Introduction to Computer Vision Introduction Zhigang Zhu, City College of New York...

Introduction to

Computer Vision IntroductionIntroduction

Zhigang Zhu, City College of New York [email protected]

CSc I6716Spring 2012

Image Enhancement

Part IFeature Extraction (1)

Introduction to

Computer Vision What are Image Features?What are Image Features?

n Local, meaningful, detectable parts of the image.

Introduction to

Computer Vision More Color WoesMore Color Woes

Squares with dots in them are the same color

Introduction to

Computer Vision TopicsTopics

n Image Enhancementl Brightness mappingl Contrast stretching/enhancementl Histogram modificationl Noise Reductionl ……...

n Mathematical Techniquesl Convolutionl Gaussian Filtering

n Edge and Line Detection and Extractionn Region Segmentationn Contour Extractionn Corner Detection

Introduction to

Computer Vision Image EnhancementImage Enhancement

n Goal: improve the ‘visual quality’ of the imagel for human viewingl for subsequent processing

n Two typical methodsl spatial domain techniques....

u operate directly on image pixelsl frequency domain techniques....

u operate on the Fourier transform of the image

n No general theory of ‘visual quality’l General assumption: if it looks better, it is betterl Often not a good assumption

Introduction to

Computer Vision Spatial Domain MethodsSpatial Domain Methods

n Transformation Tl point - pixel to pixell area - local area to pixell global - entire image to pixel

n Neighborhoods l typically rectangularl typically an odd size: 3x3, 5x5, etcl centered on pixel I(x,y)

n Many IP algorithms rely on this basic notion

T(I(x,y))

I(x,y) I’(x,y)neighborhood N

I’(x,y) = T(I(x,y))

O=T(I)

Introduction to

Computer Vision Point Transforms: General IdeaPoint Transforms: General Idea

O = T(I)

Input pixel value, I, mapped to output pixel value, O, via transfer function T.

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INPUT

OU

TP

UT

TransferFunction T

Introduction to

Computer Vision Grayscale TransformsGrayscale Transforms

n Photoshop ‘adjust curve’ command

Input gray value I(x,y)

Output gray value I’(x,y)

Introduction to

Computer Vision Point Transforms: BrightnessPoint Transforms: Brightness

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Introduction to

Computer Vision Point Transforms:ThresholdingPoint Transforms:Thresholding

n T is a point-to-point transformationl only information at I(x,y) used to generate I’(x,y)

n Thresholding

Imax if I(x,y) > tImin if I(x,y) ≤ t

I’(x,y) =

t=89

t 25500

255

Introduction to

Computer Vision Point Transforms: Linear StretchPoint Transforms: Linear Stretch

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255

INPUT

OU

TP

UT

Introduction to

Computer Vision Linear ScalingLinear Scaling

n Consider the case where the original image only utilizes a small subset of the full range of gray values:

n New image uses full range of gray values.n What's F? {just the equation of the straight line}

I'(x,y)

I(x,y) ImaxImin0

0 K

K

Output image I'(x,y) = F [ I(x,y)] Desired gray scale range: [0 , K]

Input image I(x,y) Gray scale range: [I , I ]min max

Introduction to

Computer Vision Linear ScalingLinear Scaling

n F is the equation of the straight line going through the point (Imin , 0) and (Imax , K)

n useful when the image gray values do not fill the available range.

n Implement via lookup tables

I' = mI + b

I'(x,y) = I(x,y) - minIKI max min- I

KI max min- I

Introduction to

Computer Vision Scaling Discrete ImagesScaling Discrete Images

n Have assumed a continuous grayscale.n What happens in the case of a discrete grayscale with

K levels?

?

Empty!

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Input Gray Level

Out

put

Gra

y L

evel

Introduction to

Computer Vision Non-Linear Scaling: Power LawNon-Linear Scaling: Power Law

n < 1 to enhance contrast in dark regionsn > 1 to enhance contrast in bright regions.

O = Ig

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=1g

<1g

>1g

Introduction to

Computer Vision Square Root Transfer: g=.5Square Root Transfer: g=.5

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Introduction to

Computer Vision

g=3.0

g=3.0

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Introduction to

Computer Vision ExamplesExamples

n Technique can be applied to color imagesl same curve to all color bandsl different curves to separate color bands:

Introduction to

Computer Vision Point Transforms:ThresholdingPoint Transforms:Thresholding

n T is a point-to-point transformationl only information at I(x,y) used to generate I’(x,y)

n Thresholding

Imax if I(x,y) > tImin if I(x,y) ≤ t

I’(x,y) =

t=89

t 25500

255

Introduction to

Computer Vision Threshold SelectionThreshold Selection

n Arbitrary selectionl select visually

n Use image histogram

Threshold

Introduction to

Computer Vision HistogramsHistograms

n The image shows the spatial distribution of gray values.n The image histogram discards the spatial information and

shows the relative frequency of occurrence of the gray values.

