Before We Begin. Introduction to Spatial Domain Filtering. Introduction to Digital Image Processing. Overview (1): Administrative Details (1):

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1 Overview (): Before We Begin Administrtive detils Review some questions to consider Winter 2006 Imge Enhncement in the Sptil Domin: Bsics of Sptil Filtering, Smoothing Sptil Filters, Order Sttistics Filters Tuesdy, Ferury Sptil Filtering Introduction Review Bsics of D nd 2D sptil filtering Smoothing Sptil Filters Introduction Smoothing liner filters, Administrtive Detils (): Before We Begin L Four Should e rther strightforwrd l to complete A l report is required for this l nd is due in two weeks (e.g., Ferury ) Reding Week Next week no lecture or l Test Ferury rrive on time s test will egin t 8:0m shrp! No considertion given if you re lte! Some Questions to Consider (): Wht is liner opertor nd non-liner opertor? How do we show n opertor is liner or non-liner? Wht is the sptil domin? Wht is imge enhncement in the sptil domin? Wht is sptil filtering? Wht is smoothing sptil filter? Wht is n verging filter? Wht is histogrm? Cn you think of how to enhnce n imge y using its histogrm? Introduction to Sptil Domin Filtering

2 Introduction (): Sptil Domin The ggregte of pixels comprising n imge Sptil Domin Methods Procedures tht operte directly on the imge pixels Denoted y the following expression g(x,y) = T[f(x,y)] f(x,y) input imge g(x,y) output imge T opertor defined over some neighorhood of (x,y) Introduction (2): Defining Neighorhood Aout x,y Squre (rectngulr) su-imge re centered t x,y Center of su-imge is moved from pixel to pixel Opertor T is pplied t ech loction x,y to yield output g Opertor T utilizes only pixels in re of imge f spnned y neighorhood Introduction (3): Defining Neighorhood Aout x,y (cont.) Squre (rectngulr) su-imge re centered t x,y Also known s msk, templte, filter or window Ech entry of the su-imge hs its own vlue ech vlue known s coefficient Msk does not lwys hve to e two-dimensionl Cn lso hve one-dimensionl msk (more lter ) Bsics of Sptil Filtering Introduction 2D Msks (): Introduction 2D Msks (2): Grphicl Illustrtion Msk (kernel, templte ) with its coefficients Templte, Kernel, Msk Coefficients denoted y w Origin is in the middle Aritrry size dimensions do not need to e odd implying no true center (origin) w(-,-) w(-,0) w(-,) w(0,-) w(0,0) w(0,) w(,-) w(-,0) w(,) Pixels of su-imge under msk Exmple of 3x3 templte with its coefficients

3 Introduction 2D Msks (3): Mechnics of Sptil Filtering Moving the templte over ech pixel of the imge At ech pixel (x,y) the response (e.g., output vlue) is determined using some pre-defined reltionship For liner sptil filtering, response is given y sum of the products of the filter coefficients (denoted y w) nd the corresponding imge pixels under the re of the templte Mthemticlly, response g t (x,y) given s g(x,y) = w(-,-)f(x-,y-) + w(-,0)f(x-,y) + + w(0,0)f(x,y) + + w(,0)f(x+,y) + w(,)f(x+,y+) Introduction 2D Msks (4): Mechnics of Sptil Filtering (cont ) Generl filtering expression y) = w( s, t) f ( x + s, y + t) s= t= m row dimension n column dimension m = 2+ nd n = 2+ nd, re non-negtive integers or = (m-)/2 nd = (n-)/2 But this gives output for one pixel loction (x,y) only! Introduction - 2D Msks (5): Mechnics of Sptil Filtering (cont ) To generte complete output imge, the ove eqution (process) must e pplied to ech pixel of input imge e.g., for ech x = 0 M- nd y = 0 N- where M,N re the numer of rows nd columns of the input imge Similr to frequency domin concept clled convolution (more on this in the future ) Hence sometimes referred to s convolving msk with n imge nd the msk is often clled convolution msk Introduction 2D Msks (6): Mechnics of Sptil Filtering (cont ) When we re not interested in processing entire imge ut rther only prticulr pixel (x,y) we cn use the following (shorter) mthemticl definition: R = w z + w 2 z w mn z mn + w w 2 w 3 = mn wi zi i= w 4 w 5 w 6 w 7 w 8 w 9 where, the w s re the filter coefficients nd z s re the imge gry-levels corresponding to the coefficients nd mn is totl numer of coefficients in templte Introduction 2D Msks (7): Importnt Considertions Wht hppens when kernel plced over order pixel? Severl pproches for deling with this. Ignore (don t hndle) order pixels ltogether or ny pixels which led to kernel (or portions of the kernel) flling out of imge rnge 2. Wrp-round 3. Zero-pd Keep in mind, some pproches my led to n output imge whose size is not equl to input imge! Introduction D Msks (): Similr to 2D Cse Except Now D w w 2 w 3 Msk Size of msk cn e odd or even If even how do you define center? Cn convert to odd y prepending zero entry Now origin is middle entry 2 0 2

