3 Image Enhancement in the Spatial Domain

Transcription

1 3 Image Enhancement in the Spatial Domain Chih-Wei Tang 唐之瑋 ) Department o Communication Engineering National Central Universit JhongLi, Taiwan 013 Spring

2 Outline Gra level transormations Histogram processing Enhancement using arithmetic operations Spatial iltering C.E., NCU, Taiwan

3 Spatial domain methods Background operate directl on the piels g, ) = T[, )] T operates over some neighborhood o,) neighborhood shape: square & rectangular arras are the most predominant due to the ease o implementation mask processing/iltering masks/ilters/kernels/templates/windows e.g., image sharpening the center o the window moves rom piel to piel the simplest orm: gra-level transormation s=tr) T can operate on a set o input images e.g., sum o input images or noise reduction C.E., NCU, Taiwan 3

4 Point Processing Eamples contrast stretching: produce an image o higher contrast than the original darken the levels below m & brighten the levels above m the limiting case o contrast stretching: thresholding unction Contrast Stretching Thresholding Function C.E., NCU, Taiwan 4

5 Some Basic Gra Level Transormations C.E., NCU, Taiwan 5

6 Gra Level Transormations Image Negatives Reverse the intensit levels s = L 1 r Enhance white/gra detail embedded in dark regions o an image, especiall when the black areas are dominant in size Original image Negative image C.E., NCU, Taiwan 6

7 Gra Level Transormations Log Transormations s=clog1r), c: constant, r 0 Map a narrow range o low gra-level values in the input image into a wider range o output levels Compress the higher-level values An eample: Fourier spectrum o wide range 0 ~ 10 ) a signiicant degree o detail would lost in the displa 6 Displa in a 8-bit displa Fourier spectrum 6 0 ~ ) Ater log transormation c=1 s:0~6.) C.E., NCU, Taiwan 7

8 Gra Level Transormations Power-Law Transormations s = cr γ, c & γ: positive constants ractional γ : map a narrow range o dark input values into a wider range o output values a amil o possible transormation curves: varing γ A variet o devices used or image capture, printing & displa respond according to a power law device-dependent value o gamma C.E., NCU, Taiwan 8

9 Eample #1 o Power-Law Transormations Gamma Correction Correct the power-law response phenomena an eample: The intensit-to-voltage response o CRT devices is a power unction eponents : 1.8~.5) darker than intended γ =.5 γ =1 /.5 = 0.4 C.E., NCU, Taiwan 9

10 Eample # o Power-Law Transormations Contrast Manipulation Original image c=1, γ=0.6 c=1, γ=0.4 c=1, γ=0.3 Magnetic resonance MR) image o a ractured human spine 10

11 Eample #3 o Power-Law Transormations Contrast Manipulation Original image c=1, γ=3.0 c=1, γ=4.0 c=1, γ=5.0 A Washed-out appearance: a compression o gra levels is desirable 11

12 Piecewise-Linear Transormation Functions Advantage: arbitraril comple Disadvantage: the speciication requires considerabl more user inputs Eample: contraststretching transormation increase the dnamic range o the gra levels o the input image r1,s1)=rmin,0) r1=r=m r,s)= rma,l-1) Thresholding unction) 1

13 Gra-Level Slicing Highlight a speciic range o gra levels is oten desired Various was to accomplish this highlight some range and reduce all others to a constant level highlight some range but preserve all other levels Original image C.E., NCU, Taiwan Transormed image b a) 13

14 Bit-plane Slicing Instead o highlighting gra-level ranges, highlighting the contribution made to total image appearance b speciic bits Higher-order bits: the majorit o the visuall signiicant data Lower-order bits: subtle details Obtain bit-plane images thresholding gra-level transormation unction More signiicant C.E., NCU, Taiwan 14

15 Outline Gra level transormations Histogram processing Enhancement using arithmetic operations Spatial iltering C.E., NCU, Taiwan 15

16 Histogram Processing Histogram: A plot k vs. r k : kth gra level n k : number o piels in the image having gra level r k k=0,1,..,l-1 Normalized histogram p r ) = Purposes image enhancement image compression segmentation Simple to calculate economic hardware implementation proper or real-time image processing r H r k ) = n k k nk / n C.E., NCU, Taiwan 16

