Image Enhancement in the Spatial Domain

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2 Image Enhancement in the Spatial Domain Algorithms for improving the visual appearance of images Gamma correction Contrast improvements Histogram equalization Noise reduction Image sharpening Optimality is in the eye of the observer: Ad hoc. February 17, / 104

3 What is a histogram Each individual histogram entry is defined as for all 0 i < K, that is h(i)= Number of pixels in l with the intensity value i h(i)=card{(u,v) I(u,v)=i} h(0) is the number of pixels with the value 0, h(1) the number of pixels with the value 1, and so forth. February 17, / 104

4 Histograms Used to conclude if an image is properly exposed. Digital cameras often provide a real-time histogram overlay on the viewfinder. Used to improve the visual appearance of an image. Used as a forensic tool for determining what type of processing has been applied to an image. February 17, / 104

5 Histograms Three very different images with identical histograms. February 17, / 104

6 Image Acquisition- Exposure Histograms make classic exposure problems apparent. Large span of the intensity range at one end is largely unused other end is crowded with high-value peaks represents poorly exposed image. (a) Underexposed, (b) properly exposed, (c) overexposed February 17, / 104

7 Image Acquisition- Contrast Range of intensity values effectively used. Difference between the image s maximum and minimum pixel values. A full contrast image makes effective use of the entire range of available intensity values from a=a min a max = 0 K 1 (black to white). (a) Low contrast, (b) Normal contrast, (c) High contrast February 17, / 104

8 Image Acquisition- Dynamic Range Number of distinct pixel values in an image. Ideal case: Image uses all K usable pixel values. (a) High dynamic range, (b) low dynamic range with 64 intensity values, (c) extremely low dynamic range, 6 intensity values February 17, / 104

9 Image Defects-Saturation Under or over-exposed images have illumination values that lye outside of the sensor s range which are mapped to its minimum or maximum values. (a) Saturation of high intensities, (b) histogram gaps caused by increase in contrast, (c) histogram spikes resulting from a reduction in contrast February 17, / 104

10 Color Quantization Original image converted to a 256 color GIF image. (a) Original histogram, (b) histogram after GIF conversion. (c) RGB image is scaled by 50%, some of the lost colors are recreated by interpolation, but the results of the GIF conversion remain clearly visible in the histogram. February 17, / 104

11 Compression Effect of JPEG compression. The original image (a) with only two different gray values, (b) histogram. JPEG compression, results in additional gray values histogram (d). February 17, / 104

12 Color Image Histograms Histograms of an RGB color image: (a) original image, (b) luminance histogram h Lum,(c e) RGB color components, and associated component histograms h R,h G,h B (f h). All three color channels have saturation problems. The spike in the distribution resulting from this is found in the middle of the luminance histogram (b). February 17, / 104

13 Point Operations Linear histogram equalization. (a) Original image I and (b) modified image I, corresponding histograms h, h (c, d), and cumulative histograms H, H (e, f). The resulting cumulative histogram H (f) approximates a uniformly distributed image. Notice that new peaks are created in the resulting histogram h (d) by merging original histogram cells, particularly in the lower and upper intensity ranges. February 17, / 104

14 Basic Gray Level Transformations Image negatives: Easier visualization of detail embedded in dark regions Left: original digital mammogram. Right: negative image obtained using the negative transformation. February 17, / 104

15 Spatial Domain Processing Utilize neighborhood operations g(x,y)=t[f(x,y)] Simple case: point operations s = T(r) Contrast stretching Thresholding February 17, / 104

16 Effects of thresholding upon the histogram The threshold value is a th. The original distribution (a) is split and merged into two isolated entries at a 0 and a 1 in the resulting histogram (b). February 17, / 104

17 Gray Level Transformation Curves More general transformations Log Inverse Log n th power n th root Used to map narrow dark (log/root) or bright (inverse log/power) range to a greater dynamic range February 17, / 104

18 Fourier Spectrum Example The spectrum has a large dynamic range Example: 0 to Compress range to view visually (db s) Left: Fourier spectrum. Right: Result of applying the log transformation. February 17, / 104

19 Power-Law (Gamma) Transformations Power-law transformation: s = cr γ Many devices require gamma correction CRTs have power function intensity-to-voltage responses Monitor specific Applied in color planes February 17, / 104

20 Gamma Correction Example Monitor Top Left: linear-wedge gray-scale image. Top right: response of monitor to linear wedge. Bottom left: Gamma-corrected wedge. Bottom right: Output of monitor February 17, / 104

21 Gamma Correction Example MR γ=0.6 γ=0.4 γ=0.3 Top Left: MRI of a fractured human spine. Results of applying Gamma correction with c = 1 and γ = 0.6,0.4,0.3 respectively. February 17, / 104

22 Gamma Correction Example - Aerial γ=3 γ=4 γ=8 Top Left: aerial image. Top right, Bottom left and bottom right: results of applying Gamma correction with c = 1 and γ = 3,4,5 respectively. February 17, / 104

