CompPhoto: Pyramids

Image Pyramid

A multi-scale image representation that stores images at progressively lower resolutions to analyze different levels of detail.

Gaussian Pyramid

A pyramid created by repeatedly smoothing (blurring) and downsampling an image; captures coarse image structure.

Laplacian Pyramid

A pyramid formed from the differences between successive levels of a Gaussian pyramid; captures fine image details.

Purpose of Pyramids

Allows operations like blending, compression, and texture analysis at multiple resolutions.

Reduce Operation

Applies a low-pass filter (blur) and then downsamples the image by a factor of 2.

Expand Operation

Performs the inverse of reduce by upsampling and interpolating between known pixel values.

Relationship Between Levels

Each level gₖ = REDUCE(gₖ₋₁), where higher k corresponds to a coarser (lower resolution) image.

Gaussian Filter in Pyramids

Used to smooth images before downsampling to prevent aliasing.

Parameter a in Gaussian Pyramid

Determines the shape of the Gaussian kernel; typically between 0.3 and 0.6.

Gaussian Pyramid Formula

gₖ = h ✶ g(k−1), where h is the Gaussian kernel and ✶ represents convolution.

Laplacian Pyramid Formula

Lₗ = gₗ − EXPAND(gₗ₊₁); each level stores details lost between Gaussian levels.

Fine vs Coarse Levels

Fine levels store high-frequency details; coarse levels store low-frequency, large-scale information.

Pyramid Construction Process

1) Blur the image, 2) Subsample to create Gaussian pyramid, 3) Subtract to get Laplacian pyramid.

Pyramid Blending

A technique that combines images smoothly by blending across multiple pyramid levels.

Goal of Pyramid Blending

To avoid visible seams and ghosting by mixing images at multiple frequency scales.

Blending Process Steps

1) Build Laplacian pyramids of images A and B, 2) Build Gaussian mask pyramid of region R, 3) Combine pyramids using the mask, 4) Collapse to final image.

Pyramid Blending Equation

LO(i,j) = GR(i,j) × LA(i,j) + (1−GR(i,j)) × LB(i,j), where GR is the Gaussian mask and LA/LB are Laplacian pyramids.

Coarse-to-Fine Blending

Blending starts from low-frequency (coarse) levels and moves up to finer details for seamless transitions.

Optimal Window Size (Blending)

Window = size of largest prominent feature; helps minimize visible seams.

Avoiding Ghosting in Blending

Keep window ≤ 2× size of smallest prominent feature to prevent overlapping misalignments.

Frequency Domain Blending

Decompose images into frequency bands (octaves) using FFT and blend corresponding bands.

Burt and Adelson (1983)

The researchers who introduced multi-resolution image blending using Gaussian and Laplacian pyramids.

Octave in Frequency Domain

A range of frequencies differing by a factor of two; used in multi-band image blending.

Sampling in Pyramids

Downsampling reduces resolution; proper sampling prevents aliasing artifacts.

Aliasing

Occurs when image frequencies exceed half the sampling rate, causing visual distortions.

Good Sampling Practices

Sample often or sample wisely to capture sufficient detail and prevent aliasing.

Practical Applications of Pyramids

Image compression, texture mapping, focus stacking, and multi-scale feature extraction.