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Chapter3 Image Enhancement in the Spatial Domain

Chapter3 Image Enhancement in the Spatial Domain. 3.0 Introduction 3.1 Background 3.2 Some Basic Gray Level Transformations 3.3 Histogram Processing 3.4 Enhancement Using Arimethic/Logic Operations 3.5 Basics of Spatial Filtering 3.6 Smoothing Spatial Filters

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Chapter3 Image Enhancement in the Spatial Domain

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  1. Chapter3Image Enhancement in the Spatial Domain • 3.0 Introduction • 3.1 Background • 3.2 Some Basic Gray Level Transformations • 3.3 Histogram Processing • 3.4 Enhancement Using Arimethic/Logic Operations • 3.5 Basics of Spatial Filtering • 3.6 Smoothing Spatial Filters • 3.7 Sharpening Spatial Filters • 3.8 Combining Spatial Enhancement Methods

  2. 3.0 INTRODUCTION • 1. Objective • to process an image so that the result is more suitable for specific applications. • 2. Categories • a. spatial domain methods. • b. frequency domain methods. • c. combinations of above two types.

  3. 3.1 BACKGROUND • 1. Operate directly on an image f by the following way: g(x, y) = T [ f (x, y)] where g is the processed image; and T is an operator over an n  n neighborhood of f • For simplicity in notation： S = T（r） r and s is the gray level of f（x,y）and g（x,y）

  4. 3.1 BACKGROUND

  5. 3.1 BACKGROUND • 2. Point processing -- for n = 1, i.e., neighborhood = the pixel itself • 3. Mask processing -- also called filtering; for n  1; neighborhood = n  n pixels; using masks.

  6. 4.2 ENHANCEMENT BY POINT PROCESSING • 4.2.1 Some Simple Intensity Transformations • 4.2.2 Histogram Processing • 4.2.3 Image Subtraction • 4.2.4 Image Averaging

  7. 4.2.1 Some Simple Intensity Transformations…(1) • 1. Image negatives • given a pixel gray level ( g. l. ) r, output pixel g. l. s is L-1 T s L-1 = largest g. l. r L-1 0

  8. L-1 (r2,s2) s (r1,s1) 0 r L-1 4.2.1 Some Simple Intensity Transformations…(2) • 2. Contrast stretching • (1) Stretching is useful for improving image contrast. • (2) General transformation diagram. L-1 = largest g. l.

  9. 4.2.1 Some Simple Intensity Transformations…(3) • (3) cases : • a. no change -- if r1 = s1 , r2 = s2 . • b. thresholding -- if r1 = r2, s1 = 0, and s2 = L-1 ; threshod value = r1 = r2. • c. stretching -- if r1 r2 & s1 s2 ( to keep monotonicity of transformation ) • d. Why dies stretching improve image contrast? • e. How to stretch is problem - dependent

  10. L-1 (r2, s2) =(r0,L-1) T(C) s S=T(r) T(B) (r1, s1) =(r0,0) T(A) r r0 L-1 0 0 A B C 4.2.1 Some Simple Intensity Transformations…(4) Mappings: A=>T(A) (longer => smaller) L-1 = largest g. l. B => T(B) (smaller => larger ) CONTRAST IMPROVED) C => T(C) (larger => smaller)

  11. 4.2.1 Some Simple Intensity Transformations…(5) • 3. Compression of dynamic ranges • (1) Gray levels may be out of the display range after certain transformations. • (2) One way to compress is S = T( r ) = c log(1+ | r | ) • where c is a constant to make s to lie between 0 and L-1

  12. 4.2.1 Some Simple Intensity Transformations…(6) • 4. Gray level slicing • (1) Highlighting a specific g. l. range • (2) Useful for many applications. • (3) Two approaches: • a. use high values for desired ranges and low values for others; • b. use high values for desired ranges and keep the values for others. • (4) See Fig 4.7

  13. Fig 4.7

  14. 4.2.1 Some Simple Intensity Transformations…(7) • 5. Bit-plane slicing • (1) Highlighting “ contributions made by specific bits to image appearance. • (2) More important information is included in higher- order bit planes; details in others (see Fig. 4.8 ) • (3) Bit - plane 7 ( highest - order ) = result of thresholding with threshold = 128 • (4) See Fig. 4.9

  15. Nk/n …. r 1 2 3 k L-1 4.2.2Histogram Processing…(1) • 1.Definition and properties of histogram • (1) the histogram of a given image f is a function P( rk ) = nk / n where rk = the kth g. l.; nk = the no. of pixels with g.l. rk n = the total no. of pixels in f. • (2) Diagram of histogram

