Digital Image Processing Chapter 3: Image Enhancement in the Spatial Domain

Transcription

1 Digital Image Processing Chapter 3: Image Enhancement in the Spatial Domain Principle Objective o Enhancement Process an image so that the result will be more suitable than the original image or a speciic application. Techniques are problem oriented. A method which is quite useul or enhancing an image ma not necessaril be the best approach or enhancing another images No general theor on image enhancement eists.

2 domains Spatial Domain image plane: Techniques are based on direct manipulation o piels in an image. Gra level transormations. Histogram processing. Arithmetic/Logic operations. Filtration techniques. Frequenc Domain : Techniques are based on modiing the Fourier transorm o an image Good images For human visual The visual evaluation o image qualit is a highl subjective process. It is hard to standardize the deinition o a good image. For machine perception The evaluation task is easier. A good image is one which gives the best machine recognition results. A certain amount o trial and error usuall is required beore a particular image enhancement approach is selected.

3 Spatial Domain Procedures that operate directl on piels. g, = T[,] where, is the input image g, is the processed image T is an operator on deined over some neighborhood o, Mask/Filter, Neighborhood o a point, can be deined b using a square/rectangular common used or circular subimage area centered at, The center o the subimage is moved rom piel to piel starting at the top o the corner

4 Point Processing Neighborhood = 11 piel g depends on onl the value o at, T = gra level or intensit or mapping transormation unction s = Tr Where r = gra level o, s = gra level o g, Contrast Stretching Produce higher contrast than the original b darkening the levels below m in the original image Brightening the levels above m in the original image

5 Thresholding Produce a two-level binar image Mask Processing or Filter Neighborhood is bigger than 11 piel The value o the mask coeicients determine the nature o the process Used in techniques Image Sharpening Image Smoothing

6 3 basic gra-level transormation unctions Output gra level, s Negative Log nth root nth power Linear unction Negative and identit transormations Logarithm unction Log and inverse-log transormation Power-law unction Identit Inverse Log n th power and n th root transormations Input gra level, r Identit unction Output gra level, s Negative Log nth root nth power Output intensities are identical to input intensities. Is included in the graph onl or completeness. Identit Inverse Log Input gra level, r

7 Image Negatives Output gra level, s Negative nth root Log nth power Identit Inverse Log Input gra level, r An image with gra level in the range [, L-1] L where L = n ; n = 1, Negative transormation : s = L 1 r Reversing the intensit levels o an image. Suitable or enhancing white or gra detail in dark background. Eample o Negative Image Original Image showing a small lesion Negative Image : gives a better vision to analze the image

8 Log Transormations Output gra level, s Negative Log Identit nth root nth power Inverse Log s = c log 1+r c is a constant and r Log curve maps a narrow range o low gra-level values in the input image into a wider range o output levels. Epand dark value to enhance details o dark area. Input gra level, r Log Transormations It compresses the dnamic range o images with large variations in piel values Eample o image with dnamic range: Fourier spectrum image It can have intensit range rom to 1 6 or higher. We can t see the signiicant degree o detail as it will be lost in the displa.

9 Eample o Logarithm Image Fourier Spectrum with range = to Result ater appl the log transormation Power-Law Transormations Output gra level, s Input gra level, r Plots o s = cr γ or various values o γ c = 1 in all cases s = cr γ c and γ are positive constants Power-law curves with ractional values o γ map a narrow range o dark input values into a wider range o output values, with the opposite being true or higher values o input levels. c = γ = 1 Identit unction

10 Gamma correction Gamma correction γ =1/.5 =.4 Monitor γ =.5 Monitor Cathode ra tube CRT devices have an intensit-to-voltage response that is a power unction, with γ varing rom 1.8 to.5 The picture will become darker. Gamma correction is done b preprocessing the image beore inputting it to the monitor with s = cr 1/γ Another eample : MRI a c b d a a magnetic resonance image The picture is dark b result ater power-law transormation with γ =.6, c=1 c transormation with γ =.4 best result d transormation with γ =.3 under acceptable level

11 Eect o decreasing gamma When the γ is reduced too much, the image begins to reduce contrast to the point where the image started to have ver slight washout look, especiall in the background Another eample a c b d a image has a washed-out appearance, it needs a compression o gra levels needs γ > 1 b result ater power-law transormation with γ = 3. suitable c transormation with γ = 4. suitable d transormation with γ = 5. high contrast, the image has areas that are too dark, some detail is lost

