the basc lterng structure s the same. Adaptaton mposes the exstence o a objectve, quanttatve measurement procedure, o the processng eects (desred or not) nduced n the result by the modcaton o the lter parameters. What can be changed? The neghborhood (the shape o the lterng wndow). The weghts correspondng to the T transorm. Whatever s moded n the lter rom one locaton to another, the resultng global processng s not longer spatal nvarant, and thus s non-lnear. Flter parameter are deduced by the mnmzaton o some error measures. 1

2 Adaptaton o the shape o the lterng wndow (neghborhood) Smoothng s evaluated by the derences nduced as compared wth the ltered mage; the derences are too mportant, there s the possblty that the lter would blur mage contours. The colateral assumpton s the nosy s weaker than the mage contours, and the varatons dues to nose are smaller than the derences between adjacent regons. The lters orm ths category are usng a swtch between several possble lters, correspondng to an varaton acceptance threshold. 1. Average / All-pass lter Let the nosy mage and g the ltered mage. ( l, ( l, ( l, < T g( l, = ( l, otherwse The new pxel value s the average o the values rom ts neghborhood only the average s not too derent rom the ntal value. The swtchng s perormed between a neghborhood V averagng lter and an all-pass lter. Example nosy mage SNR = 17.3 db averagng SNR = 1.5 db adaptve lter SNR = 19 db T = 13 usng 50% o ltered pxels

3 . Lnear orented lters A set o derent orented, lnear-shaped neghborhoods s used. Let g be the lterng result accordng to V, orented accordng to θ. Ex. Example averagng SNR = 1.5 db V 1 V V 3 V 4 g ( l, wth k = arg mn{ ( l, g ( l, )} g( l, = c k θ k s the drecton accordng to whch the smoothng s the sotest, nducng mnmal derences w.r. to the ntal mage. nosy mage SNR = 17.3 db adaptve lter SNR = 19.9 db Ideally, the smoothng lterng wndow has to be parallel to the local contour, such that t should select values placed nsde a unque regon. 3. Nagao lter Uses a set o orented neghborhoods. sotropc lterng wndow, blurs the contour lnear lterng wndow, perpendcular to the contour; blurs the contour lnear lterng wndow, parallel to the contour; preserves the contour V 1 V V 3 V 4 V 5 V 6 V 7 V 8 V 9 3

4 3. Nagao lter Let g the lterng result accordng to V. Let σ the varance o values wthn V. { σ } g( l, = gk ( l, wth k = arg mn V k s the neghborhood provdng the sotest smoothng, nducng mnmal derence w.r. to the ntal mage, due to the act that the neghborhood s the most unorm (selected values are the most smlar). Adaptng the weghts (parameters) o the lters Adaptng the lnear lters The weghts o the lter are moded accordng to mage values, at each locaton (at each locaton there s the potental o obtanng a derent lter). Seeks the reducton o blurrng n contour regons. Example : the Lee lter (LLMMSE - Locally Lnear Mnmal Mean Squared Error) Idea : the ltered mage s the lnear convex combnaton o the ntal (degraded) mage and o the locally averaged mage. g = α + ( 1 α g = α + ( 1 α whte, addtve nose (uncorrelated wth the mage) = 0 + z z = 0, 0 z = 0 + (1 α ) σ 0 z Lee lter LLMMSE g = α 0 + z + (1 α )( 0 + z ) = α 0 + (1 α )( 0 + The approxmaton error o the correct mage 0 by the ltered mage g s: ε = 0 g = α( 0 0 ) (1 α )z The mean squared error s: ε = ( α( ) (1 α )z) 0 0 z ) ε = α ( 0 0 ) + (1 α ) z α (1 α )( 0 0 )z ε = α σ 4

5 Mnmzng the mean squared error : But ε = 0 α ε = ασ 0 α Equvalently : 0 + z z (1 α ) σ = 0 α = σ = σ = σ + σ 0 z σ z α = σ σ z σ z g = + (1 σ σ Lee lter LLMMSE σ z + σ 0 z g = α + ( 1 α Partcular (lmt) cases : I. II. σ z σ z + σ 0 z σ α = = σ σ z = 0 α = 0 σ << σ z α = 1 g = In noseless or contour regons the lter s an all-pass lter z σ >> σ Lee lter LLMMSE g = In hgh nose or unorm regons the lter s a low-pass lter (local averagng). Lee lter LLMMSE : Example Double-wndow Lee lter The Lee lter needs the power o the nose aectng the mage. Idea: use or the processng o mage pxel two derent wndows. org. Gaussan nose avg. LLMMSE a small wndow or the nose estmaton a larger wndow, or a classcal Lee lterng assumes that n the area selected by the wndow the mage s deally constant (equal to the average o values n the wndow); the varatons are due to addtve nose. 5

6 Double-wndow Lee lter pxel to be processed wndow or nose estmaton Adaptaton o non-lnear rank-order lters Automatcally computng the rank o the output order statstcs. wndow or actual Lee lterng [Zamperon] The rank o the output order statstcs or a length-k wndow lter s: K 1 x ( ) x(1) j = + = 1 x( K ) x(1 ) g l, = rank { x, x,..., } ( j 1 x K Adaptaton o non-lnear rank-order lters Inverse contrast rato mappng Computng the replcaton weghts or weghted rank-order lters. σ ( l, g( l, = ( l, local dsperson o mage values locala average o mage values Computng the weghts or L-lters g(l, values are rescaled n order to preserve the global mean o the mage. 6

7 Local extremes x( K ) + x x, (, ) (, ) = ( ) l c g l c K x(1), otherwse (1) 7