17 Histogram Equalization/ Histogram Linearization I) Having gra-level values automaticall cover the entire gra scale controlling the probabilit densit unction pd) o its gra levels via the transormation unction Tr), where k k n j sk = T rk ) = pr rj ) =, k = 0,1,,..., L 1 j= 0 j= 0 n Continuous transormation S=Tr): assume Tr) satisies a.1) singled-valued: guarantee that the inverse transormation will eist a.) monotonicall increasing: preserves the increasing order rom black to white in the output image not monotonicall increasing: at least a section o the intensit range being inverted b) 0 T r 1or 0 r 1: the output gra levels will be in the same range as the input levels C.E., NCU, Taiwan 17

18 Histogram Equalization II) According to an elementar probabilit theor, i p r r) and Tr) are known and T 1 s) satisies condition a), then the probabilit densit unction p s s) o the transormed variable s can be obtained b p s) = p r s r ) The probabilit densit unction o the transormed variable s is determined b the gra-level pd o the input image and b the chosen transormation unction dr ds C.E., NCU, Taiwan 18

19 Histogram Equalization III) Consider the transormation unction s = T r) = r 0 p r w) dw ds dt r) d r = = [ p w) dw] p dr d r) dr 0 r = 1 ps s) pr r) = [1] pr r) 1 r r) = 1 r= T s) r= T s) = 1, 0 s 1 1. Perorming the transormation ields a random variable s characterized b a uniorm probabilit densit unction.. Tr) depends on p r r), but the resulting p s s) alwas is uniorm, independent o r). p r C.E., NCU, Taiwan 19

20 Results o Histogram Equalization 1/) Transormation unctions C.E., NCU, Taiwan 0

21 Results o Histogram Equalization /) C.E., NCU, Taiwan 1

22 Histogram Matching Histogram Speciication) 1/) Motivation sometimes, it is useul to speci the shape o the histogram o the processed image instead o uniorm histogram the original image & histogram speciication are given Histogram Continuous gra level r & z Equalization Give input image r, s = T Give p z z), deine G z) r r) = 0 z = p t) 0 p r z dt = w) dw s Obtain Tr) Obtain Gz) G z) = T z) z = G 1 s) = G 1 [ T r)] Discrete gra level z k 1 = G s ), k = 0,1,,..., L 1 k C.E., NCU, Taiwan

23 Histogram Matching /) Inverse mapping rom s to the corresponding z C.E., NCU, Taiwan 3

24 Motivation Local Enhancement to enhance details over small areas in an image the computation o a global transormation does not guarantee the desired local enhancement Solution deine a square/rectangular neighborhood move the center o this area rom piel to piel histogram equalization in each neighborhood region Original Image Ater Global Enhancement Ater Local Enhancement 4

25 Use o Histogram Statistics or Image Enhancement Provide simple while powerul image enhancement Histogram statistics global mean a measure o average gra level in an image global variance a measure o average contrast local mean local variance m µ m σ L = 1 ri p r i ) i= 0 L 1 r) = ri m) p r i ) i= 0 s S r) = nth moment o r µ n r) = s, t) S = rs, t p rs, t ) s, t) S [ r L 1 i= 0 s, t r i m m) S ] n p r p r s, t i ) ) C.E., NCU, Taiwan 5

26 Outline Gra level transormations Histogram processing Enhancement using arithmetic operations Spatial iltering C.E., NCU, Taiwan 6

27 Enhancement Using Arithmetic Operations Image Subtraction g, ) =, ) h, The enhancement o the dierences between images to bring out more details, contrast stretching can be perormed histogram equalization power-law transormation ) Original image Zero the 4 lower-order bit plane C.E., NCU, Taiwan 7

28 Image Subtraction Mask Mode Radiograph The mask is an X-ra image o a region o a patient s bod The image to be subtracted is taken ater injection o a contrast medium into the bloodstream The bright arterial paths carring the medium are unmistakabl enhanced The overall background is much darken than the mask image Mask image Ater inject o a contrast medium into the bloodstream 8 C.E., NCU, Taiwan 8

29 The Range o the Image Values Ater Image Subtraction For 8-bit image: -55~55 Two principle was to scale a dierent image add 55 to ever piel & divide b limitation: the ull range o the displa ma not be utilized the value o the minimum dierence is obtained & its negative added to all the piels in the dierence image scale to the interval [0, 55] more comple C.E., NCU, Taiwan 9

