Discrete Images (Chapter 7)
Fourier Transform on discrete and bounded domains.
Sampling function
Given an image:
1. Zero boundary condition
2. Periodic boundary condition
(tiling images)
Main point: Fourier transform is still
valid with each condition and depends
only
Mathematics in Chapter 7 can look heavy at times….
But don’t get bogged down by the symbols and
notations !!
For example, the Fourier transform of a periodic function is
discrete.
Sampling Theorem
If the Fourier transform of a function is bandlimited, then, it
can be reconstructed from samples on a regular grid.
If
Then f(u, v) can be recovered by knowing the values
for all k, l
Conversely
If the signal is known to be bandlimited with
of the highest frequency present
the wavelengh
Then the sampling interval should be less than
In particular, if
is the sampling interval, then the signal
can contain frequencies only up to the Nyquist frequency
If it is to be faithfully reconstructed from Samples.
Edge and Edge Detection
February 6
Edges (or Edge points) are pixels at or around which the image
values undergo a sharp variation.
Edge Detection: Given an image corrupted by acquisition noise,
locate the edges most likely to be generated by scene elements,
not by noise.
Edge Formation:
1. Occluding Contours
1. Two regions are images of two different surfaces
2. Discontinuity in surface orientation or reflectance properties
Mathematical Model of edges and noise
More Realistically, due to blurring and noise, we generally
have
Image Gradients
The gradient of a differentiable function I gives the direction
in which the values of the
function change most rapidly.
Its magnitude gives you the rate
of change.
Approximating Derivatives
Approximating Derivatives
The discrete Laplacian is given as
1
1 -4 1
1
Local Operators (Differential Operators)
accentuate noise !! (why?)
Therefore, need smoothing before computing image
gradients.
Motivation: Smoothing removes local intensity
variation and what remains are the prominent edges.
Gaussian Smoothing with exponential kernel function
Examples
Three Steps of Edge Detection
1. Noise Smoothing: Suppress as much of the image noise as
possible. In the absence of specific information, assume the
noise white and Gaussian
2. Edge Enhancement: Designe a filter responding to edges. The
filter’s output is large at edge pixels and low elsewhere. Edges
can be located as the local maxima in the filters’ output.
3. Edge Localization: Decide which local maxima in the filter’s
output are edges and which are just caused by noise.
1. Thinning wide edges to 1-pixel with (nonmaximum suppression);
2. Establishing the minimum value to declare a local
maximum an edge (thresholding)
Edge Descriptors (The output of an edge detector)
1. Edge normal: The direction of the maximum
intensity variation at the edge point.
2. Edge direction: The direction tangent to the edge.
3. Edge Position : The location of the edge in image
4. Edge strength: A measure of local image contrast.
How significance the intensity variation is across
the edge.
Canny Edge Detector (smoothing and enhancement)
CANNY_ENHANCER
Given image I
1. Apply Gaussian Smoothing to I.
2. For each pixel (i, j):
1. Compute the gradient components
2. Estimate the edge strength
3. Estimate the orientation of the edge normal
Canny Edge Detector (Nonmaximum suppression)
The input is the output of CANNY_ENHANCER. We need
to thin the edges. Given Es, Eo, the edge strength and
orientation images. For each pixel (i, j),
1. Find the direction best approximate the direction Eo(i,
j).
2. If Es(i, j) is smaller than at least one of its two
neighbors along this direction, suppress this pixel.
The output is an image of the thinned edge points after
suppressing nonmaxima edge points.
Canny Edge Detector (Hysteresis Thresholding)
Performs edge tracking and reduces the probability of false
contours.
Input I is the output of nonmaximum_suppression, Eo and
two threshold parameters
Scan I in a fixed order:
1.
Locate the next unvisited edge pixel (i, j) such that I(i, j)
2.
Starting from (I, j), follow the chains of connected local maxima in both
directions perpendicular to the edge normal as long as I
3.
Marked all visited points and save a list of the locations of all points in the
connected contour found.
Threshold Results
Using Second Derivatives