Filtering Images Work in the spatial domain Convert the filter into a matrix, the filter mask eg 3x3 box filter performs averaging of a neighborhood:

Filtering Algorithm If Iinput is the input image, and Ioutput is the output image, M is the filter mask and k is the mask size: Care must taken at the boundary Make the output image smaller Extend the input image in some way

Box Filter Box filters smooth by averaging neighbors In frequency domain, clearly a low-pass filter

Bartlett Filter Triangle shaped filter in spatial domain In frequency domain, product of two box filters, so attenuates high frequencies more than a box

Constructing Masks: 2D Sample the filter function at matrix “pixels” eg 2D Bartlett Can go to edge of pixel or middle of next: results are slightly different 1 1 3 1 5 1 2 1 1 2 1 4

Constructing Masks: 3D Multiply 2 2D masks together using outer product M is 3D mask, m is 2D mask 0.2 0.6 0.2 0.2 0.04 0.12 0.04 0.6 0.12 0.36 0.12 0.2 0.04 0.12 0.04

Guassian Filter Attenuates high frequencies even further In 2d, rotationally symmetric, so fewer artifacts

Constructing Gaussian Mask Use the binomial coefficients Central Limit Theorem (probability) says that with more samples, binomial converges to Gaussian 1 1 2 1 4 1 1 4 6 4 1 16 1 1 6 15 20 15 6 1 64

High-Pass Filters A high-pass filter can be obtained from a low-pass filter If we subtract the smoothed image from the original, we must be subtracting out the low frequencies What remains must contain only the high frequencies High-pass masks come from matrix subtraction: eg: 3x3 Bartlett

Fixing Negative Values The negative values in high-pass filters can lead to negative image values Most image formats don’t support this Solutions: Truncate: Chop off values below min or above max Offset: Add a constant to move the min value to 0 Re-scale: Rescale the image values to fill the range (0,max)

Edge Enhancement High-pass filters give high values at edges, low values in constant regions Adding high frequencies back into the image enhances edges One approach: Image = Image + [Image – smooth(Image)] Low-pass High-pass

Image Warping An image warp is a mapping from the points in one image to points in another f tells us where in the new image to put the data from x in the old image Simple example: Translating warp, f(x) = x+o, shifts an image

Reducing Image Size Warp function: f(x)=kx, k < 1 Problem: More than one input pixel maps to each output pixel Solution: Filter down to smaller size Apply the filter, but not at every pixel, only at desired output locations eg: To get half image size, only apply filter at every second pixel

2D Reduction Example (Bartlett)

Ideal Image Size Reduction Reconstruct original function using reconstruction filter Resample at new resolution (lower frequency) Clearly demonstrates that shrinking removes detail Expensive, and not possible to do perfectly in the spatial domain

Enlarging Images Warp function: f(x)=kx, k > 1 Problem: Have to create pixel data More pixels in output than in input Solution: Filter up to larger size Apply the filter at intermediate pixel locations eg: To get double image size, apply filter at every pixel and every half pixel New pixels are interpolated from old ones Filter encodes interpolation function

Enlargement

Ideal Enlargement Reconstruct original function Resample at higher frequency Original function was band-limited, so resampling does not add any extra frequency information

Image Morphing Process to turn one image into another Define parameterized path from each point in the original image to its destination in the output image Animate points along paths, while interpolating colors from source to destination

2D Morphing Example Final resample

Morphing Issues User interface to specify pixel mapping and paths is important Different interpolation schemes are possible to get different effects Simple morphs, like cross dissolve, are key to video effects: Cross dissolve simply interpolates colors between the two images. Pixels move on straight paths