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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

Neural Networks代做 神经网络代写 代写机器学习 machine learning 人工智能

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

Week 2 COMP9517 2022 T2 36
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