23 Gamma Correction Compensates the output signal of a camera with gamma value γ c Gamma correction is applied with γ c = 1 γ c. Corrected signal b is proportional to the received light intensity B. February 17, / 104

24 Gamma Correction in the Digital Imaging Flow Images are processed in a linear intensity space, where gamma correction is used to compensate for the transfer characteristic of each input and output device. February 17, / 104

25 Piecewise-Linear Transformations Contrast stretching Poor illumination Sensor dynamic range Lens aperture settings Feature extraction Medical imaging: bone, soft tissue. February 17, / 104

26 Histogram Processing Light, dark, and low contrast images have concentrated histograms Images with uniform histograms Contain the full range of gray values Have high contrast Better general visual appearance February 17, / 104

27 Histogram Equalization We focus on the (normalized) scalar mapping s = T(r) 0 r 1 where the following are satisfied: T(r) is single-valued and monotonically increasing in [0, 1] 0 T(r) 1 for 0 r 1 The single-valued condition allows the inverse transformation to be defined r = T 1 (s) 0 s 1 February 17, / 104

28 Probability Density Function Let the PDF of r be the p r (r) The CDF is P r (r)= r 0 p r(w)dw Note CDFs are monotonically increasing and have range [0, 1] Defined the RV s = T(r) The PDF of a RV function is p s (s)=p r (r) dr ds February 17, / 104

29 CDF Distribution Set T(r)=P r (r) ds dr d[ r 0 p r(w)dw] dr = dt(r) dr = p r (r) Thus the PDF of s is p s (s) = p r (r) dr ds 1 p r (r) = 1 p r (r) The CDF is uniformly distributed. February 17, / 104

30 Histogram Equalization February 17, / 104

31 Histogram Equalization A 3-bit image (L=8) of size 64 64(MN = 4096) has the intensity distribution in table. Intensity levels are integers in [0,L 1]=[0,7]. February 17, / 104

32 Histogram Equalization Values of the histogram equalization are obtained using: Similarly, s 0 = T(r 0 )=7 0 j=0 P r (r j )=7p r (r 0 )=1.33 s 1 = T(r 1 )=7 1 j=0 P r (r j )=7p r (r 0 )+7p r (r 1 )=3.08 and s 2 = 4.55, s 3 = 5.67, s 4 = 6.23, s 5 = 6.65, s 6 = 6.86, s 7 = 7.00 February 17, / 104

33 The s values still have fractions because they were generated by summing probability values, so we round them to the nearest integer: s 0 = s 1 = s 2 = s 3 = s 4 = s 5 = s 6 = s 7 = February 17, / 104

34 Histogram Equalization CDF mapping of gray values Yields uniformed histogram Simple, parameter-free Discrete case Results not strictly uniform Implementation issues February 17, / 104

35 Histogram EQ Mappings February 17, / 104

36 Histogram Equalization Result Image Histograms Cumulative Histograms Original image and corresponding histograms. Notice that new peaks are created in the resulting histogram h (d) by merging original histogram cells. February 17, / 104

37 Noise Reduction Simple observation model: g = f + η Reduce noise by averaging across (fixed) images Note that (for zero mean noise) g =(1/K)Σ K i=1 g i Eg σ 2 g = f = (1/K)σ 2 η Images must be registered to avoid blurring February 17, / 104

38 Astronomy Example Applicable for sensor noise Additive Gaussian noise Increasing the number of images averaged reduces the noise variance (shown: K = 8,16,64, and 128) Histogram equalization improves detail visualization February 17, / 104

39 Filters What is a filter? Main difference between filters and point operations is that filters generally use more than one pixel to compute each new pixel input. No point operation can blur or sharpen an image. February 17, / 104

40 Filters Each new pixel value I (x,y) is computed as a function of the pixels in a corresponding region in the original image I. February 17, / 104

41 Spatial Filtering Spatial filtering is based on a moving window or mask In the linear filtering case: g(x,y)=σ a s= aσ b t= b w(s,t)f(x+s,y+t) More compactly R = Σ m i=1 nw iz i Must take into account border effects February 17, / 104

42 Filtering To sharpen or filter an image that was taken out-of-focus and is blurred, each pixel is replaced with a linear combination of its neighbors. February 17, / 104

43 Filtering February 17, / 104

44 Simple Smoothing Masks Simplest linear filter: spatial average: reduces noise, but introduces blurring Distance weight samples Centrally located samples are more important Reduces blurring (somewhat) Example above: simple integer arithmetic Alternative approach: utilize spectral characteristics February 17, / 104

45 February 17, / 104

46 Border geometry Filter can be applied only at locations (x,y) where the filter matrix H of size (2K + 1) (2L+1) is fully contained in the image. February 17, / 104

47 Linear Filters Examples of linear filters. The box filter (a) and the Gauss filter (b) are both smoothing filters with all-positive coefficients. The Laplace or Mexican hat filter (c) is a difference filter. February 17, / 104

