Lectures 6&7: Image Enhancement

Transcription

1 Lectures 6&7: Image Enhancement Leena Ikonen Pattern Recognition (MVPR) Lappeenranta University of Technology (LUT) 1

2 Content Background Spatial domain methods Frequency domain methods Enhancement by point processing Spatial filtering Enhancement in the frequency domain 2

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 3

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 image g(x,y) T the processed image 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) 4

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) Fourier transform: G(u,v) = H(u,v)F(u,v) a linear, postion invariant operator H(u,v) the transfer function of the process Inverse Fourier transform: g(x,y) = F -1 [H(u,v)F(u,v)] 5

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 Thresholding function => binary image (two values only) 6

7 Contrast stretching 7

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 8

9 Gray-level slicing Original image (top) Thresholded (left) Gray-level slicing (right) 9

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

11 Enhancement by point processing: Histogram processing Histogram of the image: p(r k ) = n k /n where r k is the kth gray-level n k is the number of pixels with that gray-level n is the total number of pixels in the image k = 0, 1, 2,, L-1 L is the number of gray-levels 11

12 Histogram of an image 12

13 Histogram equalization 13

14 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 r represents gray-level values normalized to interval [0,1] (r=0=black, r=1=white) s = T(r) is the new equalized gray-value for gray-value 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

15 Enhancement by point processing: Histogram processing (cont.) 15

16 Enhancement by point processing: Histogram processing (cont.) 16

17 Enhancement by point processing: Histogram processing (cont.) Example: p r (r) = -2r + 2 when 0<=r<=1 0 elsewhere What transformation function creates uniform density? r s T ( r) ( 2w 2) dw r 2 2r 0 r T 1( s) 1 1 s,0 r 1 r 1 1 s 17

18 Enhancement by point processing: Histogram processing (cont.) In discrete form, probabilities: p r (r k ) = n k /n where 0 r k 1, k = 0, 1,, L-1 n is the total number of pixels in the image n k is the number of pixels with gray-value r k L is the total number of possible gray-levels in the image Transformation function: s k = T(r k ) = p r (r j ) = n j /n for j=0,...,k where 0 r k 1 and k=0,1,,l-1 Note that probability p r (r j ) is simply the fraction of pixels with grayvalue r j out of the total number of pixels The new gray-value is the gray-level closest to the sum of probabilities up to the original value k: round((l-1) s k ) 18

19 Enhancement by point processing: Histogram processing (cont.) 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 around each pixel 19

20 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) The use of a mask image (pixelwise subtraction) Applications in medical image processing: The mask is a normal image which is subtracted from a sample image to point out regions of interest, e.g. object that has moved between frames/images (see next slide) Remember also the regular image subtracted from the original to detect irregularities (e.g. missing dots) 20

21 Image subtraction 21

22 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 have zero average value! Do NOT use averaging for salt and pepper noise! Example: noisy microscope images 22

23 Spatial filtering: Background Spatial filtering: the use of spatial filters Spatial filters: Lowpass filters Highpass filters Bandpass filters The mask: w1 w2 w3 w4 w5 w6 w7 w8 w9 Smoothing filters, sharpening filters 23

24 Spatial filtering: Smoothing filters For blurring and noise reduction Lowpass spatial filtering: /9 x 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

25 Spatial filtering: Averaging vs. median Original image (upper left) Original + noise (upper right) Smoothed image (lower right) Median smoothing (lower left) 25

26 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

27 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: /9 x The sum of the coefficients is zero The filter eliminates the zero frequency term => reduced global contrast of the image Scaling and/or clipping for negative values to map the range [0, L-1] 27

28 Spatial filtering: High-boost filtering Highpass = Original Lowpass Low frequencies are lost High-boost or high-frequency-emphasis filter: High boost = (A)(Original) Lowpass = (A-1)(Original) + Original Lowpass = (A-1)(Original) + Highpass. Looks like original image, with edge enhancement by A fourier.eng.hmc.edu/e161/lectures/gradient/node2.htm l 28

29 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): /9 x -1 w

30 Spatial filtering: Derivative filters For sharpening an image (averaging vs. differentiation) The gradient of f(x,y): df = f/ x f/ y The magnitude is the basis for image differentiation methods: mag(df)= (( f/ x) 2 + ( f/ y) 2 ) (-1/2) 30

31 Spatial filtering: Derivate filters (cont.) Roberts: Prewitt: Sobel:

32 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 Discrete 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

33 Fourier transform: Image power Radius (pixels) % Image power Distance from point (u,v) to the origin: D(u,v) = (u 2 + v 2 ) (-1/2) 33

34 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) 34

35 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. 35

36 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). 36

37 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. 37

38 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 38