Seite - 7 - in Document Image Processing

J. Imaging 2018,4, 68 for i=1,...,N,where‖.‖0 is the 0-norm,whichcounts thenon-zeroelements inx, andthedictionary updatestage for theXobtainedfromthesparsecodingstage D=argmin D ‖Y−DX‖2F . (3) Dictionary learningalgorithmsareoftensensitive to thechoiceofm. Theupdatestepcaneither be sequential (oneatomata time) [51,52], orparallel (all atomsatonce) [53,54]. Adictionarywith sequentialupdate,althoughcomputationallyabitexpensive,willgenerallyprovidebetterperformance than theparallel update, due to thefiner tuningof eachdictionaryatom. In sequential dictionary learning, thedictionaryupdateminimizationproblem(3) is split intoK sequentialminimizations, byoptimizingthecost function(3) foreach individualatomwhilekeepingfixedtheremainingones. Most of the proposed algorithmshave kept the two stage optimizationprocedure, the difference appearingmainly in thedictionaryupdate stage,with someexceptionshavingadifference in the sparsecodingstageaswell [43]. In themethodproposedin[51],whichhasbecomeabenchmark in dictionary learning,eachcolumndkofDanditscorrespondingrowofcoefficientsxrowk areupdated basedonarank-1matrixapproximationof theerror forall thesignalswhendk is removed {dk,xrowk } = arg mindk,xrowk ‖Y−DX‖2F = arg min dk,xrowk ‖Ek−dkxrowk ‖2F, (4) whereEk =Y−∑Ki=1,i =kdixrowi . Thesingularvaluedecomposition (SVD)ofEk =UΔV isused to findtheclosest rank-1matrixapproximationofEk. Thedkupdate is takenas thefirst columnofU, andthexrowk update is takenas thefirst columnofVmultipliedbythefirstelementofΔ. Toavoid the lossof sparsity inxrowk thatwouldbecreatedbythedirectapplicationof theSVDonEk, in [51], itwas proposedtomodifyonly thenon-zeroentriesofxrowk resultingfromthesparsecodingstage. This is achievedbytaking intoaccountonly thesignalsyi thatuse theatomdk inEquation(4),or,by taking, insteadof theSVDofEk, theSVDofERk =EkIwk,wherewk= {i|1≤ i≤N;xrowk (i) = 0}, and Iwk is theN×|wk| submatrixof theN×N identitymatrixobtainedbyretainingonly thosecolumnswhose indexnumbersare inwk. 3.Bleed-ThroughIdentification Thealgorithmusedtorecognise thepixels thatbelongto thebleed-throughpatternmakesuseof bothsidesof thedocument, i.e., therectoandtheverso images,andsuitablycompares their intensities inapixel-by-pixelmodality.Hence, it is essential that twocorresponding,oppositepixelsexactly refer to the samepieceof information. Inotherwords, at location (i, j), to thepixel in a side, let us say ableed-throughpixel,mustcorrespond, in theoppositeside, the foregroundpixel thathasgenerated it, and vice versa. In order to ensure thismatching, one of the two images needs to be reflected horizontally,andthenthe twoimagesmustbeperfectlyaligned[55]. Thewayinwhichweperformthecomparisonbetweenpairsofcorrespondingpixels ismotivated bysomeconsiderationsabout thephysicalphenomenon. Indeed, throughexperience,weobserved that, in themajority of themanuscripts examined, due topaperporosity, the seeped inkhas also diffusedthroughthepaperfiber.Hence, ingeneral, thebleed-throughpattern isasmearedandlighter version of the opposite text that has generated it. Note that this assumptiondoes notmean that, on thesameside,bleed-throughis lighter thanthe foregroundtext. In fact,oneachside, the intensity ofbleed-through isusuallyveryvariable,which ishighlynon-stationary, andsometimescanbeas darkas the foregroundtext. Otherconsiderationscanbemadebyreasoningintermsof“quantityof ink”.Indeed, it isapparent that thequantityof ink shouldbezero in thebackground, i.e., theunwrittenpaper, nomatter the colorof thepaper,maximuminthedarkerandsharper foregroundtext,andminimuminthe lighter 7