4 Introduction (): Smoothing Sptil Filters Purpose of Smoothing Sptil Filters Used for lurring, prticulrly in pre-processing Removl of smll detil prior to extrction of lrge oject(s) in imge Bridging ( closing ) of smll gps in lines or curves Also used for noise reduction Cn e chieved with liner or non-liner filter Smoothing Liner Filters (): Essentilly n Averging Filter Output of smoothing filter is simply the verge of pixels in the neighorhood of filter msk (kernel) Also known s low pss filter Elimintes high frequency components (we will descrie this further in lter lectures) Ide of smoothing filter Rndom noise typiclly consists of shrp trnsitions in gry levels Smoothing Liner Filters (2): Ide of smoothing filter (cont ) By replcing the vlue of every pixel y the verge of its neighors, we essentilly reduce the shrp trnsitions in gry levels Most ovious ppliction of smoothing filter is noise reduction However, ewre! Not ll shrp trnsitions re d nd un-wnted! Edges which re typiclly very importnt nd wnted fetures of n imge re lso defined s shrp trnsitions in gry levels Smoothing Liner Filters (3): However, ewre! (cont ) Since smoothing filter removes shrp trnsitions in gry level, verging (smoothing) filters lur edges! Severl other pplictions of smoothing (verging) filters in ddition to noise reduction Smoothing of flse contours which result when not using lrge numer of gry levels Removing irrelevnt detils in n imge pixel regions tht re smll in comprison to the size of the filter kernel Smoothing Liner Filters (4): Smoothing Filter Kernels Exmple of 3 x 3 smoothing filter R = Aove kernel (filter) produces verge of the pixels under the msk Notice tht coefficients re equl to nd not /9! One division insted of nine more efficient! 9 9 zi i=

5 Smoothing Liner Filters (5): Smoothing Filter Kernels (cont ) Another exmple of 3 x 3 smoothing filter zi i= This is n exmple of weighted verge filter Coefficients re not ll the sme vlue! R Some pixels multiplied y higher vlues thus giving those pixels more importnce in verge = 6 9 Smoothing Liner Filters (6): Another exmple of 3 x 3 smoothing filter (cont ) Center coefficient is highest mening center pixel is given most importnce Other coefficients re reduced inversely s function of distnce from the center coefficient Digonl terms re further wy thn the edge neighors nd thus the corresponding pixels in imge provide less importnce to verge Smoothing Liner Filters (7): Another exmple of 3 x 3 smoothing filter (cont ) Mny other types of coefficient msks re lso ville depending on ppliction ut typiclly try to keep the sum of the coefficients n integrl power of 2 (e.g., 6 s in previous exmple) In generl, hrd to notice differences etween imges filtered y oth these filter exmples Smoothing Liner Filters (8): Generl Filter Implementtion Recll the filtering expression for filtering M x N imge with weighted verging filter of size m x n y) = w( s, t) f ( x + s, y + t) s = t = The generl expression eqution cn now e stted s (gin, given n N x M imge with m x n filter where m,n re odd) y ) s = t = = t ) f ( x + s, y + t ) s = s = t ) Smoothing Liner Filters (9): Generl Filter Implementtion (cont ) Smoothing Liner Filters (0): Grphicl Exmples y ) s = t = = t ) f ( x + s, y + t ) s = s = t ) Originl imge 3 x 3 filter Complete filtered imge is otined y pplying ove eqution for ech x = 0,,2,, M- nd y = 0,,2,, N- Denomintor is sum of the msk (kernel) coefficients nd is constnt (e.g., computed once!) Typiclly division pplied once to output imge rther thn t ech stge (pixel output) 5 x 5 filter 9 x 9 filter 5 x 5 filter 35 x 35 filter

6 Smoothing Liner Filters (): Some Notes Regrding Averging Filters For smll filter (e.g., 3 x 3) slight generl lurring of entire imge occurs ut detils tht re out sme size of filter re ffected considerly Noise is less pronounced Jgged orders re plesntly smoothed As filter size increses, lurring is more pronounced Smoothing Liner Filters (2): Imge Blurring Importnt ppliction of verging filter is lurring to get gross representtion of ojects of interest Smller ojects lend into the ckground Lrger ojects ecome more lo-like nd esy to detect Size of msk (kernel) determines size of ojects to e lended into ckground smller msk, smller ojects lended to ckground Smoothing Liner Filters (3): Imge Blurring Grphicl Exmple Imge otined with Hule spce telescope Order-Sttistics Filters (): Non-liner Sptil Filters Response of filter is sed on ordering (rnking) the pixels contined in imge re encompssed y filter nd then replcing center pixel with vlue determined y rnking result One of the most known exmples is the medin filter Originl imge Filtered with 5 x 5 verging msk Threshold imge smll ojects hve disppered Order-Sttistics Filters (2): Medin Filter Replces the vlue of pixel with medin of the gry levels in the neighorhood of tht pixel Cn provide excellent noise reduction with less lurring thn verging filters Prticulrly effective for impulsive noise lso known s slt-nd-pepper noise due to its ppernce s white nd lck dots in the imge Wht is the medin? Given set of vlues, the medin ξ of the set is chosen such tht hlf the vlues of set re less thn ξ nd hlf re more Order-Sttistics Filters (3): Medin Filter (cont ) To perform medin filtering:. Sort the vlues of the pixel in question t sptil loction (x,y) nd its neighors 2. Determine the medin vlue ξ 3. Assign intensity vlue t loction (x,y) the medin vlue Given 3 x 3 msk medin vlue is the 5 th lrgest vlue, 5 x 5 msk medin is 3 th lrgest vlue

7 Order-Sttistics Filters (4): Medin Filter (cont ) Principle function of medin filter is to force points with distinct gry levels to e more like their neighors Isolted clusters of pixels tht re light or drk with respect to their neighors re eliminted Order-Sttistics Filters (5): Medin Filter Grphicl Exmple Comprison etween verging filter Originl imge corrupted y sltnd-pepper noise Imge filtered with 3 x 3 verge filter Imge filtered with 3 x 3 medin filter