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Image CountGray Value

Rel. Freq.

Sum= 36 1.00

Introduction to

Computer Vision Image HistogramImage Histogram

n The histogram typically plots the absolute pixel count as a function of gray value:

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ount

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For an image with dimensions M by N

MNiHI

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)(

Introduction to

Computer Vision Probability InterpretationProbability Interpretation

n The graph of relative frequency of occurrence as a function of gray value is also called a histogram:

n Interpreting the relative frequency histogram as a probability distribution, then:

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tive F

requency

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P(I(x,y) = i) = H(i)/(MxN)

Introduction to

Computer Vision Cumulative Density FunctionCumulative Density Function

n Interpreting the relative frequency histogram as a probability distribution, then:

n Curve is called the cumulative distribution function

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requency

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P(I(x,y) = i) = H(i)/(MxN)

i

k

kPiQ0

)()(

i

k

kHiCH0

)()(

Introduction to

Computer Vision ExamplesExamples

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Introduction to

Computer Vision Color HistogramsColor Histograms

Introduction to

Computer Vision Histogram EqualizationHistogram Equalization

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n Image histograms consist of peaks, valleys, and low plains

n Peaks = many pixels concentrated in a few grey levels

n Plains = small number of pixels distributed over a wider range of grey levels

Introduction to


n The goal is to modify the gray levels of an image so that the histogram of the modified image is flat.l Expand pixels in peaks over a wider range of gray-levels.l “Squeeze” low plains pixels into a narrower range of gray

levels.n Utilizes all gray values equallyn Example Histogram:

l Note low utilization of small gray values

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Gray Value

Introduction to

Computer Vision Desired HistogramDesired Histogram

n All gray levels are used equally.n Has a tendency to enhance contrast.

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Gray Value

Introduction to

Computer Vision Brute Force Brute Force

0 0 256 22 from 7, 234 from 8 1 0 256 1 from 8, 255 from 9 2 0 256 195 from 9, 61 from 10 3 0 256 256 from 10 4 0 256 256 from 10 5 0 256 256 from 10 6 0 256 256 from 10 7 22 256 256 from 10 8 235 256 256 from 10 9 450 256 256 from 10

10 1920 256 67 from 10, 189 from 11 11 212 256 23 from 11, 10 from 12, 223 from 13 12 10 256 10 from 13, 246 from 14 13 233 256 256 from 14 14 672 256 170 from 14, 86 from 15 15 342 256 256 from 15

Gray Actual Desired How to get it Scale Count Count

Sum 4096 4096

How are the gray levels in the original image changed to produce the enhanced image?

Method 1. Choose points randomly.

Method 2. Choice depends on the graylevels of their neighboring points.

Computationally expensive.

Approximations.

Introduction to


n Mapping from one set of grey-levels, I, to a new set, O.n Ideally, the number of pixels, Np, occupying each grey

level should be:

n To approximate this, apply the transform

l Where CH is the cumulative histogram (see next slide)l j is the gray value in the source imagel i is the gray value in the equalized image

Np = M*N

G

i = MAX 0, roundCH(j)

Np

-1

Introduction to

Computer Vision ExampleExample

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j H(j) CH(j) i

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100900160021002200230024002400

CH(j) = S H(i)i=0

j

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Introduction to

Computer Vision ExampleExample

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Introduction to

Computer Vision ComparisonComparison

Original >1

Histogram equalization

Introduction to

Computer Vision Why?Why?

n Given MxN input image I with gray scale p0….pk and histogram H(p).

n Desire output image O with gray scale q0….qk and uniform histogram G(q)

n Treat the histograms as a discrete probability density function. Then monotonic transfer function m = T(n) implies:

n The sums can be interpreted as discrete distribution functions.

From Image Processing, Analysis, and Machine Vision, Sonka et al.

S G(qi) = S H(pj)i=0 j=0

m n

Introduction to

Computer Vision Why? continuedWhy? continued

n The histogram entries in G must be:(because G is uniform)

n Plug this into previous equation to get:

n Now translate into continuous domain (can’t really get uniform distributions in the discrete domain)

MxNqk - q0

G(i) =

S = S H(pi)i=0 i=0

m n MxNqk - q0

MxNqk - q0

= H(s)dsp0

pn

q0

qm

ds

MN(qm-q0)qk - q0

Introduction to

Computer Vision Why? continuedWhy? continued

n Solve last equation for q to get:

n Map back into discrete domain to get:

n Leads to an algorithm…..

