CS292 Computational Vision and

LanguageImage processing and transform

Image processing and transform

Objectives:

• To enhance features that are meaningful to applications

• Obtain key representations (silent features) of image content

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

Monadic, Point-by-point Operators

• Monadic point-to-point operator

Brightness Adjustment

Add a constant to all values

g(x,y) = f(x,y) + k

(k = 50)

Intensity Shift

g(x,y) <=

Where k is a user defined variable

0 a(x,y)+k < 0

f(x,y) +k 0<=f(x,y) +k <=W

W W<f(x,y) +k

Contrast Adjustment

Scale all values by a constant

g(x,y) = a* f(x,y)

(a = 1.5)

General expression for brightness and contrast modification

g(x,y) = a * f(x,y) +bIf we do not want to specify a gain and a bias, but

would rather map a particular range of grey levels, [f1,f2], onto a new range, [g1,g2]. This form of mapping is accomplished using

(g(x,y)-g1)/(f(x,y)-f1)=(g2-g1)/(f2-f1)i.e.

g(x,y) = g1 +((g2-g1)/(f2-f1))[f(x,y)-f1]

linear mapping

Linear and non-linear mapping• If ‘a’ is a constant, then it is a linear mapping;• When ‘a’ is a function, then it is a non-linear

mapping• Non-linear mapping functions have a useful

properties: the gain, ‘a’, applied to input grey level, can vary. Thus the way in which contrast is modified depends on the input grey level.

• If a range of grey level is mapped to a wider ranger of grey level, the contrast is enhanced

• If a range of grey level is mapped to a narrower range of grey level, the contrast is reduced.

Non-linear mapping

•Generally, logarithmic mapping is for to enhance details in the darker regions of the image, at the expense of detail in the brighter regions.

•Exponential mapping has a reverse effect, contrast in the brighter parts of an image is increased at the expense of contrast in the darker parts.

Image Histogram

• Measure frequency of occurrence of each grey/colour value

Grey Value Histogram

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1 16 31 46 61 76 91 106

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Grey Value

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Calculation of an image histogram

Create an array histogram with 2b elements

for all grey levels, i, dohistogram[i] = 0

end for

for all pixel coordinates, x and y, do

Increment histogram [f(x,y)] by 1

end for

Histogram Manipulation

• Modify distribution of grey values to achieve some effect

Cumulative Histogram

• Records the cumulative frequency distribution of grey levels in an image.

• A cumulative histogram is a mapping that counts the cumulative number of pixels in all of the bins up to the specified bin. That is, the cumulative histogram Hk of a histogram hk is defined as:

• (k’ can start from 0, if the range is from 0-255)

Histogram Equalisation

• Histogram equalisation is based on the argument that the image’s appearance will be improved if the distribution of pixels over the available grey level is even.

• A non-linear mapping of grey level, specific to that image, that will yield an optimal improvement in contrast.

• Redistributes grey levels in an attempt to flatten the frequency distribution.

• More grey levels are allocated where there are most pixels, fewer grey levels where there are fewer pixels.

• This tends to increase contrast in the most heavily populated regions of the histogram, and often reveals previously hidden detail.

Histogram Equalisation• If we are to increase contrast for the most frequently

occurring grey levels and reduce contrast in the less popular part of the grey level range, then we need a mapping function which has a steep slope (a>1) at grey levels that occur frequently, and a gentle slope (a<1) at unpopular grey levels.

• The cumulative histogram of the image has these properties.

• The mapping function we need is obtained simply by rescaling the cumulative histogram so that its values lie in the range 0-255.

Calculating histogram equalisation

Compute a scaling factor, a=255/number of pixels

Calculate histogram (see previous algorithm)

c[0] = a*histogram[0]

for all remaining grey levels, i, do

c[i] = c[i-1]+a*histogram[i]

end for

for all pixel coordinates, x and y, do

g(x,y) = c[f(x,y)]

end for

Equalisation/Adaptive Equalisation

• Specifically to make histogram uniform

Grey Value Histogram

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1 16 31 46 61 76 91 106

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Threshold

• This is an important function, which converts a grey scale image to a binary format. Unfortunately, it is often difficult, or even impossible to find satisfactory values for the user defined integer threshold value

• Transform grey/colour image to binary

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

else 0• How to find T?

Threshold

• C(x,y) <=

for (int y=0; y<height; y++) for(int x=0; x<width; x++)

if (input.getxy(x,y) < threshold)output.setxy(x,y, BLACK);

elseoutput.setxy(x,y,WHITE);

W a(x,y)>=threshold

0 otherwise

Threshold Value

• Manual– User defines a threshold

• P-Tile – if we know the proportion of the image is occupied by the object, define the threshold as that grey level which has the correct proportion

• Mode - Threshold at the minimum between the histogram’s peaks

• Other automatic methods

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

(such operations will be practised in the labs)

Local Transforms

• Consider neighbourhood information

• Convolution

• Applications– smoothing– sharpening– Matching

• Very useful, but computationally costly

Local Operators

• The concept behind local operators is that the intensities of several pixels are combined together in order to calculate the intensity of just one pixel. Amongst the simplest of the local operators are those which use a set of nine pixels arranged in a 3 x 3 square region. It computes the value for one pixel on the basis of the intensities within a region containing 3 x 3 pixels. Other local operators employ larger windows.

Local Operators

A sub-image

(x-1, y-1) (x, y-1) (x+1, y-1)

(x-1, y) (x,y) (x+1,y)

(x-1,y+1) (x, y+1) (x+1, y+1)

Convolution Definition

Place template on image

Multiply overlapping values in image and template

Sum 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

Applications

• Usefulness of convolution is the effects generated by changing templates– Smoothing

• Noise reduction

– Sharpening• Edge enhancement

Smoothing

• Aim is to reduce noise• What is “noise”?

– Noise is deviation of a value from its expected value

• How is it reduced– Addition– Adaptively– Weighted

Original Smoothed

Median Smoothing

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.

•Significant: in relation to the proportional change in image intensity across the edge

•Local: Neighbourhood relationship matters

Algorithms for Edge Detection

• Algorithms for detecting edges - edge detectors– differentiation based

• estimated the derivatives of the image intensity function, the idea being that large image derivatives reflect abrupt intensity changes.

– Model based• determine whether the intensities in a small area

conform to some model for the edges that we have assumed.

First Derivative, Gradient Edge Detection

• If an edge is a discontinuity

• Can detect it by differencing

Roberts Cross Edge Detector

• Simplest edge detector

-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

Example Results

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

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','

Summary

• Point transforms– scaling, histogram manipulation,thresholding

• Local transforms– edge detection, smoothing

• Global transforms– Fourier, Hough, Principal Component, Wavelet

• Geometrical transforms