Page - 10 - in Document Image Processing

J. Imaging 2018,4, 68 Then,wehaveyj=Rj(I),whereRj(.) isanoperatorthatextracts thepatchyj fromtheimageI , andits transpose, denotedbyRTj (.), is able toputbackapatch into the j-thposition in the reconstructed image. Considering that patches are overlapped, the recovery ofC from {yˆj} canbeobtainedby averagingall theoverlappingpatches,as follows: C= P ∑ j=1 RTj (yˆj)./ P ∑ j=1 RTj (1ps). (9) 4.1.GroupBasedBleed-ThroughPatch Inpainting Traditionalsparsepatchinpainting,wherethemissingpixelvaluesareestimatedusingtheknown pixels from the correspondingpatchonly, ignores the relationshipbetweenneighbouringpatches whenestimatingthemissingpixels [31]. Incorporating informationofsimilarneighbouringpatches assists in theestimationofmissingpixelsandguarantees smooth transitionbyexploiting the local similarity typical of natural images. Following this line, weused a non-local group basedpatch inpainting approachhere. For eachpatch to be inpainted,we search for similar patcheswithin a limitedneighborhoodusingEuclideandistanceassimilaritycriterion, calculatedasgivenbelow: distpatch= √ (Pxref−Pxnew)2+(Pyref−Pynew)2, wherePxref ,Pyref andPxcur,Pycur represents thehorizontalandverticalpositionofcentralpixel in thereferenceandcurrentpatch, respectively. For each patch ywith bleed-throughpixels, we select L non-local similar patcheswithin an Ns×Ns neighbouringwindow. Thesimilarpatchesaregrouped together inamatrix,yG ∈Rps×L. Ineachpatch,wehaveknownpixelsandmissingorbleed-throughpixels. LetΩbeanoperator that extracts theknownpixels inapatchand Ω¯anoperator thatextracts themissingpixels, so thatΩ(y) represents theknownpixelsand Ω¯(y) represents themissingpixels inapatchy. An illustrationof suchpixels’ extraction isgiven inFigure2. Figure2.Extractingknownandbleed-throughpixels inapatch. Similarly, for a group of patches,Ω(yG) extracts the knownpixels of all patches, averaging multiple entries at the same pixel location, and Ω¯(yG) represents the missing pixels. Given a well-trained dictionaryD, the sparse reconstruction of patcheswith bleed-throughpixels can be formulatedas xˆ=argmin x ‖Ω(yG)−Ω(Dx)‖2+α‖ x ‖0, (10) where α is a small constant. The first term of Equation (10) represents the data-fidelity and the secondtermis thesparseregularization.Afterobtainingthesparsecoefficients xˆusingEquation(10), anestimate for thebleed-throughpixelscanbeobtainedusing Ω¯(y)= Ω¯(Dxˆ). (11) Using thereconstructedpatches,anestimated,bleed-throughfree image isobtainedbymeansof patchaveraging,accordingtoEquation(9). 10