Image Enhancement - National Chiao Tung Universityvlab.ee.nctu.edu.tw/.../2015/01/Image-Enhancement-2015.pdfDigital Image Processing (Fall, 2015) NCTU EE Image Enhancement 3 Bit-Plane

Digital Image Processing (Fall, 2015)

NCTU EE Image Enhancement 1

Image Enhancement

Intensity Transformation

(x,y)

k

kT

f(x,y) g(x,y)

mask

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

Point Processing (K=1)

Log Transformations

e.g. g(x,y) = c log( 1 + | f(x,y)| )

f(x,y)

g(x,y)

Power-Law Transformations

crs



Piecewise-Linear Transformation Functions low-contrast images result from poor illumination, lack of dynamic range in the maging sensor, wrong setting of a lens aperture, etc.



Bit-Plane Slicing

7 6 5 4 3 2 1 0

1 byte

Bit-plane 7(most significant)

Bit-plane 0(least significant)

One 8-bit byte



Histogram Processing

intensity

count

.. .................

position

intensity

Dark Image Bright Image Low-contrast Image High-contrast Image



Histogram Equalization Pr(r)

r

Ps(s)

s

T

r

s

rmaxrmin rmaxrmin

Two conditions for T:

(1) T(r) is single-valued and monotonically increasing. (2) r T r r for r r rmin max min max ( )

dr

ds

r

s

Pr(r)

r

ds

s

Ps(s

)

dr

Pr(r) dr = Ps(s) dss = T(r)

dT(r)/dr = ds/dr = Pr(r)/Ps(s)

If Ps(s) = constant = K = 1/( rmax - rmin )

T(r) = Pr(r) dr + rmin

r1K rmin

for rmin r rmax.



Histogram Matching (Specification)

T1

r

s

Pr(r)

rrmaxrmin

Ps(s)

srmaxrmin

T2

Pq(q)

qrmaxrmin

q

s

T1 T2-1

q = T2 (T1(r))-1



Use of Histogram Statistics

Let Sxy denote a neighborhood centered at (x, y), xySp be the histogram

of the pixels in Sxy, and L be the number of possible intensity values in the image.

1

0

)(L

iiSiS rprm

xyxy

1

0

22 )()(L

iiSSiS rpmr

xyxyxy

otherwise y)f(x,

DKDk AND ),(),( G2SG10 xy

GS MkmifyxfEyxg xy



Spatial-Domain Filtering

Correlation w(x,y)☆f(x,y) =

a

as

b

bt

t)ys,t)f(xw(s,

Convolution w(x,y)★f(x,y) =

a

as

b

bt

t)-ys,t)f(xw(s,

Linear Filter

R = gij f(x+i,y+j)i = -m j = -n

nm

g(x)

G(f)

lowpass highpass bandpass Nonlinear Filter

e.g. min filter, max filter, median filter



Smoothing

smoothing

Purpose: blurring & noise reduction

lowpass spatial filter



Order-Statistics Filters -- median filter ( nonlinear filter )

The median of a set A is the value M, M A, s.t.

Pr ( {x | x M, x A } ) = Pr ( { x | x M, x A } ).

e.g. the median of { 10, 20, 20, 20, 15, 20, 20, 25, 100 } is 20.

Remarks: 1. Given N samples x1, x2, …, xN,

the sample mean, x , minimizes

N

iixG

1

2)( ;

the sample median, x~ , minimizes

N

iixG

1

)(

2. The sample mean is the maximum likelihood (ML) estimator of location of a constant parameter in Gaussian noise. The sample median is the maximum likelihood (ML) estimator of location of a constant parameter in Laplacian noise.



Sharpening Purpose: highlight or enhance fine detail.

1st-order derivative

)()1( xfxfx

f

2nd-order derivative

)(2)1()1(2

2

xfxfxfx

f

First Derivatives (Gradient)

gradient of f at (x,y): f f x

f y

T[ , ]

gradient magnitude:

y

f

x

f

y

f

x

f

22 )()(

1 0

- 10

1

- 1 0

0

Roberts Prewitt Sobel

0 0 0

1 2 1

-1 -2 -1 -1 -1 -1

0 0 0

1 1 1

-1

1 0

0

0

-1

-1

1

1

-2

1 0

0

0

-1

-1

2

1

2nd-order Derivatives

-- Laplacian Filter

2

2

2

22

y

f

x

ff





--- Unsharp Masking & High-Boost Filtering

),(),(),( yxfyxfyxgmask

),( yxf : a blurred version of f(x,y)

