Image Transforms

Image Transforms

Transforming images to images

Image Processing and Computer Vision: 2 2

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


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

Grey Value Histogram

0

2000

4000

6000

8000

10000

12000

1 16 31 46 61 76 91 106

121

136

151

166

181

196

211

226

241

256

Grey Value

Fre

qu

ency


Histogram Manipulation

Modify distribution of grey values to achieve some effect


Equalisation/Adaptive Equalisation

Specifically to make histogram uniform

Grey Value Histogram

0

2000

4000

6000

8000

10000

12000

1 16 31 46 61 76 91 106

121

136

151

166

181

196

211

226

241

256

Grey Value

Fre

qu

ency


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.


Automated Methods

Find a threshold such that

(Start at = 0 and work upwards.)

Average l Average h

l

h2


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 imageMultiply overlapping values in image

and templateSum products and normalise(Templates usually small)

x yyx,tycx,rgcr,g'


ExampleImage Template Result

1 1 11 2 11 1 1

… . . . . . ...… 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 . …… . . . . . ...… . . . . . ...

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

Laplacian template

Separated kernels

0 –1 0-1 4 –1 0 –1 0

-1 2 -1 -12-1


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

CBACBA


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 (x + n) = (x) + (n) = (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 = 42 3 4 5 6 7 median = 4.5

evennnn

average

oddnn

2

1,2

2

1


Original Smoothed

Median Smoothing


Gaussian Smoothing

To reduce ringing Weighted smoothing Numbers from Gaussian (normal)

distribution are weights.


Sharpening

What is it? Enhancing discontinuities Edge detection

Why do it? Perceptually important Computationally important


Edge Definition

An edge is a significant local change in image intensity.


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

-1 0

0 1

0 -1

1 0


Prewitt/Sobel Edge Detector-1 -1 -1

0 0 0

1 1 1

-1 0 1

-1 0 1

-1 0 1


Edge Detection

Combine horizontal and vertical edge estimates

h

vandvhMag tan 122


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

otherwise

xifxC

N

jy

N

ixyxpixeljCiCNjiDCT

Nx

x

Ny

y

1

02

1

2

12cos

2

12cos,2,

1

0

1

0


Fourier All periodic signals can be represented by a

sum of appropriately weighted sine/cosine waves

eW

f

f

WWWW

F

F

N

nij

in

NNNN

N

N

N

and

2

,

1

0

1,10,1

1,00,0

1

0

1


Transformed section of BT Building image.


Frequency Domain Filtering

Convolution Theorem:

Convolution in spatial domainis equivalent to

Multiplication in frequency domain


Smoothing

Suppress high frequency components


Sharpening

Suppress low frequency components


Example


Wavelet

A hierarchical representation

detail


Hough Transform

To detect curves analytically Example

straight lines


Straight Line

y = mx + cgradient m, intercept c

c = -mx + ygradient -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


Uses

Hierarchical representation Multiresolution processing Coding


Geometric Transformations

Definitions Affine and non-affine transforms

Applications Manipulating image shapes


Affine TransformsScale, Shear, Rotate, Translate

Change values of transform matrix elements according to desired effect.

a, e scalingb, d shearing

a, b, d, e rotationc, f translation

=x’y’1[ ] a b c

d e fg h i[ ] x

y1[ ]

Length and areas preserved.


Affine Transform Examples


Warping ExampleAnsell Adams’ Aspens

xyxyxyx 22 2.12.1','


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

Date post:	06-Jan-2016
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Image Transforms

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