Image Transforms

Presentation Transcript

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

Grey Scale Manipulation

Brightness modifications
Contrast modifications
Histogram manipulation

Brightness Adjustment

Add a constant to all values

g’ = g + k

(k = 50)

Contrast Adjustment

Scale all values by a constant

g’ = g*k

(k = 1.5)

Image Histogram

Measure frequency of occurrence of each grey/colour value

Histogram Manipulation

Modify distribution of grey values to achieve some effect

Equalisation/Adaptive Equalisation

Specifically to make histogram uniform

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.

Thresholding

Transform grey/colour image to binary

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

else 0

How to find T?

Threshold Value

Manual

User defines a threshold

P-Tile
Mode
Other automatic methods

P-Tile

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

Mode

Threshold at the minimum between the histogram’s peaks.

Average l

Average h

Automated Methods

Find a threshold  such that

(Start at  = 0 and work upwards.)

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

Local Transforms

Convolution
Applications

Smoothing
Sharpening
Matching

Convolution Definition

Place template on image

Multiply overlapping values in image and template

Sum products and normalise

(Templates usually small)

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 . …

… . . . . . ...

1 1 1

1 2 1

1 1 1

Divide by template sum

Separable Templates

Convolve with n x n template

n2 multiplications and additions

Convolve with two n x 1 templates

2n multiplications and additions

Example

0 –1 0

-1 4 –1

0 –1 0

Laplacian template
Separated kernels

-1

2

-1

-1 2 -1

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

Applications

Usefulness of convolution is the effects generated by changing templates

Smoothing

Noise reduction

Sharpening

Edge enhancement

Template matching

A later lecture

Smoothing

Aim is to reduce noise
What is “noise”?
How is it reduced

Addition
Adaptively
Weighted

Noise Definition

Noise is deviation of a value from its expected value

Random changes

x  x + n

Salt and pepper

x  {max, min}

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

Adaptive Smoothing

Compute smoothed value, s

Output = s if |s – x| > T

x otherwise

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

Original

Smoothed

Median Smoothing

Gaussian Smoothing

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

Sharpening and Edge Enhancement

What is it?

Enhancing discontinuities
Edge detection

Why do it?

Perceptually important
Computationally important

Edge Definition

Edge Types

Step edge
Line edge
Roof edge
Real edges

First Derivative, Gradient Edge Detection

If an edge is a discontinuity
Can detect it by differencing

Roberts Cross Edge Detector

Simplest edge detector
Inaccurate localisation

Prewitt/Sobel Edge Detector

Edge Detection

Combine horizontal and vertical edge estimates

Problems

Enhanced edges are noise sensitive
Scale

What is “local”?

Canny/Deriche Edge Detector

Require

Edges to be detected
Accurate localisation
Single response to an edge

Solution

Convolve image with Difference of Gaussian (DoG)

Example Results

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

Example Results

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

Cosine/Sine

A halfway solution to the Fourier Transform
Used in image coding

Fourier

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

Transformed section of BT Building image.

Frequency Domain Filtering

Convolution Theorem:

Convolution in spatial domain

is equivalent to

Multiplication in frequency domain

Smoothing

Suppress high frequency components

Sharpening

Suppress low frequency components

Hough Transform

To detect curves analytically
Example

Straight lines

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

Analytic Curve Finding

Alternative representation

To avoid infinities

Other curves

Higher dimensional accumulators

Performance Improvement Techniques

Look at pairs of points
Use edge orientation

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

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

Uses

Compact representation of variable data
Object recognition

detail

Wavelet

A hierarchical representation

Example

Uses

Hierarchical representation
Multiresolution processing
Coding

Geometric Transformations

Definitions

Affine and non-affine transforms

Applications

Manipulating image shapes

affine transforms scale shear rotate translate

[

]

x’

y’

1

[

]

[

]

a b c

d e f

g h i

x

y

1

Affine TransformsScale, Shear, Rotate, Translate

Length 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

Affine Transform Examples

Warping Example

Ansell Adams’ Aspens

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

You Should Know…

Point transforms

Scaling, histogram manipulation, thresholding

Local transforms

Edge detection, smoothing

Global transforms

Fourier, Hough, Principal Component, Wavelet

Geometrical transforms

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Image Transforms