),(),(),( yxgkyxfyxg mask

k = 1 : Unsharp Masking k > 1: High-Boost Filtering



Combining Spatial Enhancement Methods



Frequency-Domain Filtering

x

y

u

v

-1

spatial domain frequency domain Frequency-domain methods

Principles of Frequency-Domain Analysis Linear, Shift-Invariant System

linear: T[a x1(t) + b x2(t)] = a T[x1(t)] + b T[x2(t)]

shift-invariant: if y(t) = T[x(t)], then y(t-t0) = T[x(t-t0)]

1-D Convolution

f(x) g(x) = f( )g(x - )d ( Continuous Case )

f(x) g(x) = f m g x m ( Discrete Case )

-

( ) ( )m

T

h(t)

x(t) y(t) = h( )x(t - )d-

1-D Fourier Transform

xX

A

f(x)

0

-2/X - 1/X 0 1/X 2/X

AX

F u( )

u

1

Fourier Transform Pair

{ }

{ }

( ) ( )

( )

( ) ( )

( )

( )

( )

( )

f x F(u) = f x dx

F(u) f(x) = F(u) du

F u R(u) + iI(u) = F(u)

F(u) = R u I u Fourier Spectrum

(u) = tan Phase Angle

P(u) =

-

-

-1 I u

R u

e

e

e

i ux

i ux

i u

2

1 2

2 2

F(u) Power Spectrum2



1-D Convolution Theorem

f(x) g(x) F(u)G(u) f(x) g(x) F(u) G(u)

Sampling

Whittaker-Shannon Sampling Theorem

x

f(x)

F(u)

u

x

f(x)

......

x

f(x) s(x)s(x)

S(u)F(u) S(u)

F(u)

x

x x

x 1/2w

Reconstruction



2-D Fourier Transform

Fourier Transform Pair

{ }

{ }

{ }

( ) ( , )

( ) ( , )

( )

(

( )

( )

( , )

f x,y F(u,v) = f x y dxdy

F(u,v) f(x,y) = F(u, v) dudv

Note: F x,y f x y

F u, v R(u, v) + iI(u,v) = F(u,v)

F(u, v) = R

=

=

=

e

e

e

i ux vy

i ux vy

i u v

2

1 2

u, v I u,v Fourier Spectrum

(u, v) = tan Phase Angle

P(u, v) = F(u,v) Power Spectrum

-1 I u, v

R u, v

) ( )

( )( )

( )

2 2

2

2-D Convolution & Convolution Theorem

f(x, y) g(x, y) = f( , )g(x - , y - )d d

f(x, y) g(x, y) = f m, n g x m,y - n n=-m

( ) ( )

*

(x,y)

G(u,v) = H(u,v) F(u,v) g(x,y) = h(x,y) f(x,y) *

optical transfer function point spread function (spatial convolution mask)

H(u,v)

h(x,y) f(x,y) g(x,y)

F(u,v) G(u,v)



2-D Sampling

2Wu

2Wv u

v

u

1/y

1/xv

2-D sampling function

y

x

y

x

z

Finite Sampling

x

f(x) s(x)

x

h(x)

X

u

F(u) S(u) H(u)

u

x

f(x) s(x) h(x)

X

u u

x

(DFT)

1/x

F(u) S(u) H(u)



1-D Discrete Fourier Transform (DFT)

f(4)

f(3)

f(2) f(1) f(0)

x0 x1 x2 x3 x4

x

F(4) F(3) F(2) F(1)

F(0)

u0 u1 u2 u3 u4

u

u = 1/ (Nx)

f(x) = F(u)

x =

i2 ux

N

u=0

N-1

0, 1, 2, .. , N - 1

e

F(u) = 1

Nf(x)

u =

-i2 ux

N

x=0

N-1

0, 1, 2, .. , N - 1

e

2-D DFT pair:

F(u,v) = 1

Nf(x,y)

f(x,y) = 1

NF(u,v)

-i2 (ux+vy

N

y=0

N-1

x=0

N-1

i2 (ux+vy

N

v=0

N-1

u=0

N-1

e

e

)

)

F(u,v) = 1

MNf(x,y)

f(x,y) = F(u,v)

-i2 (ux

M

vy

N

y=0

N-1

x=0

M-1

i2 (ux

M

vy

N

v=0

N-1

u=0

M-1

e

e

)

)

If M = N, we may also use



Properties of 2-D DFT Separability

F(u, v) = 1

Nf(x, y)

= 1

NN

1

Nf(x, y)

-i2 (ux+vy

N

y=0

N-1

x=0

N-1

-i2 (ux

N-i2 (

vy

N

y=0

N-1

x=0

N-1

e

e e

)

) )

