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## Image Transforms

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### Image Transforms

Transforming images to images

Classification of Image Transforms

- Point transforms
- Modify individual pixels
- Modify pixels’ locations
- Local transforms
- Output derived from neighbourhood
- Global transforms
- Whole image contributes to each output value

Image Processing and Computer Vision: 2

Point Transforms

- Manipulating individual pixel values
- Brightness adjustment
- Contrast adjustment
- Histogram manipulation
- Equalisation
- Image magnification

Image Processing and Computer Vision: 2

Grey Scale Manipulation

- Brightness modifications
- Contrast modifications
- Histogram manipulation

Image Processing and Computer Vision: 2

Brightness Adjustment

Add a constant to all values

g’ = g + k

(k = 50)

Image Processing and Computer Vision: 2

Contrast Adjustment

Scale all values by a constant

g’ = g*k

(k = 1.5)

Image Processing and Computer Vision: 2

Image Histogram

- Measure frequency of occurrence of each grey/colour value

Image Processing and Computer Vision: 2

Histogram Manipulation

- Modify distribution of grey values to achieve some effect

Image Processing and Computer Vision: 2

Equalisation/Adaptive Equalisation

- Specifically to make histogram uniform

Image Processing and Computer Vision: 2

Equalisation Transform

- Equalised image has n x m/l pixels per grey level
- Cumulative to level j
- jnm/l pixels
- Equate to a value in input cumulative histogram C[i]
- C[i] = jnm/l
- j = C[i]l/nm
- Modifications to prevent mapping to –1.

Image Processing and Computer Vision: 2

Thresholding

- Transform grey/colour image to binary

if f(x, y) > T output = 1

else 0

- How to find T?

Image Processing and Computer Vision: 2

Threshold Value

- Manual
- User defines a threshold
- P-Tile
- Mode
- Other automatic methods

Image Processing and Computer Vision: 2

P-Tile

- If we know the proportion of the image that is object
- Threshold the image to select this proportion of pixels

Image Processing and Computer Vision: 2

Average h

Automated MethodsFind a threshold such that

(Start at = 0 and work upwards.)

Image Processing and Computer Vision: 2

Image Magnification

- Reducing
- New value is weighted sum of nearest neighbours
- New value equals nearest neighbour
- Enlarging
- New value is weighted sum of nearest neighbours
- Add noise to obscure pixelation

Image Processing and Computer Vision: 2

Local Transforms

- Convolution
- Applications
- Smoothing
- Sharpening
- Matching

Image Processing and Computer Vision: 2

Convolution Definition

Place template on image

Multiply overlapping values in image and template

Sum products and normalise

(Templates usually small)

Image Processing and Computer Vision: 2

Example

Image

Template

Result

… . . . . . ...

… 3 5 7 4 4 …

… 4 5 8 5 4 …

… 4 6 9 6 4 …

… 4 6 9 5 3 …

… 4 5 8 5 4 …

… . . . . . ...

… . . . . . ...

… . . . . . ...

… . 6 6 6 . …

… . 6 7 6 . …

… . 6 7 6 . …

… . . . . . ...

… . . . . . ...

1 1 1

1 2 1

1 1 1

Divide by template sum

Image Processing and Computer Vision: 2

Separable Templates

- Convolve with n x n template
- n2 multiplications and additions
- Convolve with two n x 1 templates
- 2n multiplications and additions

Image Processing and Computer Vision: 2

Example

0 –1 0

-1 4 –1

0 –1 0

- Laplacian template
- Separated kernels

-1

2

-1

-1 2 -1

Image Processing and Computer Vision: 2

Composite Filters

- Convolution is distributive
- Can create a composite filter and do a single convolution
- Not convolve image with one filter and convolve result with second.
- Efficiency gain

Image Processing and Computer Vision: 2

Applications

- Usefulness of convolution is the effects generated by changing templates
- Smoothing
- Noise reduction
- Sharpening
- Edge enhancement
- Template matching
- A later lecture

Image Processing and Computer Vision: 2

Smoothing

- Aim is to reduce noise
- What is “noise”?
- How is it reduced
- Addition
- Adaptively
- Weighted

Image Processing and Computer Vision: 2

Noise Definition

- Noise is deviation of a value from its expected value
- Random changes
- x x + n
- Salt and pepper
- x {max, min}

Image Processing and Computer Vision: 2

Noise Reduction

- By smoothing

S(x + n) = S(x) + S(n) = S(x)

- Since noise is random and zero mean
- Smooth locally or temporally
- Local smoothing
- Removes detail
- Introduces ringing

Image Processing and Computer Vision: 2

Adaptive Smoothing

- Compute smoothed value, s

Output = s if |s – x| > T

x otherwise

Image Processing and Computer Vision: 2

Median Smoothing

Median is one value in an ordered set:

1 2 3 4 5 6 7 median = 4

2 3 4 5 6 7 median = 4.5

Image Processing and Computer Vision: 2

Gaussian Smoothing

- To reduce ringing
- Weighted smoothing
- Numbers from Gaussian (normal) distribution are weights.