  16. 4.2.2Histogram Processing…(2) • (3) Concept : histogram = p.d.f ( probability density function ) • (4) A histogram gives the global appearance of an image . • (5) Histograms of images with high and low contrasts (see Fig. 4.10 )

  17. 4.2.2Histogram Processing…(3) • 2. Histogram equalization • (1) Equalization transformation of a given image f : where Pr(w) is the histogram of f • (2) S above is exactly the c. d. f. of r ( c. d. f. = cumulative distribution function )

  18. 4.2.2Histogram Processing…(4) • (3) It can be shown by probability theory (see textbook) that the new image with g. l. s has a uniform distribution, i.e., • (4) the transformation illustration Pr(r) Ps(s) Equalization r S Image with low contrast Image with higher contrast

  19. 4.2.2Histogram Processing…(5) • (5) Read the example in pp. 176-177. • (6) Histogram equalization is also called histogram flatting or linearization. • (7) Discrete form of histogram equalization

  20. 1 2 3 4 5 6 7 4.2.2Histogram Processing…(6) • (8) A computation example 1 3 9 17 23 24

  21. 4.2.2Histogram Processing…(7) • (9) A major advantage of histogram equalization for image contrast enhancement is that it can be applied automatically. • (10) See Fig. 4.14 for a real example. • 3. Histogram specification • See textbook • 4. Local enhancement • See textbook

  22. 4.2.3Image Subtraction • 1. Computes the difference g of two images f and h : • 2. For application example, see Fig. 4.17

  23. 4.2.4Image Averaging • 1.Reduces noise by averaging several copies of and identical image • 2. Method : where gi(x, y) is one of copies of origin image • 3. Why work ? Noise standard deviation can be reduced to 1/n of origin • 4. See Fig. 4.18

  24. 4.3Spatial Filtering • 4.3.1 Background • 4.3.2 Smoothing Filtering • 4.3.3 Sharpening Filtering

  25. 4.3.1 Background…(1) • 1. Spatial filtering is also called mask processing. • 2. Masks ( also called spatial filters ) are used • 3. High - frequency components in images: • noise, edges, sharp details, etc. • 4. Low - frequency components in images: • uniform regions, slow - changing background g. l., etc. • 5. Type of spatial filtering : • (1) lowpass filtering • useful for image blurring (smoothing)

  26. 4.3.1 Background…(2) • (2) Highpass filtering • useful for image sharpening • (3) Bandpass filtering • mostly used in image restoration • 6. See Fig 4.19 for the above 3 types. • 7. Linear mask operation :

  27. 4.3.1 Background…(2) • Operation : replace z5 by • R = MN = z1w1 + z2w2 +……. + z9w9 • 8. Nonlinear mask operation : • R is computed nonlinearly using information of the neighborhood of current pixel as well as the mask. Z5 = g. l. of current pixel Mask M Neighborhood N

  28. 4.3.2Smoothing Filters…(1) • 1. Used for image blurring & noise reduction. • 2. Useful for removing small details and bridging small gaps in lines or curves. • 3. Lowpass spatial filtering : • (1) Mask for 33 neighborhood

  29. 4.3.2Smoothing Filters…(2) • (2) Operation -- replace z5 by • (3) Also called neighborhood averaging • (4) See Fig. 4.22 for effect • 4. Median filtering • (1) Reducing noise without blurring images

  30. 4.3.2Smoothing Filters…(3) • (2) Operation -- • replace g. l. at (x, y) with the median of all the g. l. of neighborhood. • (3) Meaning of median • value r such that • (i.e., r = (1/2)tile of the p.d.f. ) P(x) r = median x Area = 1/2

  31. 4.3.2Smoothing Filters…(4) • (4) An example: given g. l. z5 = 15 is replaced by median = 20 • (5) For effect of median filtering, see Fig. 4.23

  32. 4.3.3Sharpening Filters…(1) • 1. Objective • highlighting or enhancing fine details in images. • 2. Applications • (1) electronic printing; • (2) medical imaging; • (3)industrial inspection; • (4) target detection; • etc.

  33. 4.3.3Sharpening Filters…(2) • 3. Basic highpass spatial filtering (HPSF) • (1) Mask for 33 neighborhood • (2) See Fig. 4.25 for effect of filtering • 4. High best filtering • (1) Also called high-frequency emphasis filtering.