{ }

1-D

1-D y

x

f(x,y)

v

x

F1(x,v)

v

u

F(u,v) N2 NN2 N

Periodicity f(x,y) = f(x+kN,y+lN), k, l = 0, 1, 2, .. F(u,v) = F(u+kN,v+lN)

Rotation f(r, + ) F(w, + ) 0 0



Translation f(x-x0,y-y0) F(u,v)e

-i2 (ux +vy

N0 0 )

f(x,y)ei2 (

u x+v y

N0 0 )

F(u-u0,v-v0)

e.g. if u0 = v0 = N/2

f x y F(u - , v - )(x+y)N / 2 N / 2( , )( )1

Conjugate Symmetry If f(x,y) is real, F(u,v) = F (-u,-v)

Distributivity f (x,y) + f (x,y) F (u,v) + F (u,v)

( f (x,y) f (x,y) F (u,v) F (u,v) )1 2 1 2

1 2 1 2

Scaling a f(x, y) a F(u,v)

f(ax,by) 1

abF(

u

a,v

b)

Average Value

f (x,y) = 1

Nf x y =

1

NF(0,0) 2

y=0

N-1

x=0

N-1

( , )



(Ref: http://www.cs.unm.edu/~brayer/vision/fourier.html)







Frequency-Domain Filtering





Smoothing Ideal Lowpass Filters



Butterworth Lowpass Filters

nDvuDvuH

20 ]/),([1

1),(



Gaussian Lowpass Filters 22 2/),(),( vuDevuH



Sharpening Ideal Highpass Filter

0

0

Dv)D(u, if 1

Dv)D(u, if 0v)H(u,

Butterworth Highpass Filter

2n0 v)]/D(u,[D1

1v)H(u,

Gaussian Highpass Filter

20

2 2/),(1v)H(u, DvuDe



Laplacian Filter

v))F(u,v+(u- y)f(x, 222

Unsharp Masking, High-Boost Filtering, and High-Frequency Emphasis Filters

),(1),(),(),(),( vuHvuHyxfyxfyxf lphplphp

),()1(),(

),(),()1(),(),(),(

vuHAvuH

yxfyxfAyxfyxAfyxf

hphb

lplphb

),(),( vubHavuH hphfe

original blurred image result

(Ref: http://www.astropix.com/HTML/J_DIGIT/USM.HTM)



Homomorphic Filter

normalsource

observeri(x,y)

r(x,y)

ln FFT H(u,v) FFT-1 expf(x,y) g(x,y)

f(x,y) = i(x,y) r(x,y) I(u,v) R(u,v)z(x,y) = ln f(x,y) = ln i(x,y) + ln r(x,y) LNI(u,v) + LNR(u,v)

slow spatial variation vary abruptly

H

H(u,v)

D(u,v)

L

H > 1L < 1

contrast enhancement

dynamic range compression

Original image Processed image

(Ref: http://www.vision.ee.ethz.ch/~pcattin/SIP/5-Enhancement.html )



Noise Suppression Noise Models

Usually image noise is assumed to be white and to be uncorrelated with the image.

Some Important Noise Probability Density Functions Gaussian Noise

22 2/)(

2

1)(

zezp

Remark: such as electronic circuit noise and sensor noise due to poor illumination and/or high temperature.

Rayleigh Noise

az

azeazbzp

baz

for 0

for )(2

)(/)( 2

4

)4(

4/

2

b

ba

Remark: such as noise in range imaging.

Erlang (Gamma) Noise

0for 0

0for )!1()(

1

z

zeb

zazp

azbb

22

a

ba

b

Remark: such as noise in laser imaging.

Exponential Noise (a special case of the Erlang pdf)

0for 0

0for )(

z

zaezp

az

22 1

1

a

a

Remark: such as noise in laser imaging. Uniform Noise



otherwise

bzabzp

0

afor 1

)(

12

)(

22

2 ab

ba

Remark: useful as the basis for numerous random number generation. Impulse (Salt-and-Pepper; Shot; Spike) Noise

otherwise 0

for

for

)( bzP

azP

zp b

a

Remark: found in situations where quick transients, such as faulty switching, take place during imaging.





Period Noise

Estimation of Noise Parameters If the imaging system is available, capture a set of images of “flat” environments. When only images are available, we may use small patches of reasonably constant gray level.