Image Processing and Computer Vision: 2

Sharpening and Edge Enhancement

- What is it?
- Enhancing discontinuities
- Edge detection
- Why do it?
- Perceptually important
- Computationally important

Image Processing and Computer Vision: 2

Edge Definition

Image Processing and Computer Vision: 2

First Derivative, Gradient Edge Detection

- If an edge is a discontinuity
- Can detect it by differencing

Image Processing and Computer Vision: 2

Roberts Cross Edge Detector

- Simplest edge detector
- Inaccurate localisation

Image Processing and Computer Vision: 2

Prewitt/Sobel Edge Detector

Image Processing and Computer Vision: 2

Problems

- Enhanced edges are noise sensitive
- Scale
- What is “local”?

Image Processing and Computer Vision: 2

Canny/Deriche Edge Detector

- Require
- Edges to be detected
- Accurate localisation
- Single response to an edge
- Solution
- Convolve image with Difference of Gaussian (DoG)

Image Processing and Computer Vision: 2

Example Results

Image Processing and Computer Vision: 2

Second Derivative Operators Zero Crossing

- Model HVS
- Locate edge to subpixel accuracy
- Convolve image with Laplacian of Gaussian (LoG)
- Edge location at crossing of zero axis

Image Processing and Computer Vision: 2

Example Results

Image Processing and Computer Vision: 2

Global Transforms

- Computing a new value for a pixel using the whole image as input
- Cosine and Sine transforms
- Fourier transform
- Frequency domain processing
- Hough transform
- Karhunen-Loeve transform
- Wavelet transform

Image Processing and Computer Vision: 2

Cosine/Sine

- A halfway solution to the Fourier Transform
- Used in image coding

Image Processing and Computer Vision: 2

Fourier

- All periodic signals can be represented by a sum of appropriately weighted sine/cosine waves

Image Processing and Computer Vision: 2

Transformed section of BT Building image.

Image Processing and Computer Vision: 2

Frequency Domain Filtering

- Convolution Theorem:

Convolution in spatial domain

is equivalent to

Multiplication in frequency domain

Image Processing and Computer Vision: 2

Hough Transform

- To detect curves analytically
- Example
- Straight lines

Image Processing and Computer Vision: 2

Straight Line

y = mx + c

gradient m, intercept c

c = -mx + y

gradient -x, intercept y

ALL points in (x, y) transform to a straight line in (c, m)

Can therefore detect collinear points

Image Processing and Computer Vision: 2

Analytic Curve Finding

- Alternative representation
- To avoid infinities
- Other curves
- Higher dimensional accumulators

Image Processing and Computer Vision: 2

Performance Improvement Techniques

- Look at pairs of points
- Use edge orientation

Image Processing and Computer Vision: 2

Karhunen-Loeve (Principal Component)

- A compact method of representing variation in a set of images
- PCs define a co-ordinate system
- 1st PC records most of variation
- 2nd PC records most of remainder
- etc

Image Processing and Computer Vision: 2

Method

- Take a set of typical images
- Compute mean image and subtract from each sample
- Transform images into columns
- Group images into a matrix
- Compute covariance matrix
- Compute eigenvectors
- These are the PCs
- Eigenvalues show their importance

Image Processing and Computer Vision: 2

Uses

- Compact representation of variable data
- Object recognition

Image Processing and Computer Vision: 2

Example

Image Processing and Computer Vision: 2

Uses

- Hierarchical representation
- Multiresolution processing
- Coding

Image Processing and Computer Vision: 2

Geometric Transformations

- Definitions
- Affine and non-affine transforms
- Applications
- Manipulating image shapes

Image Processing and Computer Vision: 2

]

x’

y’

1

[

]

[

]

a b c

d e f

g h i

x

y

1

Affine TransformsScale, Shear, Rotate, TranslateLength and areas preserved.

=

Change values of transform matrix elements according to desired effect.

a, e scaling

b, d shearing

a, b, d, e rotation

c, f translation

Image Processing and Computer Vision: 2

Affine Transform Examples

Image Processing and Computer Vision: 2

Image Resampling

- Moving source to destination pixels
- x’ and y’ could be non-integer
- Round result
- Can create holes in image
- Manipulate in reverse
- Where did warped pixel come from
- Source is non-integer
- Interpolate nearest neighbours

Image Processing and Computer Vision: 2

You Should Know…

- Point transforms
- Scaling, histogram manipulation, thresholding
- Local transforms
- Edge detection, smoothing
- Global transforms
- Fourier, Hough, Principal Component, Wavelet
- Geometrical transforms

Image Processing and Computer Vision: 2

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