today early vision on a single image
TRANSCRIPT
Midterm Exam
Starting from 12:00am EST, Tuesday, March 9th in Blackboard
Due: 11:59pm EST, Wednesday, March 10th
A calculator is allowed
Cover everything till the class on Thursday, March 4th
Open book and open notes
Early Vision on One Image
So far, we talked about image formation. Next, we will discuss early vision on one image
โข Linear and nonlinear filters
โข Linear and nonlinear filters for noise reduction
โข Linear filters for differentiation
โข Edge detection
โข Features
โข Edges, lines, curves, corners etc.
Early Vision on One Image: Two Important
Topics
โข Image noise: intrinsic property of the sensor (CCD) and independent of scene
โข Intensity noise โ quantization and sensor
โข Positional noise โ spatial sampling
Early Vision on One Image: Two Important
Topics
โข Features: characterizing the shape/appearance of the objects in the image
โข Edges, lines, curves, corners etc.
โข Image statistics: mean, variance, histogram etc.
โข Complex features
โข Widely used in many problems of computer vision including camera calibration, stereo, object detection/tracking/recognition, etc.
Properties of Noise
โข Spatial properties
โข Spatially periodic noise
โข Spatially independent noise
โข Frequency properties
โข White noise โ noise containing all frequencies within a bandwidth
Image Noise
แ๐ผ(๐ฅ, ๐ฆ) = ๐ผ(๐ฅ, ๐ฆ) + ๐(๐ฅ, ๐ฆ)
Image noise
Observed image intensity Ideal image intensity
Some Important Noise Model
๐(๐ง) =1
2๐๐๐โ(๐งโ ว๐ง)2
2๐2
โข Due to electronic circuit
โข Due to the image sensor
โข poor illumination
โข high temperature
Most popular noise model
Assumption: the image noise is identically and independently.
Gaussian noise model
Some Important Noise Model
๐(๐ง) = แ2
๐(๐ง โ ๐)๐โ
(๐งโ๐)2
๐ ๐ง โฅ ๐
0 ๐ง < ๐๐(๐ง) = เต
๐๐๐ง๐โ1
(๐ โ 1)!๐โ๐๐ง ๐ง โฅ 0
0 ๐ง < 0
Rayleigh noise
โข range imaging
โข Background model for Magnetic
Resonance Imaging (MRI) images
Gamma noise
โข laser imaging
Figure from โDigital Image Processingโ, Gonzalez and Woods
Some Important Noise Model
๐(๐ง) =
๐๐ for ๐ง = ๐๐๐ for ๐ง = ๐
0 Otherwise
๐(๐ง) = แ๐๐โ๐๐ง ๐ง โฅ 00 ๐ง < 0 ๐(๐ง) = แ
1
๐ โ ๐๐ โค ๐ง โค ๐
0 ๐๐กโ๐๐๐ค๐๐ ๐
Exponential noise
โข laser imaging
Impulse noise
โข salt and pepper noise
โข A/D converter error
โข bit error in transmission
Uniform noise
Figure from โDigital Image Processingโ, Gonzalez and Woods
An Example (cont.)
Figure from โDigital Image Processingโ, Gonzalez and Woods
ExponentialRayleigh GammaGaussian Uniform Salt & Pepper
Periodical Noise
Image is corrupted by a set of
sinusoidal noise of different
frequencies
Figure from โDigital Image Processingโ, Gonzalez and Woods
Estimation of Noise Parameters
Take a small stripe of the background, do statistics for the mean and variance.
Figure from โDigital Image Processingโ, Gonzalez and Woods
Image Filtering
Modifying the pixels in an image based on some function of a local neighborhood of the pixels
10 30 10
20 11 20
11 9 1
p
N(p)
f(p)
๐ ๐
f(p):
โข Linear functionโข Correlation
โข Convolution
โข Nonlinear functionโข Order statistic (median)
Linear Filters
General process:
โข Form new image whose pixels are a weighted sum of original pixel values, using the same set of weights at each point.
Properties
โข Output is a linear function of the input
โข Output is a shift-invariant function of the input (i.e. shift the input image two pixels to the left, the output is shifted two pixels to the left)
Example: smoothing by averaging
โข form the average of pixels in a neighborhood
Example: smoothing with a Gaussian
โข form a weighted average of pixels in a neighborhood
Example: finding an edge
Linear Filtering
The output is the linear combination of the neighborhood pixels
The coefficients of this linear combination combine to form the โfilter-kernelโ
๐ ๐ =
๐๐โ๐ ๐
๐๐๐๐
1 3 0
2 10 2
4 1 1
Image
1 0 -1
1 0.1 -1
1 0 -1
Kernel
= 5
Filter Output
โ
Convolution
๐ ๐, ๐ = ๐ผ โ ๐ป =
๐
๐
๐ผ ๐, ๐ ๐ป ๐ โ ๐, ๐ โ ๐
๐ผ = Image๐ป = Kernel
H7 H8 H9
H4 H5 H6
H1 H2 H3
H9 H8 H7
H6 H5 H4
H3 H2 H1
H1 H2 H3
H4 H5 H6
H7 H8 H9
๐ป
๐ โ ๐๐๐๐
๐ โ ๐๐๐๐
I1 I2 I3
I4 I5 I6
I7 I8 I9
โ
๐ผ โ ๐ป = ๐ผ1๐ป9 + ๐ผ2๐ป8 + ๐ผ3๐ป7
+ ๐ผ4๐ป6 + ๐ผ5๐ป5 + ๐ผ6๐ป4
+ ๐ผ7๐ป3 + ๐ผ8๐ป2 + ๐ผ9๐ป1
๐ผ