Szii

Szii

i

i

zpz

zpz

)()(

)(

22



Noise Suppression Mean Filters

Arithmetic Mean Filter

xySts

tsgmn

yxf),(

),(1

),(ˆ

Geometric Mean Filter

mn

Sts xy

tsgyxf1

),(

]),([),(ˆ

Harmonic Mean Filter

xySts tsg

mnyxf

),( ),(

1),(ˆ

Remark: works well for salt noise, but fails for pepper noise. Contraharmonic Mean Filter

xy

xy

Sts

Q

Sts

Q

tsg

tsg

yxf

),(

),(

1

),(

),(

),(ˆ

Remark: Q is called the order of the filter Positive Q eliminate pepper noise Negative Q eliminate salt noise Q = 0 arithmetic filter Q = -1 harmonic filter





Order-Statistics Filters Median Filter

)},({),(ˆ),(

tsgmedianyxfxySts

Remark: work well for both bipolar and unipolar impulse noise.

Max and Min Filters )},({max),(ˆ

),(tsgyxf

xySts )},({min),(ˆ

),(tsgyxf

xySts

Remark: Max filter works well for pepper noise. Min filter works well for salt noise.

Midpoint Filter

)}],({min)},({max[2

1),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

Remark: work well for Gaussian noise and uniform noise.

Alpha-trimmed Mean Filter Suppose we delete the d/2 lowest and the d/2 highest gray-level values of g(s,t) in the neighborhood Sxy. Let gr(s,t) represent the remaining mn-d pixels.

xySts

r tsgdmn

yxf),(

),(1

),(ˆ

Remark: useful in situations involving multiple types of noise, such as a combination of salt-and-pepper and Gaussian noise.





Bilateral Filter Proposed by C. Tomasi and R. Manduchi, 1998. Based on geometric closeness and photometric similarity.

Linear filter

where

f(x): original image

c(,x): measure the geometric closeness between x and a nearby point

Bilateral filter

where

s(f(); f(x)): measure the photometric similarity between the pixel at x

and that of a nearby point . Example:



})),(

(2

1exp{),( 2

d

dc

x

x where xx ),(d

})))(),((

(2

1exp{),( 2

r

ffs

x

x where ff ),(

(Ref: http://www.cs.duke.edu/~tomasi/papers/tomasi/tomasiIccv98.pdf )



Periodic Noise Reduction Bandreject Filters

-- Ideal bandreject filter

2

Dv)D(u, if 1

2),(

2D if 0

2

Dv)D(u, if 1

),(

0

00

0

W

WDvuD

W

W

vuH

D(u,v): distance from the origin.

-- Butterworth

n

DvuD

WvuDvuH

220

2]

),(

),([1

1),(

-- Gaussian 2

20

2

]),(

),([

2

1

1),( WvuD

DvuD

evuH



Bandpass Filters ),(1),( vuHvuH brbp

Notch Filters

-- Ideal notch reject filter

otherwise 1

),(Dor ),(D if 0),( 0201 DvuDvu

vuH

2/120

202

2/120

201

])2/()2/[(),(

])2/()2/[(),(

vNvuMuvuD

vNvuMuvuD

-- Butterworth notch reject filter

n

vuDvuD

DvuH

]),(),(

[1

1),(

21

20

-- Gaussian notch reject filter 2

20

21 ]),(),(

[2

1

1),( D

vuDvuD

evuH



-- Notch pass filter ),(1),( vuHvuH nrnp



Optimum Notch Filtering

),(),(),( vuGvuHvuN

H(u,v): notch pass filter to pass only components associated with the interference pattern.

G(u,v): Fourier transform of the corrupted image.

)},(),({),( 1 vuGvuHyx

),(),(),(),(ˆ yxyxwyxgyxf

Objective: to select w(x,y) so that the variance of the estimate is minimized over a specified neighborhood of every point (x,y).

a

as

b

bt

yxftysxfba

yx 22 )],(ˆ),(ˆ[)12)(12(

1),(

where

a

as

b

bt

tysxfba

yxf ),(ˆ)12)(12(

1),(ˆ

2

2

]}),(),(),([

)],(),(),({[)12)(12(

1),(

yxyxwyxg

tysxtysxwtysxgba

yxa

as

b

bt

Assume that w(x,y) remains essentially constant over the neighborhood. ),(),( yxwtysxw

),(),(),(),( yxyxwyxyxw

2

2

)]},(),(),([

)],(),(),({[)12)(12(

1),(

yxyxwyxg

tysxyxwtysxgba

yxa

as

b

bt

0),(

),(2

yxw

yx

),(),(

),(),(),(),(),(

22 yxyx

yxyxgyxyxgyxw



Image Enhancement - National Chiao Tung Universityvlab.ee.nctu.edu.tw/.../2015/01/Image-Enhancement-2015.pdfDigital Image Processing (Fall, 2015) NCTU EE Image Enhancement 3 Bit-Plane

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