Gaussian-Laplacian Pyramid Image Coding

• Down-sampling:
  • Convolution with Gaussian (low-pass):
    
    $$f'_{l+1}=g * f_l$$
    
    

    All images so obtained by Gaussian (low-passed) filtering for a Gaussian pyramid.

  • Difference image (band-pass):
    
    $$h_{l+1}=f_l-f'_{l+1}$$
    
    

  • Down-sampling of $f'_{l+1}$:
    
    $$f_{l+1}[n]=f'_{l+1}[2n]$$
    
    

  Note that image $h_{l+1}=f_l-f'_{l+1}$ is the difference between two images convolved by gaussian kernals of difference sizes, and is a bandpass filtered image. All images so obtained form a Laplician pyramid.

  

  

• Feature Extraction

• Image Compression

• Up-sampling:
  • Interlacing of zeros (rows and columns)

    
    

    $$f'_l[n]=\left\{ \begin{array}{ll} f_l[n/2] & n=0,2,4,\cdots \\ 0 & \mbox{otherwise} \end{array} \rignt.$$
    
    

  • Convolution with Gaussion (low-pass):
    
    $$f_{l-1}=g*f'_l+h_l$$