12 Piecewise-Linear Transormation Functions Advantage: Allow more control on the compleit o Tr. Disadvantage: Their speciication requires considerabl more user input Contrast stretching. Gra-level slicing. Bit-plane slicing. Contrast Stretching Increase the dnamic range o gra levels. a Transormation Function b a low-contrast image : result rom poor illumination, lack o dnamic range in the imaging sensor, or even wrong setting o a lens aperture o image acquisition c result o contrast stretching d result o thresholding

13 Gra-level slicing Highlighting a speciic range o gra levels Displa a high value o all gra levels in the region o interest and a low value or all other gra levels a transormation highlights range [A,B] o gra level and reduces all others to a constant level b transormation highlights range [A,B] but preserves all other levels c An image.d Result o using thetransormationin a. Bit-plane slicing One 8-bit bte Bit-plane 7 most signiicant Bit-plane least signiicant Highlighting the contribution made to total image appearance b speciic bits Suppose each piel is represented b 8 bits Higher-order bits contain the majorit o the visuall signiicant data Useul or analzing the relative importance plaed b each bit o the image

14 Eample The binar image or bit-plane 7 can be obtained b processing the input image with a thresholding gra-level transormation. Map all levels between and 17 to Map all levels between 19 and 55 to 55 An 8-bit ractal image 8 bit planes Bit-plane 7 Bit-plane 6 Bitplane 5 Bitplane 4 Bitplane 3 Bitplane Bitplane 1 Bitplane

15 Histogram Processing Histogram o a digital image with gra levels in the range [,L-1] is a discrete unction Where r k : the k th gra level hr k = n k n k : the number o piels in the image having gra level r k hr k : histogram o a digital image with gra levels r k Normalized Histogram dividing each o histogram value at gra level r k b the total number o piels in the image, n pr k = n k / n For k =,1,,L-1 pr k gives an estimate o the probabilit o occurrence o gra level r k The sum o all components o a normalized histogram is equal to 1

16 Histogram Processing Used eectivel or image enhancement Inormation inherent in histograms also is useul in image compression and segmentation Data-dependent piel-based image enhancement method. hr k or pr k Eample r k Dark image Components o histogram are concentrated on the low side o the gra scale. Bright image Components o histogram are concentrated on the high side o the gra scale.

17 Eample Low-contrast image histogram is narrow and centered toward the middle o the gra scale High-contrast image histogram covers broad range o the gra scale and the distribution o piels is not too ar rom uniorm, with ver ew vertical lines being much higher than the others

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19 Eample beore ater Histogram equalization Eample beore ater Histogram equalization The qualit is not improved much because the original image alread has a broaden gra-level scale

20 Histogram Equalization: Implementation 1. Obtain the histogram o the input image.. For each input gra level k, compute the cumulative sum. 3. For each gra level k, scale the sum b ma gra level/number o piels. 4. Discretize the result obtained in Replace each gra level k in the input image b the corresponding level obtained in 4. Eample image Gra scale = [,9] No. o piels Gra level histogram

21 Gra Levelj No. o piels k n j j= s = k j= n j n 6 / / / / / / / / 16 s Eample Output image Gra scale = [,9] No. o piels Gra level Histogram equalization

22 Note It is clearl seen that I the cumulative numbers o gra levels are slightl dierent, the will be mapped to little dierent or same gra levels as we ma have to approimate the processed gra level o the output image to integer number Thus the discrete transormation unction can t guarantee the one to one mapping relationship Histogram Equalization A gra-level transormation method that orces the transormed gra level to spread over the entire intensit range. Full automatic, Data dependent, Contrast enhanced. Usuall, the discrete-valued histogram equalization algorithm does not ield eact uniorm distribution o histogram. In practice, one ma preer histogram speciication.

23 Histogram Matching Speciication Histogram equalization has a disadvantage which is that it can generate onl one tpe o output image. With Histogram Speciication, we can speci the shape o the histogram that we wish the output image to have. It doesn t have to be a uniorm histogram Procedure Conclusion Indirect Method: 1. Obtain the transormation unction Tr b calculating the histogram equalization o the input image s = T r. Obtain the transormation unction Gz b calculating histogram equalization o the desired densit unction v = Gz

24 Procedure Conclusion 3. Set v = s to obtain the inversed transormation unction G -1 z = G -1 s = G -1 [Tr] 4. Obtain the output image b appling the processed gra-level rom the inversed transormation unction to all the piels in the input image Histogram Matching: Eample Consider an 8-level image with the shown histogram Match it to the image with the histogram

25 Histogram Matching: Eample 1. Equalize the histogram o the input image using transorm s =Tr. Histogram Matching: Eample. Equalize the desired histogram v = Gz.

