Image Pyramids (Page 2 of PDF)

• Structure: Represent images at multiple resolutions. A collection of decreasing resolution images arranged in a pyramid shape.
• Highest Resolution: At the pyramid base (Level $J$).
• Resolution and Size: Decrease as you move up the pyramid.
• Base Level: Size $2^J\times 2^J$.
• General Level $j$ size: $2^j\times 2^j$, $0\le j\le J$.

• Number of Pixels: $N^2(1+\frac{1}{4}+\frac{1}{4^2}+...+\frac{1}{4^P})\le \frac{4}{3}{N}^2$

DIP_Lecture9, p.2

• Building Image Pyramids (Iterative Process)

  1. Compute Reduced-Resolution Approximation: Filter and downsample (by a factor of 2) the level $j$ input image. Place the result at level $j-1$ of the approximation pyramid.
  2. Create Estimate of Level j Input: Upsample and filter the reduced resolution approximation from step 1. The resulting prediction image has the same dimensions as the level $j$ input.
  3. Compute Difference: Subtract the prediction image (step 2) from the input (step 1). Place the result in level $j$ of the prediction residual pyramid.

• After $P$ iterations: Level $J-P$ approximation output is placed in the pyramid at level $J-P$.

• Types of Pyramids (based on filters):
  • Mean pyramids (neighborhood averaging)
  • Gaussian pyramids (lowpass Gaussian filtering)
  • Subsampling pyramids (no filtering)
  • Filter can be based on: Nearest, Bilinear, Bicubic.

Upsampling and Downsampling

• Upsampling:

  • Doubles the spatial dimensions of approximation images.

  • Given a 1D sequence $f(n)$, the upsampled sequence is:

  f_{2\uparrow}(n) = \begin{cases} f(n/2) & \text{if } n \text{ is even} \ 0 & \text{otherwise} \end{cases}$$

    Insert a 0 after every value in sequence.
  

• Downsampling:

  • Halves the spatial dimensions of the prediction images.

  • Given by:

    $f_{2\downarrow }(n)=f(2n)$ Discard every other sample.