lectures 6&7: image enhancement

Machine Vision and Dig. Image Analysis

1 Prof. Heikki Kälviäinen CT50A6100

Lectures 6&7: Image Enhancement

Professor Heikki Kälviäinen

Machine Vision and Pattern Recognition Laboratory Department of Information Technology

Faculty of Technology ManagementLappeenranta University of Technology (LUT)

[email protected]://www.lut.fi/~kalviai

http://www.it.lut.fi/ip/research/mvpr/


Prof. Heikki Kälviäinen CT50A6100

2

Content

• Background.– Spatial domain methods.– Frequency domain methods.

• Enhancement by point processing.• Spatial filtering.• Enhancement in the frequency domain.



3

Background: Motivation

• For preprocessing to make the image look better, i.e., more suitable for further processing.

• Problems with – contrast, – sharpness, – smoothness, – noise, – distortions, – etc.



4

Background: Spatial domain methods

• Method: Slide the mask though the image and compute new pixel values.

• Image processing function:g(x,y) = T[f(x,y)]

f(x,y) the input imageg(x,y) the processed imageT an operator on f, defined over some neighborhood of (x,y)

• Gray-level transformation (mapping) function:s = T(r)

r denotes f(x,y) and s denotes g(x,y)



5

Background: Frequency domain methods

• Method: Multiply the Fourier transforms of the image and the mask and apply the inverse transform to the multiplication.

• Convolution:

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

h(x,y) a linear, postion invariant operator• Fourier transform:

G(u,v) = H(u,v)F(u,v)

H(u,v) the transfer function of the process• Inverse Fourier transform:

g(x,y) = F^{-1} [H(u,v)F(u,v)]



6

Enhancement by point processing: Some simple intensity transformations

• Image negatives:

s = ((L-1) – r) where L = number of gray-levels• Contrast stretching:

– Poor illumination, lack of dynamic range in the imaging sensor, wrong setting of a lens aperture during image acquisition.

– To increase the dynamic range of the gray-levels.– Piecewise linear function. – When a thresholding function => a binary image (two values

only).



7

Contrast stretching



8

Enhancement by point processing: Some simple intensity transformations (cont.)

• Compression of dynamic range: – The dynamic range exceeds the capability of the

display device. The need of brighter pixels. s = c log(1 + abs(r)) where c is a scaling constant

• Gray-level slicing: – Highlighting a specific range of gray-levels with

• “removing” or • preserving other pixels.



9

Gray-level slicing

• Original image (top).

• Thresholded (left).

• Gray-level slicing (right).



10

Enhancement by point processing: Some simple intensity transformations (cont.)

• Bit-plane slicing:– Select the specific bit planes.– For example: the image of eight 1-bit planes.– Plane 7 contains all the high-order bits:

• Higher planes contain visually significant data. • Note: digital watermarking!

– To select the plane 7 only corresponds to the image thresholded at gray-level 128.



11

Enhancement by point processing: Histogram processing

• Histogram of the image: p(r_k) = n_k/nwhere

r_k is the kth gray-level n_k is the number of pixels with that gray-leveln is the total number of pixels in the imagek = 0, 1, 2, …, L-1 L is the number of gray-levels



12

Histogram of an image



13

Enhancement by point processing: Histogram processing (cont.)

• Histogram equalization (or histogram linearization):

to obtain the uniform histogram.• Gray-level transformation function and its inverse function:

s = T(r)

where 0<=T(r)<=1 and

T(r) is single-valued and monotonically increasing in 0<=r<=1

r = T^{-1}(s) where 0<=s<=1



14

Histogram equalization



15


• Probability density function:

p_s(s) = [p_r(r) dr/ds]_{r=T^{-1}(s)}• Transformation function:

s = T(r) = integral_0^r (p_r(w)dw)

where 0<=r<=1

ds/dr=p_r(r)• Uniform density

p_s(s) = [p_r(r) 1/p_r(r)]_{r=T^{-1}(s)} = 1

where 0<=s<=1• To obtain T^{-1} analytically is not always easy!



16


• Example: p_r(r) = -2r + 2 when 0<=r<=1

0 elsewhere• What transformation function makes the uniform

density?s = T(r) = integral_0^r ((-2w + 2)dw) = -r^2 + 2r

• Proof: r = T^{-1}(s) = 1 ± sqrt(1-s), 0≤r≤1 => r = 1 - sqrt(1-s) p_s(s)=[p_r(r)dr/ds] = [(-2r+2)dr/ds] = 1 0≤s≤1



17


• In discrete form, probabilities:

p_r(r_k) = n_k/n

where 0≤r_k≤1 and k = 0, 1, …, L-1• Transformation function:

s_k = T(r_k) = ∑ n_j/n j=0,...,k

= ∑ p_r(r_j) j=0,...,k

where 0≤r_k≤1 and k=0,1,…,L-1• The new value n is the gray-level closest to the sum of probabilities

up to the original value k:

n = round((L-1) x s_k)



18


• Histogram specification: – To apply another transformation function than an

approximation to a uniform histogram.• Local enhancement:

– Local processing instead of the whole image.– For example, histogram equalization of a 7x7

neighborhood about each pixel.



19

Enhancement by point processing:Image subtraction

• The difference between two images f(x,y) and h(x,y):

g(x,y) = f(x,y) – h(x,y)

done by pixelwise subtraction.• The use of the mask image. • Applications in medical image processing:

The mask is a normal image which is subtracted from a sample image to point out regions of interest.