30 C.E., NCU, Taiwan 30 Image Averaging Nois image assumption: at ever pair o coordinates,), the noise is uncorrelated & has zero average value uncorrelated: covariance i an image is ormed b averaging K dierent nois images as K increases: the variabilit noise) o the piel values at each location,) decreases ), ), ), g η = 0 )] ) [ = j j i i m m E ), g = = K i i g K g 1 ), 1 ), { } ), ), g E = ), ), 1 g K η σ = σ

31 An Eample o Image Averaging An important application o image averaging: astronom motivation: imaging with ver low light levels is routine, causing sensor noise requentl to render single images virtuall useless or analsis solution: image averaging C.E., NCU, Taiwan 31

32 An Eample o Image Averaging Dierence Images The vertical scale: number o piels [0,.6 10 The horizontal scale: gra level [0, 55] The mean & standard deviation o the dierence images decrease as K increases the average image should approach the original as K increases The eect o a decreasing mean: the dierence images become darker as K increases 3 4 ]

33 Outline Gra level transormations Histogram processing Enhancement using arithmetic operations Spatial iltering C.E., NCU, Taiwan 33

34 Basics o Spatial Filtering 1/) The concept o iltering comes rom the use o the Fourier transorm or signal processing in the requenc domain. At each point,), the response o the ilter at that point is calculated using a predeined relationship. Linear spatial iltering: the response is the sum o products R = w 1, 1) w 0,0) 1, 1) w 1,0) 1, )..., ) L w 1,0) 1, ) w 1,1) 1, 34 1)

35 Basics o Spatial Filtering /) Linear spatial iltering oten is reerred to as convolving a mask with an image. iltering mask / convolution mask / convolution kernel Nonlinear spatial iltering the iltering operation is based conditionall on the values o the piels in the neighborhood under consideration, but not eplicitl use coeicients in the sum-o-products manner e.g., median ilter or noise reduction: compute the median gra-level value in the neighborhood C.E., NCU, Taiwan 35

36 When the Center o the Filter Approaches the Border o the Image? One or more rows/columns o the mask will be located outside the image plane Solution limit the ecursions o the center o the mask to be at a distance no less than n-1)/ piels rom the border the resulting iltered image is smaller than the original partial ilter mask: ilter all piels onl with the section o the mask that is ull contained in the image Padding add rows & columns o 0 s or other constant gra level), or replicate rows/columns the padding is stripped o at the end o the process side eect: an eect near the edges that becomes more prevalent as the size o the mask increases C.E., NCU, Taiwan 36

37 Smoothing Linear Filters 1/) Averaging ilters/low pass ilters: replace the value o ever piel in an image b the average o the gra levels o the piels contained in the neighborhood o the ilter mask Reduce sharp transitions in gra levels side eect: blur edges Reduce irrelevant detail in an image and get a gross representation o objects o interest irrelevant: piel regions that are small with respect to the size o the ilter mask Applications noise reduction smooth alse contouring C.E., NCU, Taiwan 37

38 Smoothing Linear Filters /) Computationall eicient: instead o being 1/9 1/16), the coeicients o the ilter are all 1 s Bo ilter: a spatial averaging ilter in which all coeicients are equal Weighted average: give more importance weight) to some piels reduce blurring in the smoothing process the piel at the center o the mask is multiplied b a higher value than an other the other piels are inversel weighted as a unction o their distance rom the center o the mask The attractive eature o 16 : integer power o or computer implementation C.E., NCU, Taiwan 38

39 Eamples o Smoothing Linear Filters 1/) n=3: general slight blurring through the entire image, but details are o approimatel the same size as the ilter mask are aected considerabl more n=9: the signiicant urther smoothing o the nois rectangles n=15 & 35: generall eliminate small objects rom an image n=35: black order due to zero padding 39

40 Eamples o Smoothing Linear Filters /) Goal: get a gross representation o objects o interest, such that the intensit o smaller objects blends with the background and larger objects become bloblike the size o the mask Smoothing linear iltering thresholding to eliminate objects based on their intensit 40

41 Order-Statistics Filters Nonlinear spatial ilters whose response is based on ordering ranking) the piels in the area to be iltered Eamples: ma ilter, min ilter & median ilter Median ilter replace the value o a piel b the median o the gra levels in the neighborhood o that piel orce points with distinct gra levels to be more like their neighbors or 33 neighborhood 10,0,0,0,15,0,0,5,100) -> median is 0 ecellent noise-reduction capabilities, with considerabl less blurring than linear smoothing ilters o similar size particular eective or impulse noise salt-and-pepper noise C.E., NCU, Taiwan 41