26 Histogram Matching: Eample 3. Set v = s to obtain the composite transorm z = G -1 s = G -1 [Tr] Eample Image o Mars moon Image is dominated b large, dark areas, resulting in a histogram characterized b a large concentration o piels in piels in the dark end o the gra scale

27 Image Equalization Transormation unction or histogram equalization Histogram o the result image Result image ater histogram equalization The histogram equalization doesn t make the result image look better. As a consequence, the output image is light and has a washed-out appearance. Solve the problem Since the problem with the transormation unction o the histogram equalization was caused b a large concentration o piels in the original image with levels near Histogram Equalization Histogram Speciication a reasonable approach is to modi the histogram o that image so that it does not have this propert

28 Result image and its histogram The output image s histogram Original image Ater histogram Speciication Notice that the output histogram s low end has shited right toward the lighter region o the gra scale as desired.

29 Note Histogram speciication is a trial-anderror process There are no rules or speciing histograms, and one must resort to analsis on a case-b-case basis or an given enhancement task.

30 Enhancement using Arithmetic/Logic Operations Arithmetic/Logic operations are perormed on piel b piel basis between two or more images ecept NOT operation which perorm onl on a single image Logic Operations Logic operation is perormed on gralevel images, the piel values are processed as binar numbers NOT operation = negative transormation

31 Eample o AND Operation original image AND image mask result o AND operation Eample o OR Operation original image OR image mask result o OR operation

32 Image Subtraction g, =, h, enhancement o the dierences between images Image Subtraction a c b d a. original ractal image b. result o setting the our lower-order bit planes to zero reer to the bit-plane slicing the higher planes contribute signiicant details the lower planes contribute more to ine detail image b. is nearl identical visuall to image a, with a ver slightl drop in overall contrast due to less variabilit o the gra-level values in the image. c. dierence between a. and b. nearl black d. histogram equalization o c. perorm contrast stretching transormation

33 Note We ma have to adjust the gra-scale o the subtracted image to be [, 55] i 8- bit is used Subtraction is also used in segmentation o moving pictures to track the changes ater subtract the sequenced images, what is let should be the moving elements in the image, plus noise Image Averaging Consider a nois image modeled as: g, =, + η, Where, is the original image, and η, is an uncorrelated zero-mean noise process Objective: to reduce the noise content b averaging a set o nois images

34 Image Averaging Deine an image ormed b averaging K dierent nois images: 1 g, = K It ollows that: K i= 1 g, i = epected value o g output ater averaging = original image,

35 Image Averaging Note: the images g i, nois images must be registered aligned in order to avoid the introduction o blurring and other artiacts in the output image. Eample a c e b d a original image b image corrupted b additive Gaussian noise with zero mean and a standard deviation o 64 gra levels. c. -. results o averaging K = 8, 16, 64 and 18 nois images

36 Spatial Filtering Use ilter can also be called as mask/kernel/template or window The values in a ilter subimage are reerred to as coeicients, rather than piel. Our ocus will be on masks o odd sizes, e.g. 33, 55,

37 Spatial Filtering Process simpl move the ilter mask rom point to point in an image. at each point,, the response o the ilter at that point is calculated using a predeined relationship. R = w1 z1 + w z w = mn i= 1 w z i i mn z mn

38 Smoothing Spatial Filters used or blurring and or noise reduction blurring is used in preprocessing steps, such as removal o small details rom an image prior to object etraction bridging o small gaps in lines or curves noise reduction can be accomplished b blurring with a linear ilter and also b a nonlinear ilter reducing the rapid piel-to-piel variation high requenc in gra values. Smoothing Linear Filters output is simpl the average o the piels contained in the neighborhood o the ilter mask. called averaging ilters or lowpass ilters. sharp details are lost.

39 Smoothing Linear Filters reduce the "sharp" transitions in gra levels. sharp transitions random noise in the image edges o objects in the image thus, smoothing can reduce noises desirable and blur edges ma be undesirable 33 Smoothing Linear Filters bo ilter weighted average the center is the most important and other piels are inversel weighted as a unction o their distance rom the center o the mask reduce blurring in the smoothing process

40 Eample a c e b d a. original image 55 piel b. -. results o smoothing with square averaging ilter masks o size n = 3, 5, 9, 15 and 35, respectivel. Note: big mask is used to eliminate small objects rom an image. the size o the mask establishes the relative size o the objects that will be blended with the background. Eample original image result ater smoothing result o thresholding with 1515 averaging mask we can see that the result ater smoothing and thresholding, the remains are the largest and brightest objects in the image.