20

Image subtraction



21

Enhancement by point processing:Image averaging

• Consider a noisy image g(x,y) formed by the addition of noise η(x.y) to an original image image f(x,y):

g(x,y) = f(x,y) + η(x.y)• By averaging noisy images, noise is reduced.• Noise must be uncorrelated and must has the zero

average value!• Example: noisy microscope images.



22

Spatial filtering: Background

• Spatial filtering: the use of spatial filters. • Spatial filters:

– Lowpass filters.– Highpass filters.– Bandpass filters.

• The mask: w1 w2 w3w4 w5 w6

w7 w8 w9• Smoothing filters, sharpening filters.



23

Spatial filtering: Smoothing filters

• For blurring and noise reduction.• Lowpass spatial filtering:

1 1 1

1/9 x 1 1 1

1 1 1– Neighborhood averaging.

• Median filtering: replace the gray-level of each pixel by the median of the gray-levels in a neighborhood of that pixel. – Removes noise, but preserves details such as edges. – Filter size?, weighted median filtering?



24

Spatial filtering: Averaging vs. median

• Original image (upper left).• Original + noise (upper right).• Smoothed image (lower right).• Median smoothing (lower left).



25

Spatial filtering: Sharpening filters

• For highlighting fine detail in an image or enhance detail that has been blurred.

• Filters:– Basic highpass spatial filter.– High-boost filtering.– Derivative filters.



26

Spatial filtering: Basic highpass spatial filtering

• Positive coefficients near the center of a filter, negative coefficients in the outer periphery.

• 3 x 3 sharpening filter:

-1 -1 -1

1/9 x -1 8 -1

-1 -1 -1• The sum of the coefficients is zero.• The filter eliminates the zero frequency term =>

the reduced global contrast of the image. • Scaling and/or clipping for negative values to map the range [0, L-1].



27

Spatial filtering: High-boost filtering

• Highpass = Original – Lowpass.• High-boost or high-frequency-emphasis filter:

High boost = (A)(Original) – Lowpass

= (A-1)(Original) + Original – Lowpass

= (A-1)(Original) + Highpass.

• Looks like an original image, with edge enhancement by A.



28

Spatial filtering: High-boost filtering (cont.)

• Unsharp masking: to subtract a blurred image from an original image.

• In the printing and publishing industry.• The mask with w = 9A -1 (with A≥1):

-1 -1 -1

1/9 x -1 w -1

-1 -1 -1



29

Spatial filtering: Derivative filters

• For sharpening an image (averaging vs. differentiation).• The gradient of f(x,y):

∂f/∂x

df =

∂f/∂y

• The magnitude is the basis for image differentiation methods:

mag(df)= ((∂f/∂x)^2 + (∂f/∂y)^2)^(-1/2)



30

Spatial filtering: Derivate filters (cont.)

• Roberts: 1 0 0 1

0 -1 1 0• Prewitt:

-1 -1 -1 -1 0 1

0 0 0 -1 0 1

1 1 1 -1 0 1• Sobel:

-1 -2 -1 -1 0 1

0 0 0 -2 0 2

1 2 1 -1 0 1



31

Enhancement in the frequency domain

• The use of image frequencies for enhancement. • Convolution: f(x)*g(x) F(u) G(u).• The filtered image g(x,y) using the Fourier transforms of an

original image f(x,y) and a mask h(x,y):

g(x,y) = F^{-1} [H(u,v)F(u,v)]

• Lowpass filtering.• Highpass filtering.



32

Images and Their Fourier Spectra

i = sqrt(-1)F(k) also denoted F(u)

F(x,y) also denoted F(u,v)



33

Discrete Fourier Transform (DFT)



34

Fourier transform: Image power

Radius (pixels) % Image power

8 95

16 97

32 98

64 99.4

128 99.8

Distance from point (u,v) to the origin: D(u,v) = ((u^2 + v^2))^(-1/2)



35

Enhancement in the Frequency Domain: Lowpass filter

• G(u,v) = H(u,v) F(u,v).• Ideal lowpass filter:

– H(u,v) = 1 if D(u,v) ≤ D_0, or 0 if D(u,v) > D_0. – Original (left) and filtered image (right).



36

Enhancement in the Frequency Domain: Butterworth lowpass filter

• The transfer function:

H(u,v) = 1/(1 + (D(u,v)/D_0)^(2n))

where

n is the order of the filter

D_0 is the cutoff frequency locus (select!)• H(u,v) from 1 to 0. • When D(u,v) = D_0, H(u,v) = 0.5.• H(u,v) = 1/√2 commonly used.



37

Enhancement in the Frequency Domain: Highpass filter

• Ideal high pass filter:– H(u,v) = 0 if D(u,v) ≤ D_0, or 1 if D(u,v) > D_0.– Original (left) and filtered image (right).



38

Enhancement in the Frequency Domain: Butterworth highpass filter

• The transfer function:

H(u,v) = 1/(1 + (D_0/D(u,v))^(2n))

where

n is the order of the filter

D_0 is the cutoff frequency locus• H(u,v) from 0 to 1. • When D(u,v) = D_0, H(u,v) = 0.5.• H(u,v) = 1/√2 commonly used.



39

Summary

• For preprocessing to make the image look better, i.e., more suitable for further processing.

• Approaches: – Spatial domain methods.– Frequency domain methods.

• Enhancement by point processing.• Spatial filtering.• Enhancement in the frequency domain.

lectures 6&7: image enhancement

Documents