Computer Vision Spring 2012 15-385,-685 Instructor: S. Narasimhan Wean 5409 T-R 10:30am – 11:50am.

Computer Vision

Spring 2012 15-385,-685

Instructor: S. Narasimhan

Wean 5409

T-R 10:30am – 11:50am

Image Resampling and Pyramids

Lecture #8

Image Scaling

This image is too big tofit on the screen. Howcan we reduce it?

How to generate a half-sized version?

Image Sub-Sampling

Throw away every other row and

column to create a 1/2 size image- called image sub-sampling

1/4

1/8

Image Sub-Sampling

1/4 (2x zoom) 1/8 (4x zoom)1/2

Good and Bad Sampling

Good sampling:•Sample often or,•Sample wisely

Bad sampling:•Aliasing!

Aliasing

Alias: n., an assumed name

Picket fence recedinginto the distance willproduce aliasing…

Input signal:

x = 0:.05:5; imagesc(sin((2.^x).*x))

Matlab output:

WHY?

Alias!

Not enough samples

Really bad in video

Wagon-wheel effect

Stroboscopic effect

Sampling Theorem

Continuous signal:

Shah function (Impulse train):

Sampled function:

Sampling Theorem

Sampled function:

Only if

Sampling frequency

Nyquist Frequency

If

Aliasing

When can we recover from ?

Only if (Nyquist Frequency)

We can use

Then and

Sampling frequency must be greater than

Sub-Sampling with Gaussian Pre-Filtering

G 1/4

G 1/8

Gaussian 1/2

• Solution: filter the image, then subsample– Filter size should double for each ½ size reduction. Why?

G 1/4 G 1/8Gaussian 1/2

Sub-Sampling with Gaussian Pre-Filtering

Compare with...

1/4 (2x zoom) 1/8 (4x zoom)1/2

Aliasing

Canon D60 (w/ anti-alias filter) Sigma SD9 (w/o anti-alias filter)

From Rick Matthews website, images by Dave Etchells

Image Resampling

• What about arbitrary scale reduction?

• How can we increase the size of the image?

• Recall how a digital image is formed

– It is a discrete point-sampling of a continuous function

– If we could somehow reconstruct the original function, any new image could be generated, at any resolution and scale

1 2 3 4 5

Image Resampling

• So what to do if we don’t know

1 2 3 4 52.5

1

– Answer: guess an approximation– Can be done in a principled way: filtering

Resampling filters

• What does the 2D version of this hat function look like?

• Better filters give better resampled images– Bicubic is common choice

performs linear interpolation

performs bilinear interpolation

Bilinear interpolation

• A common method for resampling images

Image Rotation

Multi-Resolution Image Representation

• Fourier domain tells us “what” (frequencies, sharpness, texture properties), but not “where”.

• Spatial domain tells us “where” (pixel location) but not “what”.

• We want a image representation that gives a local description of image “events” – what is happening where.

• Naturally, think about representing images across varying scales.

Figure from David Forsyth

Multi-resolution Image Pyramids

High resolution

Low resolution

Space Required for Pyramids

Constructing a pyramid by taking every second pixel leads to layers that badly misrepresent the top layer

Even worse for synthetic images

Decimation

Expansion

Interpolation Results

The Gaussian Pyramid

• Smooth with Gaussians because– a Gaussian*Gaussian=another Gaussian

• Synthesis – smooth and downsample

• Gaussians are low pass filters, so repetition is redundant• Kernel width doubles with each level

2)*( 23 gaussianGG

1G

The Gaussian Pyramid

High resolution

Low resolution

Image0G

2)*( 01 gaussianGG

2)*( 12 gaussianGG

2)*( 34 gaussianGG

blur

blur

blur

down-sample

down-sample

down-sampleblurdown-sample

Pyramids at Same Resolution

expand

expand

expand

Gaussian Pyramid Laplacian Pyramid

The Laplacian Pyramid

0G

1G

2GnG

- =

0L

- =1L

- = 2Lnn GL

)expand( 1 iii GGL

)expand( 1 iii GLG

Applications of Image Pyramids

• Coarse-to-Fine strategies for computational efficiency.• Search for correspondence

– look at coarse scales, then refine with finer scales

• Edge tracking– a “good” edge at a fine scale has parents at a coarser scale

• Control of detail and computational cost in matching– e.g. finding stripes– very important in texture representation

• Image Blending and Mosaicing • Data compression (laplacian pyramid)

Fast Template Matching

Fast Template Matching

Blending Apples and Oranges

Image Blending and Mosaicing

Pyramid blending of Regions

Horror Photo

© prof. dmartin

Image Fusion

Multi-scale Transform (MST) = Obtain Pyramid from ImageInverse Multi-scale Transform (IMST) = Obtain Image from Pyramid

Multi-Sensor Fusion

Image Compression

Next Class

• Edge Detection