On Implicit Image Derivatives and Their Applications Alexander - PowerPoint PPT Presentation

Introduction Background From Explicit to Implicit Differentiation Schemes Discussion On Implicit Image Derivatives and Their Applications Alexander Belyaev Mohamed Omran Uni Saarland Milestones and Advances in Image Analysis, 2012 1 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Background From Explicit to Implicit Differentiation Schemes Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 2 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 3 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion The Importance of Image Derivatives differentiation: one of the most fundamental tasks of low level image processing used to detect edges and corners: perceptual building blocks necessary for a host of other image processing operations: e.g. smoothing, deblurring, segmentation Figure: From Dalal & Triggs, 2005 4 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 5 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion Contributions derivatives in image processing typically obtained using imprecise explicit schemes main contributions [2, 3]: establishing a link between implicit and explicit finite 1 differences used for gradient estimation introducing new implicit differencing schemes and evaluating 2 their properties attempting to demonstrate the usefulness and potential of 3 implicit finite differencing schemes for image processing tasks 6 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 7 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Basic Problem Background Current Approach From Explicit to Implicit Differentiation Schemes Previous Work Discussion Implicit Finite Differences common among numerical mathematicians and computational physicists seminal paper by Lele [1]: demonstrated superior performance of implicit finite difference schemes uses (among others): accurate numerical simulations of physical problems involving 1 wave propagation phenomena modelling weather phenomena 2 accurate visualisation of volumetric data 3 8 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 9 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Explicit vs. Implicit Methods categorisation of numerical schemes explicit methods: dependent variable can be obtained directly from input variables implicit methods: more complex relationship between variables, requires solving systems of linear equations 10 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 11 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Taylor Approximations Definition (Taylor Approximant of Order k) k f ( i ) ( x ) ( h ) i + R k ( h ) � f ( x + h ) = i ! i = 0 = f ( x ) + f ′ ( x )( h ) + . . . + f ( k ) ( x ) ( h ) k + R k ( h ) (1) k ! approximation of a function f, centered at x used to derive explicit finite difference schemes 12 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Padé Approximations Definition (Padé Approximant of Order m/n) m a j x j � = a 0 + a 1 x + a 2 x 2 + . . . + a m x m j = 0 R ( x ) = (2) 1 + b 1 x + b 2 x 2 + . . . + b n x n n � b k x k 1 + k = 1 approximation of a function by a rational function often more precise than the Taylor approximation later used to derive powerful implicit differentiation schemes 13 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 14 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Explicit vs. Implicit Methods Background Taylor and Padé Approximations From Explicit to Implicit Differentiation Schemes Differentiation as a Linear Operator Discussion Differentiation in the Frequency Domain differentiation is a linear operation, thus has interesting properties in the frequency domain in particular: F [ f n ) ] = ( j ω ) n F [ f ]( u ) , with ω = 2 π u 3 ideal derivative 2.5 Modified wavenumber 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 Wavenumber ω Figure: Ideal Derivative 15 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Standard Explicit Schemes Background From Explicit to Implicit Schemes From Explicit to Implicit Differentiation Schemes Advanced Implicit Schemes Discussion Outline Introduction 1 Motivation Current Approach Previous Work Background 2 Explicit vs. Implicit Methods Taylor and Padé Approximations Differentiation as a Linear Operator From Explicit to Implicit Differentiation Schemes 3 Standard Explicit Schemes From Explicit to Implicit Schemes Advanced Implicit Schemes Discussion 4 16 / 32 M. Omran On Implicit Image Derivatives and Their Applications

Introduction Standard Explicit Schemes Background From Explicit to Implicit Schemes From Explicit to Implicit Differentiation Schemes Advanced Implicit Schemes Discussion Standard Central Difference Definition (Central Difference Operator) f ′ ( x ) ≈ 1 2 h [ f ( x + h ) − f ( x − h )] (3) 1 � � − 1 0 1 (4) 2 h frequency response of the central difference operator: jsin ω for 2D: rotate the mask for differentiation in y-direction with estimates for f x and f y , we can compute gradient magnitude and gradient orientation 17 / 32 M. Omran On Implicit Image Derivatives and Their Applications