2007 Theo Schouten 1

Enhancements

Techniques for editing an image such that it is more suitable for a specific application than the original image.

Spatial domain: g(x,y) = T[f(x,y)]

Frequency domain: g(x,y) = FT-1[H(u,v) F(u,v] = h(x,y) f(x,y)

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Point processing

Gamma transformation:s = c r

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Histogram

With histogram equalization we search for a T(r) that makes the histogram as smooth as possible. The T(r) that accomplishes that is:

sk = round( L j=0 k (nj / n) )

with nk the number of pixels with gray level k, n the total

number of pixels and L the number of gray levels.

Landsat image river TaagHistogram

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Examples

Original Contrast stretched Hist. equalization Local histogram Local contrast

Local contrast enhancement:g(x,y) =(x,y) + kM (f(x,y) - (x,y))/(x,y)

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Smoothing

This is used for the blurring of an image: the removal of small details and the filling in of small gaps in lines, contours and planes, and also reduces the noise in an image.

In the frequency domain smoothing becomes: G(u,v) = H(u,v)F(u,v) : low pass filterIn the spatial domain smoothing is the removal of drastic changes by averaging the gray levels in a certain region with a positive weight.

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Smoothing Frequency domain

Butterworth LPF:Hn(u,v)=1/(1+((u2+v2)/D0)

2n )

the Exponential LPF: Hn(u,v)=exp(-(u2+v2)/D0 )

n )

the Gaussian LPF:H(u,v)=exp( - (u2+v2) / 2 D0

2) Ideal LPF, the rings of especially the rivers can clearly be seen.Right image : Butterworth LPF with n=5. Here the ringing has decreased.

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Gaussian LPF

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Smoothing spatial domainA linear filter can be shown as a convolution mask:

      | 1 1 1 1 1 |   |0 1 1 1 0|       |1 2 3 2 1|        |1  4  6  4  1|      | 1 1 1 1 1 |       |1 1 1 1 1|       |2 4 6 4 2|        |4 16 24 16  4|(1/25)| 1 1 1 1 1 | (1/21)|1 1 1 1 1| (1/81)|3 6 9 6 3| (1/256)|6 24 36 24  6|      | 1 1 1 1 1 |       |1 1 1 1 1|       |2 4 6 4 2|        |4 16 24 16  4|      | 1 1 1 1 1 |       |0 1 1 1 0|       |1 2 3 2 1|        |1  4  6  4  1|

The 2 right filters are examples of separable filters, they can be executed as a convolution with (1/9) | 1 2 3 2 1 | respectively (1/16) | 1 4 6 4 1 |  in the x direction, followed by a convolution in the y direction. The right filter is a poor approximation of the Gaussian function g(x,y) = c exp( (x2+y2) / 2  2), better ones are not separable.

With a non-linear "rank" or "order-statistics" filter, the pixel values in the neighborhood are sorted according to increasing value, the value at a fixed position in the row is chosen to replace the central pixel. Choosing a value in the middle results in a so-called median filter.

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Examples mean and median image     average filter  median filter  9 9 9 0 0 0    . . . . . .     . . . . . .   9 9 9 0 0 0    . 8 5 3 0 .     . 9 9 0 0 .   9 0 9 0 0 0    . 8 5 4 1 .    . 9 9 0 0 .   9 9 9 0 9 0    . 8 5 4 1 .    . 9 9 0 0 .  9 9 9 0 0 0    . 9 6 4 1 .    . 9 9 0 0 .   9 9 9 0 0 0    . . . . . .    . . . . . .

Original5x5 mean5x5 median

20% impulse noise5x5 mean5x5 median

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Noise models

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Noise models (2)

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SharpeningThis is used to bring fine details to the front of the image and to sharpen the edges of objects. High Pass Filter: G(u,v) = H(u,v) F(u,v).

ideal HPF: H(u,v) = 1 if  (u2+v2) > D and 0 otherwise, see fig. 4.24Butterworth HPF: H(u,v) = 1/(1+D/  (u2+v2) )2n, see fig. 4.25Exponential HPF: H(u,v) = exp(- D/  (u2+v2) )n

Gaussian HPF: H(u,v) = 1- exp( - (u2+v2) / 2 D02) see fig. 4.26

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Examples combinationsHere an example from High Frequency Emphasis, where the third order Butterworth HPF is added to the original image using the proportion 0.7:1.

Blur masking: I-LPF(I)The impact of fragments from the Shoemaker-Levy comet on Jupiter on July 19th 1994.

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Sharpening in the spatial domainDifferences in the gray levels between pixels, often as an approximation of the derivative of the image function f(x,y):   f(x,y) = ( f/ x,   f/ y )  ( ( f/ x)2 + ( f/ y)2 ) results in the gradient image.

In the simplest case, one discretely approximates:  f/ x = f[x,y] - f[x-1,y] g( f[x,y] ) = |  f/ x | + |  f/ y |

Other manners often used to determine the derivative are:|1  0| | 0 1|  |-1 -1 -1| |-1 0 1|  |-1 -2 -1| |-1 0 1||0 -1| |-1 0|  | 0  0  0| |-1 0 1|  | 0  0  0| |-2 0 2|               | 1  1  1| |-1 0 1|  | 1  2  1| |-1 0 1|

   Roberts            Prewitt              Sobel

Prewitt and Sobel take more pixels into account and are thus less sensitive to noise.

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Sobel

Original Simple derivative Sobel Gausian

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LaplacianOther sharpening operators are derived from the Laplacian:     2 f(x,y) =    2f/ x2 +  2f/ y2 Discretely, one can use masks:      |-1 -1 -1|          | 0 -1  0|(1/9) |-1  8 -1| or (1/5) |-1  4 -1|      |-1 -1 -1|          | 0 -1  0|

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Moon

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Whole body bone scan