image processing basics. what are images? an image is a 2-d rectilinear array of pixels
TRANSCRIPT
Image Processing Basics
What are images?
An image is a 2-d rectilinear array of pixels
Pixels as samples
A pixel is a sample of a continuous function
Images are Ubiquitous
Input Optical photoreceptors Digital camera CCD array Rays in virtual camera (why?)
Output TVs Computer monitors Printers
Properties of Images
Spatial resolution Width pixels/width cm and height pixels/ height cm
Intensity resolution Intensity bits/intensity range (per channel)
Number of channels RGB is 3 channels, grayscale is one channel
Image errors
Spatial aliasing Not enough spatial resolution
Intensity quantization Not enough intensity resolution
Two issues
Sampling and reconstruction Creating and displaying images while reducing
spatial aliasing errors
Halftoning techniques Dealing with intensity quantization
Sampling and reconstruction
Aliasing
Artifacts caused by too low sampling frequency (undersampling) or improper reconstruction
Undersampling rate determined by Nyquist limit (Shannon’s sampling theorem)
Aliasing in computer graphics
In graphics, two major types Spatial aliasing
Problems in individual images Temporal aliasing
Problems in image sequences (motion)
Spatial Aliasing
“Jaggies”
Spatial aliasing
Ref: SIGGRAPH aliasing tutorial
Spatial aliasing
Texture disintegration
Ref: SIGGRAPH aliasing tutorial
Temporal aliasing
Strobing Stagecoach wheels in movies
Flickering Monitor refresh too slow Frame update rate too slow CRTs seen on other video screens
Antialiasing
Sample at a higher rate What if the signal isn’t bandlimited? What if we can’t do this, say because the sampling
device has a fixed resolution?
Pre-filter to form bandlimited signal Low pass filter Trades aliasing for blurring
Non-uniform sampling Not always possible, done by your visual system,
suitable for ray tracing Trades aliasing for noise
Sampling Theory
Two issues What sampling rate suffices to allow a given
continuous signal to be reconstructed from a discrete sample without loss of information?
What signals can be reconstructed without loss for a given sampling rate?
Spectral Analysis
Spatial (time) domain: Frequency domain:
Any (spatial, time) domain signal (function) can be written as a sum of periodic functions (Fourier)
Fourier Transform
Fourier Transform
Fourier transform:
Inverse Fourier transform:
dxexfuF xui
2)()(
dueuFxf xui
2)()(
Sampling theorem
A signal can be reconstructed from its samples if the signal contains no frequencies above ½ the sampling frequency.
-Claude Shannon
The minimum sampling rate for a bandlimited signal is called the Nyquist rate
A signal is bandlimited if all frequencies above a given finite bound have 0 coefficients, i.e. it contains no frequencies above this bound.
Filtering and convolution
Convolution of two functions (= filtering):
Convolution theorem: Convolution in the frequency domain is the same as
multiplication in the spatial (time) domain, and Convolution in the spatial (time) domain is the same as
multiplication in the frequency domain
dxhfxhxfxg
)()()()()(
Filtering, sampling and image processing
Many image processing operations basically involve filtering and resampling. Blurring Edge detection Scaling Rotation Warping
Resampling
Consider reducing the image resolution:
Resampling
The problem is to resample the image in such a way as to produce a new image, with a lower resolution, without introducing aliasing.
Strategy- Low pass filter transformed image by
convolution to form bandlimited signal This will blur the image, but avoid aliasing
Ideal low pass filter
Frequency domain:
Spatial (time) domain:
x
xxsync
)sin(
)(
Image processing in practice
Use finite, discrete filters instead of infinite continous filters
Convolution is a summation of a finite number of terms rather than in integral over an infinite domain
A filter can now be represented as an array of discrete terms (the kernel)
n
nxhfxhxfxg
)()()()()(
Discrete Convolution
Finite low pass filters
Triangle filter
Finite low pass filters
Gaussian filter
Edge Detection
Convolve image with a filter that finds differences between neighboring pixels
111
181
111
filter
Scaling
Resample with a gaussian or triangle filter
Image processing
Some other filters
Summary
Images are discrete objects Pixels are samples Images have limited resolution
Sampling and reconstruction Reduce visual artifacts caused by aliasing Filter to avoid undersampling Blurring (and noise) are preferable to aliasing