高阶次Caputo型分数阶微分算子及其图像增强应用王相海1,2张文雅2邢俊宇2吕芳2穆振华11（辽宁师范大学地理科学学院辽宁大连116029）2（辽宁师范大学数学学院辽宁大连116029）（xhwang@lnnu.edu.cn）High-orderCaputoFractionalOrderDifferentialOperatorandItsApplicationinImageEnhancementWangXianghai1,2,ZhangWenya2,XingJunyu2,LüFang2,andMuZhenhua11（SchoolofGeographicalScience,LiaoningNormalUniversity,Dalian,Liaoning116029）2（SchoolofMathematics,LiaoningNormalUniversity,Dalian,Liaoning116029）AbstractUnlikethetraditionalintegercalculus,whichusuallyhasintuitivegeometricandphysicalmeaning,thedefinitionoffractionalcalculusisgenerallycomplexandpresentsdifferentforms.However,itscharacteristicssuchasmemoryandnonlocalityhavelaidagoodmathematicalfoundationforsolvingsomecomplexproblemsintheengineeringfield.Atthesametime,signalandimageprocessingbasedonfractionalcalculushavealsoattractedattentioninrecentyears.Atpresent,thecommonfractionaldifferentialoperatorsusedinimageprocessingincludeGrünwald-Letnikov(G-L)fractionaldifferential,Riemann-Liouville(R-L)fractionaldifferentialandCaputofractionaldifferential.AlthoughG-LandR-Loperatorscanenhancetheimagetoacertainextent,theircapabilitesoftheimprovementofimagecontrastanddefinitionislimited.Atpresent,Caputodifferentialmaskoperatorsaremostlylimitedtolow-orderoperatorsoforderin(0,1),andtheresearchandapplicationofhigh-orderoperatorsarerelativelyfew.Inthispaper,thehigh-orderCaputofractionaldifferentialoperatoranditsapplicationinimageenhancementarestudied.Firstly,adifferentialmaskoperatorbasedonforwarddifferenceisconstructedforCaputofractionaldifferentialwithorderin(1,2)and(2,3),anditserrorisdemonstrated;Further,thegeneralformofhigh-orderCaputofractionaldifferentialoperatorisstudied,andarepresentationbasedonmatrixisgiven.Onthisbasis,theproposedhigh-orderCaputofractionaldifferentialmaskoperatorisappliedtoimageenhancement,andthecomparativeexperimentsofimageenhancementarecarriedoutformaskoperatorsofdifferentordersandsizes.Theexperimentalresultsshowthatthep...