MNqk - q0

H(s)ds + q0p0

pn

qm = T(p) =

MNqk - q0 S H(i) + q0

qm = T(p) =i=p0

pn

Introduction to

Computer Vision Histogram Equalization AlgorithmHistogram Equalization Algorithm

n For an NxM image of G gray levels, say 0-255l Create image histograml For cumulative image histogram Hc

n Set

n Rescan input image and write new output image by setting

T(p) = round ( Hc (p))G-1NM

gq = T(gp)

Introduction to

Computer Vision ObservationsObservations

n Acquisition process degrades imagen Brightness and contrast enhancement implemented

by pixel operationsn No one algorithm universally useful n > 1 enhances contrast in bright imagesn < 1 enhances contrast in dark imagesn Transfer function for histogram equalisation

proportional to cumulative histogram

Introduction to

Computer Vision Noise ReductionNoise Reduction

n What is noise?n How is noise reduction performed?

l Noise reduction from first principlesl Neighbourhood operators

u linear filters (low pass, high pass)u non-linear filters (median)

image

+

noise

=

‘grainy’ image

Introduction to

Computer Vision NoiseNoise

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n Plot of image brightness.n Noise is additive.n Noise fluctuations are

rapidl high frequency.

Image

Noise

Image + Noise

Introduction to

Computer Vision Sources of NoiseSources of Noise

n Sources of noise = CCD chip.n Electronic signal fluctuations

in detector.n Caused by thermal energy.n Worse for infra-red sensors.n Electronicsn Transmission

Radiation from the long wavelength IR band is used in most infrared imaging applications

Introduction to

Computer Vision Noise ModelNoise Model

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n Plot noise histogramn Typical noise distribution is

normal or Gaussiann Mean(noise) = 0n Standard deviation

h(x) = exp - 1

2ps2

12

(x- )ms

2

Introduction to

Computer Vision Effect of sEffect of s

n Integral under curve is 1

=10

=20

=30

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Introduction to

Computer Vision Two Dimensional GaussianTwo Dimensional Gaussian

Introduction to

Computer Vision NoiseNoise

n Noise varies above and below uncorrupted image.

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Image

Image +

Noise

Introduction to

Computer Vision Noise Reduction - 1Noise Reduction - 1

n How do we reduce noise?

n Consider a uniform 1-d image and add noise.

n Focus on a pixel neighbourhood.

n Averaging the three values should result in a value closer to original uncorrupted data

n Especially if gaussian mean is zero

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Ai-1 Ai Ai+1 Ci

Introduction to


n Averaging ‘smoothes’ the noise fluctuations.

n Consider the next pixel Ai+1

n Repeat for remainder of pixels.

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Ai-1 Ai Ai+1 Ai+2 Ci+1 321

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AAAC

Introduction to

Computer Vision Neighborhood OperationsNeighborhood Operations

n All pixels can be averaged by convolving 1-d image A with mask B to give enhanced image C.

n Weights of B must equal one when added together.l Why?

C = A * B

B = [ B1 B2 B3 ]

Ci = Ai-1 ´B1 + Ai ´B2 + Ai+1 ´B3

B = [1 1 1]

Ci =

13

Ai-1 + Ai + Ai+1

3

Introduction to

Computer Vision 2D Analog of 1D Convolution2D Analog of 1D Convolution

n Consider the problem of blurring a 2D imagen Here’s the image and a blurred version:

n How would we do this?l What’s the 2D analog of 1D convolution?

Introduction to

Computer Vision 2D Blurring Kernel2D Blurring Kernel

n C = A * Bn B becomes

1 1 11 1 11 1 1

B = 19

Introduction to

Computer Vision ConvolutionConvolution

n Extremely important concept in computer vision, image processing, signal processing, etc.

n Lots of related mathematics (we won’t do)n General idea: reduce a filtering operation to the repeated

application of a mask (or filter kernel) to the imagel Kernel can be thought of as an NxN imagel N is usually odd so kernel has a central pixel

n In practicel (flip kernel)l Align kernel center pixel with an image pixell Pointwise multiply each kernel pixel value with corresponding

image pixel value and add resultsl Resulting sum is normalized by kernel weightl Result is the value of the pixel centered on the kernel

Introduction to

Computer Vision

Image

Kernel

ResultImage

ExampleExample

n Kernel is aligned with pixel in image, multiplicative sum is computed, normalized, and stored in result image.

n Process is repeated across image.n What happens when kernel is near

edge of input image?

