Computer Vision Image Processing II
Neural Networks | angular代写 | Machine learning代写 – 这道题目是利用angular进行的编程代写任务, 包括了Neural Networks/angular/Machine learning等方面
COMP9517 2022 T2 1
- Two main types of image processing operations:
- Spatial domain operations (in image space)
- Frequency domain operations (mainly in Fourier space)
- Two main types of spatial domain operations:
- Point operations (intensity transformations on individual pixels)
- Neighbourhoodoperations (spatial filtering on groups of pixels)
Types of image processing (recap)
Point operations
Neighbourhood operations
Recap
Spatial domain, intensity transformations (on
single pixels)
- Image thresholding
- Otsus method
- Histogram thresholding
- Multiband thresholding
- Image inversion
- Log transform
- Power-law
- Averaging
Recap
Spatial domain, intensity transformations (on
single pixels)
- Piecewise-linear transformation
- Contrast stretching
- Gray-level slicing
- Bit-plane slicing
- Histogram processing
- Histogram equalization
- Histogram matching
Spatial Filtering
- These methods use a small neighbourhood of a pixel in the input image to produce a new brightnessesvalue for that pixel
- Also called filtering techniques
- Neighbourhood of (,)is usually a square or rect angular subimage centred at ,.
- filter / mask / kernel / template / window is used to indicate the concepts of the subimageor the corresponding operators, in different contexts.
Spatial Filtering
- These methods use a small neighbourhood of a pixel in the input image to produce a new brightnessesvalue for that pixel
- Also called filtering techniques
- Neighbourhood of (,)is usually a square or rectangular subimage centred at ,.
- A linear transformation calculates a value in the output image , as a linear combination of brightnessesin a local neighbourhood of the pixel in the input image (,)weighted by coefficients :
, =
=
=
, (,)
- This is called a discrete convolution with the convolution mask/filter/kernel
Spatial Filtering
Convolution
Smoothing Spatial Filters
Neighbourhood Averaging (Mean Filter)
- The most basic filter, used for image blurring/smoothing and
noise reduction
, =
1
(,)(,)
- Replace intensity at pixel (,)with the average of the
intensities in a neighbourhood of (,)
- We can also use a weighted average , giving more importance
to some pixels over others in the neighbourhood can reduce
blurring effect
- Neighbourhood averaging blurs edges
Digital Image Processing – Chapter 3, Image Enhancement in the Spatial Domain
Smoothing Spatial Filters –
Examples
Smoothing Spatial Filters –
Examples
Digital Image Processing – Chapter 3, Image Enhancement in the Spatial Domain
Smoothing Spatial Filters
Smoothing Spatial Filters
- Aim: To suppress noise, other small fluctuations in image- may be result of sampling, quantization, transmission, environment disturbances during acquisition
- Uses redundancy in the image data
- May blur sharp edges, so care is needed
What if the filter is 0 0 0 0 0 0 0 1 0 or 0 0 1 0 0 0 0 0 0?
Gaussian Filter
,, =
1
2^2
(^2) + 2 2^2
- Replace intensity at pixel (,)with the weighted average of
the intensities in a neighbourhood of (,)
- It is a set of weights that approximate the profile of a
Gaussian function
- It is very effective in reducing noise and alsoreducing details
(image blurring)
Gaussian Filter
Gaussian Filter
Many nice properties motivate the use of the Gaussian filter
- It is the only filter that is both separable and circularly symmetric
- It has optimal space-frequency localization
- The Fourier transform of a Gaussian is also a Gaussian function
- The n -fold convolution of any low-pass filter converges to a Gaussian
- It is infinitely smooth so it can be differentiated to any desired degree
- It scales naturally (sigma) and allows for consistent scale-space theory
=0.
3 x 3 5 x 5
=1.
7 x 7
=1.
9 x 9
=2.
11 x 11
=2.
Nonlinear Spatial Filters
Referring to order-statistics filters in many cases: response based on ordering the pixels in the neighbourhood and replacing centre pixel with the ranking result
Median Filter
- Intensity of each pixel is replaced by the median of the intensities in neighbourhood of that pixel
- Median M of a set of values is the middle value such that half the values in the set are less than M and the other half greater than M
Median Filter
19
69 37 19
51 43 44
48 58 68
?
