Convolution and Filtering: The Convolution Theorem
Convolution and Filtering: The Convolution Theorem
Introduction
Linear Systems and Responses
Convolution and Filtering:
The Convolution Theorem
Input
Output
Impulse Response
Transfer Function
Relationship
More Spatial Filtering
More Spatial Filtering
Sharpening
Sharpening
More Spatial Filtering
CS 450: Introduction to Digital Signal and Image Processing
The goal of sharpening is to enhance differences,
so all sharpening kernels involve differences
some pos
Level Operations (Part 1)
Level Operations (Part 1)
Introduction
Point Processing
Level Operations (Part 1)
Simplest kind of enhancement: point operations.
Process each point independently of the others.
All you can do is remap the samples value:
CS 450:
Level Operations (Part 1)
Level Operations (Part 1)
CS 450: Introduction to Digital Signal and Image Processing
Level Operations (Part 1)
Introduction
Point Processing
Simplest kind of enhancement: point operations.
Process each point independently of the
What are Signals?
CS 450:
Signal: a function carrying information
and Image Processing
Examples:
Introduction to Digital Signal
A u d io
R a d io /T e le v is io n
Im ag es
I n tr o d u c tio n a n d A p p lic a tio n s
Why Signals?
Communications
M o d
CS 450:
Introduction to Digital Signal
and Image Processing
Introduction and Applications
What are Signals?
Signal: a function carrying information
Examples:
Audio
Radio/Television
Images
Why Signals?
Communications
Modems/Networks/Wireless
Audio
Vi
Level Operations (Part 2)
Level Operations (Part 2)
Histograms
Histograms
Level Operations (Part 2)
A histogram H (r ) counts how many times each quantized value
occurs.
Example:
CS 450: Introduction to Digital Signal and Image Processing
Level Operations
Level Operations (Part 2)
Level Operations (Part 2)
CS 450: Introduction to Digital Signal and Image Processing
Level Operations (Part 2)
Histograms
Histograms
A histogram H (r ) counts how many times each quantized value
occurs.
Example:
Level Operations
CS 450: Introduction to Digital
Signal and Image Processing
Geometric Operations
Geometric Operations
Transformations (Shift, Rotation, etc.)
Resizing
Adding/Correcting a Warp
Texture Mapping
Morphing
Example: Texture Mapping
Mapping an image
onto the su
Geometric Operations
CS 450: Introduction to Digital
Signal and Image Processing
Geometric Operations
Example: Texture Mapping
Transformations (Shift, Rotation, etc.)
Resizing
Adding/Correcting a Warp
Texture Mapping
Morphing
Example: Morphing
Warp a pai
CS 450: Introduction to Digital
Signal and Image Processing
Geometric Operations
Geometric Operations
Transformations (Shift, Rotation, etc.)
Resizing
Adding/Correcting a Warp
Texture Mapping
Morphing
Example: Texture Mapping
Mapping an image
onto the su
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Magnitude and Phase
The Fourier Transform:
Examples, Properties, Common Pairs
Remember: complex numbers can be thought of as (real,imaginar
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform:
Examples, Properties, Common Pairs
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
The Fourier Transform: Examples, Properties, C
The 2D Fourier Transform
The 2D Fourier Transform
Introduction
TwoDimensional Continous Fourier Transform
Basis functions are product of
generalized sinusoids with frequency u in the x direction
generalized sinusoids with frequency v in the y direction
CS 450: Introduction to
Digital Signal and Image
Processing
A Few More Fundamentals
Sampling Revisited
How much sampling is enough?
What happens if you sample above this?
Avoids dangers of theoretical limits
Better for intermediate processing
What happens
Sampling Revisited
CS 450: Introduction to
Digital Signal and Image
Processing
How much sampling is enough?
What happens if you sample above this?
A Few More Fundamentals
Moir Patterns
Avoids dangers of theoretical limits
Better for intermediate processin
CS 450: Introduction to Digital
Signal and Image Processing
Where Have We Been?
Where Signals Come From
How audio, images, and video are stored
Sampling
Quantization
Cameras and scanners
Noise
Apertures
Tradeoffs
Level Operations
Processing sample
Video Compression
Video Compression
Storing Video
Video Compression
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Simplest: Store fullyencoded uncompressed frames
(2D images) as a sequence
Better: Individua
Video Compression
Video Compression
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Video Compression
Storing Video
Simplest: Store fullyencoded uncompressed frames
(2D images) as a sequence
Better: Individua
Generalized Harmonic Functions
Generalized Harmonic Functions
Harmonic Functions
What does f (t ) = ei 2ut look like?
Generalized Harmonic Functions
Imaginary
CS 450: Introduction to Digital Signal and Image Processing
i
Bryan Morse
BYU Computer Science

Generalized Harmonic Functions
Generalized Harmonic Functions
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Generalized Harmonic Functions
Harmonic Functions
What does f (t ) = ei 2ut look like?
Imaginary
i

Convolution and Linear Systems
Convolution and Linear Systems
Introduction
Analyzing Systems
Convolution and Linear Systems
CS 450: Introduction to Digital Signal and Image Processing
Goal: analyze a device that turns one signal into another.
Notation:
f
Convolution and Linear Systems
Convolution and Linear Systems
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Convolution and Linear Systems
Introduction
Analyzing Systems
Goal: analyze a device that turns one
More Spatial Filtering
More Spatial Filtering
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
More Spatial Filtering
Sharpening
Sharpening
The goal of sharpening is to enhance differences,
so all sharpening ker
Spatial Filtering
Spatial Filtering
Introduction
Neighborhood Operations
Output is a function of a pixels value and its neighbors
Example (8connected neighbors):
Spatial Filtering
CS 450: Introduction to Digital Signal and Image Processing
0
f (x 1, y 1)
Spatial Filtering
Spatial Filtering
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Spatial Filtering
Introduction
Neighborhood Operations
Output is a function of a pixels value and its neighbors
Example (8con
Sampling
Sampling
Introduction
Sampling
f(t)
Sampling
Continuous
CS 450: Introduction to Digital Signal and Image Processing
t
Bryan Morse
BYU Computer Science
f(t)
Discrete
t
Sampling
Sampling
Introduction
Sampling In The Time/Spatial Domain
Sampling In
Sampling
Sampling
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Sampling
Introduction
Sampling
f(t)
Continuous
t
f(t)
Discrete
t
Sampling
Introduction
Sampling
Sampling a continuous function f to produce disc
The 2D Fourier Transform
The 2D Fourier Transform
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
The 2D Fourier Transform
Introduction
TwoDimensional Continous Fourier Transform
Basis functions are product
The Fourier Transform
The Fourier Transform
Transforms
General Idea of Transforms
Suppose that you have an orthonormal (orthogonal, unit
length) basis set of vectors cfw_ek .
The Fourier Transform
Any vector in the space spanned by this basis set can be
r