  34. 4.3.3Sharpening Filters…(3) • (2) Method : high-best = A  (original) - lowpass = (A-1)(original) + original - lowpass = (A-1)(original) + highpass where A is a selected weight. • (3) Note that part of the original image is added back. • (4) Equivalent mask • (5) See Fig. 4.27 (A=1.1 is enough) Where w=9A-1 (A =1  basic HPSF)

  35. Averaging (integration) Difference (differentiation) Blurring Sharpening 4.3.3Sharpening Filters…(4) • 5. Derivative filters • (1) Concept - • (2) The most common differentiation operation is the gradient

  36. 4.3.3Sharpening Filters…(5) • (3) Gradient • a. definition -- the gradient of a function f at a pixel (x0, y0) is • b. magnitude of gradient --

  37. 4.3.3Sharpening Filters…(6) • (4) Approximation of gradient magnitude : • a. Assume the neighborhood (nbhd) g. l. of the pixel at (x, y) are • b. in continuous form With g. l. at (x, y) = z5 or

  38. 4.3.3Sharpening Filters…(7) • C. in absolute difference form : • d. Roberts operators for 2  2 nbhd : operation = | N  MR1 | + | N  MR2 | = | z5-z9 | + | z6 - z8 | or MR1 MR2

  39. 4.3.3Sharpening Filters…(8) • d. Prewitt operators for 3  3 nbhd : operation = | N  MP1 | + | N  MP2 | = | (z7+z8+z9) - (z1+z2+z3) | = | (z3+z6+z9) - (z1+z4+z7) | • e. Sobel operators for 3  3 nbhd : operation = | N  MS1 | + | N  MS2 | = | (z7+2z8+z9) - (z1+2z2+z3) | = | (z3+2z6+z9) - (z1+2z4+z7) | MP1 MP2 MS1 MS2

  40. 4.4 Enhancement in Frequency Domain • 4.4.1 Lowpass Filtering • 4.4.2 Highpass Filtering • 4.4.3 Homomorphic Filtering

  41. 4.4.1 Lowpass Filtering…(1) • 1. Use • image blurring ( smoothing ) • 2. Goal • Want to find a transfer function H(u, v) in the frequency domain to attenuate the high frequency in the FT F(u, v) of a given image f using the inverse FT

  42. 4.4.1 Lowpass Filtering…(2) • 3. Ideal lowpass filter (ILPF) • (1) Definition -- H(u, v) = 1 if D(u, v)  D0 = 0 otherwise where D(u, v) = distance from (0, 0) to (u, v); and D0 is a constant ( called cutoff frequency ) • (2) See Fig 4.30 for the filter shape in 3-D and 2-D • (3) The ILPF cannot be implemented by analog hardware ( but can be implemented by software ) • (4) Before seeing effects of the ILPF, we need to review more properties of the FT first.

  43. 4.4.1 Lowpass Filtering…(3) • 4. Additional review of the Fourier transform • See sec.3.2~3.4 for FT, DFT, FFT • See Fig. 4.31 for an example of Fourier spectrum ( or simply called spectrum ) • 5. See Fig. 4.32 for effects of applying the ILPF using different cutoff frequencies. • 6. The ILPF produces ringing effects; see Fig. 4.32(d) for an example; and see Fig. 4.33 for the reason. Note Fig. 4.33(a) is equivalent to the top view of a 2-D sinc function.

  44. 4.4.1 Lowpass Filtering…(4) • 7. Butterworth lowpass filter ( BLPF )] • (1) A BLPF of order n with cutoff frequency at D0 is defined as • where A = 1 or 0.414 • (2) See Fig. 4.34 for the shape of the BLPF. • (3) See Fig. 4.35 for the effects of applying the BLPF with n = 1 for 5 D0 values. • 8. The BLPF produces no ringing effect due to the smoothness of its transfer function where A=1 or 0.414

  45. 4.4.2 Highpass Filtering…(1) • 1. Use • Image sharpening • 2. Ideal highpass filter (IHPF) • (1) Transfer function H(u, v) = 0 if D(u, v)  D0 = 1 otherwise • (2) See Fig. 4.37 for shapes of the IHPF.

  46. 4.4.2 Highpass Filtering…(2) • 3.Butterworth highpass filter (BHPF): • (1) Transfer function where A = 1 or 0.414 • (2) See Fig. 4.38 for shapes of the BHPF. • 4.4.3 Homomorphic Filtering • 4.5 Generation of spatial masks from frequency domain specifications • Read the textbook

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