69 37 19 51 43 44 48 58 68
19 37 43 44 48 51 58 68 69
Nonlinear Spatial Filters
Median Filter
- Intensity of each pixel is replaced by the median of the intensities in neighbourhood of that pixel
- Median filtering forces points with distinct intensities to be more like their neighbours, thus eliminating isolated intensity spikes
- Also, isolated pixel clusters (light or dark), whose area is ^2 /2are eliminated by an median filter
- Good for impulse noise (salt-and-pepper noise)
- Other examples of order-statistics filters are max and min filters
Image with impulse noise (salt-and-
pepper noise)
Median Filter
-
-
-?
Chapter 3
Image Enhancement in the
Spatial Domain
Gaussian Versus Median Filtering
Original Gaussian Median
Example 1
Example 2
Max / average / median pooling Provides translation invariance Reduces computations Popular in deep convolutional Machine learning 人工智能”> Neural Networks (deep learning) Extracting the most essential/significant information
Pooling
Sharpening Spatial Filters
Edge Detection
- Goal is to highlight fine detail, or enhance detail that has been
blurred
- Spatial differentiation is the tool; strength of response of
derivative operator is proportional to degree of discontinuity
of the image at the point where operator is applied
- Image differentiation enhances edges, and de-emphasizes
slowly varying gray-level values.
Derivative Definitions
For 1-D function f(x), the first order derivative is
approximated as:
= ( + 1) ()
The second-order derivative is approximated as:
^2
^2
= ( + 1) 2() + ( 1)
These are partial derivatives, so extension to 2D is easy
Chapter 3
Image Enhancement in the
Spatial Domain
Basic Idea
- Horizontal scan of the image
- Edge modelled as a ramp to represent blurring due to sampling
- First derivative is
- Non-zero along ramp
- zero in regions of constant intensity
- constant during an intensity transition
- Second derivative is
- Nonzero at onset and end of ramp
- Stronger response at isolated noise point
- zero everywhere except at onset and termination of intensity transition
- Thus, magnitude of first derivative can be used to detect the presence of an edge, and sign of second derivative to determine whether a pixel lies on dark or light side of an edge.
Summary
- First-order derivatives produce
thicker edges, have stronger
response to gray-level step
- Second-order derivatives
produce stronger response to
fine detail (thin lines, isolated
points), produce double
response at step changes in gray
level
Gradient Operator
First-order derivatives implemented using magnitude of the gradient
For function the gradient at (,)has components = , =
The magnitude of the gradient vector is
= ^2 +^2
This is sometimes approximated as = +
and are linear and may be obtained by using masks
We use numerical techniques to compute these, giving rise to different
masks, e.g.Roberts 2 x 2 cross-gradient operators, Sobels 3 x 3 masks
Chapter 3
Image Enhancement in the
Spatial Domain
Gradient Operator
Week 2 COMP9517 2022 T2 32
Largest gradients
(,)
(,)
(,)
(,)
32
Laplacian Operator
Second order derivatives based on the Laplacian. For a function (,)the Laplacian is defined by
This is a linear operator, as all derivative operators are. In discrete form:
and similarly in y direction.
Summing them gives us
2 (^1 , ) (^1 , )^2 ( , )
2
f x y f x y f x y
x
f
= + +
^2 (,) = + 1, + 1, + ,+ 1 + , 1 4(,)
Chapter 3
Image Enhancement in the
Spatial Domain
Laplacian Operator
Week 2 COMP9517 2022 T2 35
Zero crossings
(,)
(,)
(,)
^2 (,)
35
Gradient Versus Laplacian Edge Detection
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Edges from thresholding local maxima of the gradient magnitude image
Edges from finding the zero-crossings of the Laplacian image
= 1 = 3 = 5 = 7 = 9
= 1 = 3 = 5 = 7 = 9
36
The Laplacian
- There are other forms of the Laplacian, which can include
diagonal directions, for example
- Laplacian highlights grey-level discontinuities and produces
dark featureless backgrounds
- The background can be recovered by adding or subtracting
the Laplacian image to the original image
Chapter 3
Image Enhancement in the
Spatial Domain
Chapter 3
Image Enhancement in the
Spatial Domain
Chapter 3
Image Enhancement in the
Spatial Domain
Chapter 3
Image Enhancement in the
Spatial Domain
Padding
- When we use spatial filters for pixels on the
boundary of an image, we do not have enough
neighbours
- To get an image with the same size as input image
o Zero : set all pixels outside the source image to 0
o Constant : set all pixels outside the source image to a specified border
value
o Clamp : repeat edge pixels indefinitely
o Wrap : copy pixels from opposite side of the image
o Mirror : reflect pixels across the image edge
Padding Example
COMP9517 2022 T2 43
Szeliski, Computer Vision, Chapter 3