42 An Eample o Appling Median Filters The image processed with the averaging ilter has less visible noise, but the price paid is signiicant blurring. C.E., NCU, Taiwan 4

43 Foundations o Sharpening Spatial Filters 1/) Highlight ine detail in an image or to enhance detail that has been blurred Averaging is analogous to integration, and sharpening can be accomplished b spatial dierentiation For our use, what we require or a irst derivative are zero in lat areas nonzero at the onset o a gra-level step or ramp nonzero along ramps For our use, what we require or a second derivative zero in lat areas nonzero at the onset and end o a gra-level step or ramp zero along ramps o constant slope C.E., NCU, Taiwan 43

44 Foundations o Sharpening Spatial Filters /) The irst-order derivative o a 1-D unction ) = 1) ) The second-order derivative o ) = 1) 1) ) The two deinitions satis the conditions stated previousl C.E., NCU, Taiwan 44

45 An Eample o Appling Sharpening Spatial Filters 1/3) 45

46 An Eample o Appling Sharpening Spatial Filters /3) Ramp: the irst-order derivative is nonzero along the entire ramp, while the second-order derivative is nonzero onl at the onset and end o the ramp. irst-order derivatives produce thick edges and secondorder derivatives, much iner ones Isolated noise point: the response at and around the point is much stronger or the second- than or the irst-order derivative. second-order derivative enhances much more than a irstorder derivative does. Thin line: essentiall the same dierence between the two derivatives Gra-level step: the response o the two derivatives is the same double-edge eect: The second derivative has a transition rom positive back to negative. 46

47 Foundation o Sharpening Spatial Filters 3/3) In most applications, the second derivative is better suited than the irst derivative or image enhancement. The principle use o irst derivatives in image processing: edge etraction. C.E., NCU, Taiwan 47

48 Second Derivatives or Enhancement The Laplacian Isotropic ilters rotation invariant: rotating the image and appling the ilter gives the same result as appling the ilter to the image irst and then rotating the result The simplest isotropic derivative operator: Laplacian Laplacian is a linear operator : because derivatives o an order are linear operator = ), 1), 1), ), ) 1, ) 1, = = ), 4 1)], 1), ) 1, ) 1, [ = 48

49 Filter Masks The Laplacian 1/) Isotropic or increments o 90 deg. Isotropic or increments o 45 deg. 49

50 Filter Masks The Laplacian /) The Laplacian is a derivative operator highlights gra-level discontinuities in an image & deemphasizes regions with slowl varing gra levels Being used requentl or sharpening digital images The Lapalcian or image enhancement g, ) =,, ) ),, ) ) I the center coeicient o the Laplacian mask is negative I the center coeicient o the Lapalcian mask is positive C.E., NCU, Taiwan 50

51 Eamples o the Laplacian Enhanced Image 1/) Notice that the Laplacian image ma contain both positive and negative values scaling is desired Adding the original image to the Laplacian image small details were enhanced the background tonalit is reasonabl preserved C.E., NCU, Taiwan 51

52 Eamples o the Laplacian Enhanced Image /) The results obtainable with the mask containing the diagonal terms usuall are a little sharper than those obtained with the more basic mask C.E., NCU, Taiwan 5

53 C.E., NCU, Taiwan 53 First Derivatives or Enhancement The Gradient 1/) First derivatives in image processing are implemented using the magnitude o the gradient gradient is deined as the -D column vector the magnitude o this vector gradient) nonlinear operator isotropic = = G G 1/ 1/ ] [ ) = = = G G mag

54 First Derivatives or Enhancement The Gradient /) Problem: the computational burden o implementing the gradient The approimation o the magnitude o the gradient G G the isotropic properties o the digital gradient are preserved onl or a limited number o rotational increments 90 deg.) Digital approimation o the gradient 1 G G = = z 8 z5, G = z6 z5 9 z5, G = z8 z6 z C.E., NCU, Taiwan 54

55 Roberts cross-gradient operators z z z z ) ) ) ) z z z z z z z z z z z z From ) Since even size are awkward to implement 33 Sobel Operator A 33 Region o an Image & Masks or Computing the Gradient derivative in derivative in 55 C.E., NCU, Taiwan

56 An Eample o Using Sobel Gradient An optical image o a contact lens illuminated b a lighting arrangement to highlight imperections Sobel gradient: the edge deects are quite visible, and the slowl varing shades o gra are eliminated. simpli the computational task required or automated inspection. C.E., NCU, Taiwan 56