41 Order-Statistics Filters Nonlinear Filters Nonlinear spatial ilters whose response is based on ordering ranking the piels contained in the ilter mask and then replacing the value o the center piel with the result o the ranking operation eample median ilter : R = median{z k k = 1,,,n n} ma ilter : R = ma{z k k = 1,,,n n} min ilter : R = min{z k k = 1,,,n n} note: n n is the size o the mask Median Filters popular or certain tpes o random noise - impulse noise salt and pepper noise the provide ecellent noise-reduction capabilities, with considering less blurring than linear ilters o similat size. orces the points with distinct gra levels to be more like their neighbors.

42 Median Filtering: Eample [1,1,15,,,,,5,1] Eample : Median Filters

43 Sharpening Spatial Filters to highlight ine detail in an image or to enhance detail that has been blurred - either in error or as an eect o a method o image acquisition. Blurring vs. Sharpening as we know that blurring can be done in spatial domain b piel averaging in a neighbors since averaging is similar to integration thus, we can guess that the sharpening must be accomplished b spatial dierentiation.

44 First-order derivative 1D a basic deinition o the irst-order derivative o a one-dimensional unction is the dierence = + 1 Second-order derivative 1D similarl, we deine the second-order derivative o a one-dimensional unction is the dierence =

45 First and Second-order derivative o, D when we consider an image unction o two variables,,, at which time we will dealing with partial derivatives along the two spatial aes. + = =,,,,, + = linear operator Laplacian operator Gradient operator

46 Discrete Form o Laplacian, 1, 1, + + =, 1, 1, + + = rom ield, ], 4 1, 1, 1, 1, [ = Result Laplacian mask

47 Laplacian mask implemented an etension o diagonal neighbors Other implementation o Laplacian masks give the same result, but we have to keep in mind that when combining add / subtract a Laplacian-iltered image with another image.

48 Laplacian Operator Isotropic ilters: response is independent o direction rotation-invariant. The simplest isotropic derivative operator is the Laplacian To get a sharp image: easil b adding the original and Laplacian image. be careul with the Laplacian ilter used g,, =, +,, i the center coeicient o the Laplacian mask is negative i the center coeicient o the Laplacian mask is positive

49 Eample a. image o the North pole o the moon b. Laplacian-iltered image with c. Laplacian image scaled or displa purposes d. image enhanced b subtraction with original image Mask o Laplacian + addition to simpl the computation, we can create a mask which do both operations, Laplacian Filter and Addition the original image.

50 Mask o Laplacian + addition 1], 1, 1, 1, [, 5 ], 4 1, 1, 1, 1, [,, = = g Eample

51 Note =,,,,, g = =

52 Unsharp masking s, =,, sharpened image = original image blurred image An image can be sharpened b subtracting a blurred version o it rom the original image High-boost iltering generalized orm o Unsharp masking A 1

53 High-boost iltering, = A 1,, hb + i we use Laplacian ilter to create sharpen image s, with addition o original image s,, =, + s,, High-boost iltering ields i the center coeicient o the Laplacian mask is negative A,, = A, + hb,, i the center coeicient o the Laplacian mask is positive

54 High-boost Masks A 1 i A = 1, it becomes standard Laplacian sharpening Eample

55 Use o First Derivatives or Enhancement-The Gradient First derivatives in image processing are implemented using the magnitude o the gradient. gradient = G = G = Gradient Operator Magnitude o the gradient. = mag = [ G = + + G 1 ] 1 the magnitude becomes nonlinear commonl appro. G + G Simpler to compute Still preserves relative changes in gra levels

56 Gradient Mask simplest approimation, z 9 z 8 z 7 z 6 z 5 z 4 z 3 z z 1 and z z G z z G = = ] [ ] [ z z z z G G + = + = z z z z + Gradient Mask Roberts cross-gradient operators, z 9 z 8 z 7 z 6 z 5 z 4 z 3 z z 1 and z z G z z G = = ] [ ] [ z z z z G G + = + = z z z z +

57 Gradient Mask z 1 z z 3 z 4 z 5 z 6 z 7 z 8 z 9 Sobel operators, 33 An approimation using absolute values G G = z = z z + z G + G z + z 9 9 z 1 z 1 + z + z 4 + z + z 3 7 the weight value is to achieve smoothing b giving more important to the center point Note the summation o coeicients in all masks equals, indicating that the would give a response o in an area o constant gra level.

58 Eample Eample o Combining Spatial Enhancement Methods want to sharpen the original image and bring out more skeletal detail. problems: narrow dnamic range o gra level and high noise content makes the image diicult to enhance

59 Eample o Combining Spatial Enhancement Methods solve : 1. Laplacian to highlight ine detail. gradient to enhance prominent edges 3. gra-level transormation to increase the dnamic range o gra levels

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