Computer Vision – Enhancement(Part III) Hanyang University Jong-Il Park

Department of Computer Science and Engineering, Hanyang University The Fourier transform Definition  1-D Fourier transform  2-D Fourier transform

Department of Computer Science and Engineering, Hanyang University 1-D case Fourier series

Department of Computer Science and Engineering, Hanyang University M-point spectrum

Department of Computer Science and Engineering, Hanyang University 2D Fourier series 2-D case  is periodic : period = 1  Sufficient condition for existence of

Department of Computer Science and Engineering, Hanyang University original 256x256 lena Centered and normalized spectrum (log-scale) Eg. 2D Fourier transform

Department of Computer Science and Engineering, Hanyang University Filtering in Frequency Domain

Department of Computer Science and Engineering, Hanyang University Unitary Transforms Unitary Transformation for 1-Dim. Sequence  Series representation of  Basis vectors :  Energy conservation :

Department of Computer Science and Engineering, Hanyang University Unitary Transformation for 2-D Sequence  Definition :  Basis images :  Separable Unitary Transforms: 2D Unitary Transformation

Department of Computer Science and Engineering, Hanyang University 2-D DFT

Department of Computer Science and Engineering, Hanyang University

Department of Computer Science and Engineering, Hanyang University Separability

Department of Computer Science and Engineering, Hanyang University Transform Operations

Department of Computer Science and Engineering, Hanyang University Centered Spectrum

Department of Computer Science and Engineering, Hanyang University Generalized Linear Filtering Unitary transform Point operation Inverse transform HPF BPF LPF Zonal masks for Orthogonal(DCT, DHT etc) transforms BPF LPF HPF BPF LPF BPF LPF BPF LPF Zonal masks for DFT

Department of Computer Science and Engineering, Hanyang University Eg. Filtering - DFT

Department of Computer Science and Engineering, Hanyang University Eg. Filtering - LPF and HPF

Department of Computer Science and Engineering, Hanyang University Eg. Filtering - HPF + DC

Department of Computer Science and Engineering, Hanyang University Correspondence between Spatial Domain and Frequency Domain

Department of Computer Science and Engineering, Hanyang University Ideal LPF  NOT practical because of “ringing”

Department of Computer Science and Engineering, Hanyang University Ringing

Department of Computer Science and Engineering, Hanyang University Illustration of Ringing convolution Ideal LPF

Department of Computer Science and Engineering, Hanyang University Butterworth LPF

Department of Computer Science and Engineering, Hanyang University Ringing in BLPF

Department of Computer Science and Engineering, Hanyang University Eg. 2 nd order Butterworth LPF A good compromise between Effective LPF and Acceptable ringing

Department of Computer Science and Engineering, Hanyang University Gaussian LPF(GLPF)

Department of Computer Science and Engineering, Hanyang University Eg. GLPF No ringing!

Department of Computer Science and Engineering, Hanyang University Application of GLPF(1)

Department of Computer Science and Engineering, Hanyang University Application of GLPF(2) Soft and pleasing

Department of Computer Science and Engineering, Hanyang University Homomorphic Filtering  f(x, y) = i(x, y) r(x, y) i(x,y) : - illumination component - responsible for the dynamic range - low freq. Components r(x,y) : - reflectance component - responsible for local contrast - high frequency component  enhancement based on the image model - reduce the illumination components - enhance the reflectance components

Department of Computer Science and Engineering, Hanyang University Transform Operations Homomorphic System note log Linear System exp log exp HP LP g(x, y) f(x, y)  <1  >1

Department of Computer Science and Engineering, Hanyang University Eg. Homomorphic filtering(1)

Department of Computer Science and Engineering, Hanyang University Eg. Homomorphic filtering(2)