Introduction to

Computer Vision Border ProblemBorder Problem

n missing samples are zeron missing samples are grayn copying last linesn reflected indexing (mirror)n circular indexing (periodic)n truncation of the result

Introduction to

Computer Vision

Image size = M1 N1

Mask size = M2 N2

Convolution size = M1- M2 +1 N1-N2+1

N1

N2

N1-N2+1

Typical Mask sizes= 33, 5 5, 77,9 9, 1111

What is the convolved image size for a 128 128 image and 7 7 mask?

Convolution SizeConvolution Size

Introduction to

Computer Vision ConvolutionConvolution

Image Data:

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Mask:

1

0

-1

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1 11

1 11

1 11

k=-1 m=-1

1

1S = M(k,m)

I (i,j) = I(i+k,j+m) *M(k,m)fk=-1 m=-1

1

11S = I M*

Introduction to

Computer Vision Properties of ConvolutionProperties of Convolution

n Commutative: f1(t) * f2 (t) = f2 (t) * f1 (t)n Distributive: f1(t) * [f2(t) + f3 (t)] = f1(t) * f2 (t) + f1 (t) * f3 (t)n Associative: f1(t) * [f2(t) * f3 (t)] = [f1(t) * f2(t)] * f3 (t) n Shift: if f1(t) * f2(t) = c(t), then

l f1(t) * f2(t-T) = f1(t-T) * f2(t) = c(t-T)

n Convolution with impulse: f(t) *d(t) = f(t)n Convolution with shifted impulse: f(t) *d(t-T) = f(t-T)

Introduction to


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Uncorrupted Image

Image + Noise

Image + Noise - Blurred

Introduction to


n Technique relies on high frequency noise fluctuations being ‘blocked’ by filter. Hence, low-pass filter.

n Fine detail in image may also be smoothed.

n Balance between keeping image fine detail and reducing noise.

Introduction to


n Saturn image coarse detailn Boat image contains fine

detailn Noise reduced but fine

detail also smoothed

Introduction to

Computer Vision hmmmmm…..hmmmmm…..Image Blurred Image

- =

Introduction to

Computer Vision Noise Reduction - 2: Median FilterNoise Reduction - 2: Median Filter

n Nonlinear Filtern Compute median of points in neighborhoodn Has a tendency to blur detail less than averagingn Works very well for ‘shot’ or ‘salt and pepper’ noise

Original Low-pass Median

Introduction to


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Low-pass Median

• Low-pass: fine detail smoothed by averaging• Median: fine detail passed by filter

Introduction to

Computer Vision Edge Preserving SmoothingEdge Preserving Smoothing

n Smooth (average, blur) an image without disturbingl Sharpness orl position

Of the edgesn Nagoa-Maysuyama Filtern Kuwahara Filtern Anisotropic Diffusion Filtering (Perona & Malik, .....)n Spatially variant smoothing

Introduction to

Computer Vision Nagao-Matsuyama FilterNagao-Matsuyama Filter

n Calculate the variance within nine subwindows of a 5x5 moving window

n Output value is the mean of the subwindow with the lowest variance

n Nine subwindows used:

Introduction to

Computer Vision Kuwahara FilterKuwahara Filter

n Principle:l divide filter mask into four regions (a, b, c, d). l In each compute the mean brightness and the variance l The output value of the center pixel (abcd) in the window is the

mean value of that region that has the smallest variance.

Introduction to

Computer Vision Kuwahara Filter ExampleKuwahara Filter Example

Original

Median(1 iteration)

Median(10 iterations)

Kuwahara

Introduction to

Computer Vision Anisotropic Diffusion FilteringAnisotropic Diffusion Filtering

Note: Original Perona-Malik diffusion process is NOT anisotropic, although they erroneously said it was.

Introduction to

Computer Vision Example: Perona-MalikExample: Perona-Malik

n In general:l Anisotropic diffusion estimates an image from a noisy imagel Useful for restoration, etc.l Only shown smoothing examplesl Read the literature

Introduction to

Computer Vision Observations on EnhancementObservations on Enhancement

n Set of general techniquesn Varied goals:

l enhance perceptual aspects of an image for human observationl preprocessing to aid a vision system achieve its goal

n Techniques tend to be simple, ad-hoc, and qualitativen Not universally applicable

l results depend on image characteristicsl determined by interactive experimentation

n Can be embedded in larger vision systeml selection must be done carefullyl some techniques introduce artifacts into the image data

n Developing specialized techniques for specific applications can be tedious and frustrating.

Introduction to

Computer Vision SummarySummary

Summary

n What is noise?l Gaussian distribution

n Noise reductionl first principles

n Neighbourhoodl low-passl median

n Averaging pixels corrupted by noise cancels out the noise.

n Low-pass can blur image.

n Median may retain fine image detail that may be smoothed by averaging.

Conclusion

Introduction to Computer Vision Introduction Zhigang Zhu, City College of New York...

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