48 Spatial Filtering February 17, / 104

49 Smoothing Example Square averaging filter results Window sizes: 3, 5, 9, 15, and 35 February 17, / 104

50 NonLinear Filters Any image structure is blurred by a linear filter. Image structures such as edges (top) or thin lines (bottom) are widened, and the local contrast is reduced. February 17, / 104

51 Order-Statistic Filters Linear (weighted sum) filters Blur edges and details Are susceptible to outliers Order-statistic filters (nonlinear) Preserve edges and details Are less susceptible to outliers Spatially ordered samples: z 1,z 2,,z N Rank ordered samples: z (1),z (2),,z (N) Selection-type filter z (1) z (2) z (N) MED[z 1,z 2,,z N ] = z ((N+1)/2) February 17, / 104

52 Filtering Sometimes it is very useful to apply a non-linear filter such as the median. February 17, / 104

53 Effects of a minimum filter on various local signal structures Original signal (top) and result after filtering (bottom), where color bars indicate the extent of the filter. Step edge (a) and linear ramp (c) are shifted to the right. Narrow pulse (b) is removed. February 17, / 104

54 Minimum Filter February 17, / 104

55 Maximum Filter February 17, / 104

56 Minimum and Maximum filters applied to a grayscale image The original image is corrupted with salt and pepper noise (a). The 3 3 pixel minimum filter eliminates the bright dots and widens all dark image structures (b). The maximum filter shows the exact opposite effects (c). February 17, / 104

57 Median Filter Computation of a 3 3 median filter. The nine pixel values extracted from the window are sorted and the center value is the median February 17, / 104

58 Effects of Median Filter Effects of a 3 3 median filter on two-dimensional structures. Isolated dots are eliminated (a), as are thin lines (b). Step edge remains unchanged (c), corner is rounded off (d). February 17, / 104

59 Median Filter February 17, / 104

60 Linear smoothing filter vs median filter (a) Original image corrupted with salt-and-pepper noise. (b) Linear 3 3 pixel box filter (b) reduces noise but the entire image is blurred. (c) The median filter eliminates the noise dots and keeps the structures largely intact. February 17, / 104

61 Salt and Pepper Noise Example Left: X-ray Image of circuit board corrupted by salt and pepper noise. Middle: Noise reduction with a 3x3 averaging mask. Right: Noise reduction with a 3x3 median filter. Bad sensors and bit errors yield salt-and-pepper noise Heavy tailed noise distribution. Other examples: Laplacian, Cauchy, a-stable Outliers (generally) located in the extremes of the ordered set. Do not affect median February 17, / 104

62 Weighted Median Filters (WMF) The median filter can be generalized to weighted order statistic filters. Given x(n)=[x 1,...,x N ], and a set of weights w 1,...,w N, the WMF output is given by y(n)= MEDIAN[x 1 w 1,x 2 w 2,...,x N w N ] (1) where x i w i denotes replication: and w i Z + w i times {}}{ x i w i = x i,x i,...,x i (2) February 17, / 104

63 Weighted median example Each pixel value is replicated multiple times, as specified by the weight matrix W. For example, the value 0 from the center pixel is inserted three times. The pixel vector is sorted and the center value (2) is the weighted median. February 17, / 104

64 Center Weighted Median Filter (CWMF) A special case is the center weighted median filter (CWMF) y(n)=median[x 1,x 2,...,x c 1,x c w c,x c+1,...,x N ] (3) where c =(N+ 1)/2=N is the index of the center sample. For w c = 1, the CWMF reduces to a median filter. For w c N the CWMF is an identity filter. By varying the parameter more smoothing less 1 N w c February 17, / 104

65 Property: Let y(n) be the output of a CWMF, then y(n)=median[x 1,...,x c 1,x c w c,x c+1,...,x N ] = MEDIAN[x (k),x c,x (N k+1) ] where k =(N+ 2 w c )/2 for 1 w c N, and k = 1 for w c > N. Hence, x c if x (k) x c x (N+1 k) y(n)= x (k) if x c x (k) x (N+1+k) if x c x (N+1 k) February 17, / 104

66 Original February 17, / 104

67 Salt and pepper noise February 17, / 104

68 5 5 window - w c = 1 February 17, / 104

69 5 5 window - w c = 3 February 17, / 104

70 5 5 window - w c = 5 February 17, / 104

71 Methods for extending the image to facilitate filtering along the borders Nonexisting pixels outside the original image are either set to some constant value (a), take on the value of the closest border pixel (b), are mirrored at boundaries (c), or repeat periodically along the coordinate axes (d). February 17, / 104

72 Sharpening Filters Objective: Highlight and enhance fine detail Details may have been blurred in acquisition process Method: utilize first- and second-order derivative Derivatives identify signal changes (details/features) First-derivative requirements: Zero in flat regions Nonzero along ramps Second-derivative requirements: Zero in flat regions Zero along ramps of constant slope February 17, / 104

73 First derivative in one dimension Original image (a), horizontal intensity profile f(x) along the center image line (b), and first derivative f (x) (c). df f(x+ 1) f(x 1) (x) = 0.5 (f(x+ 1) f(x 1)). dx 2 February 17, / 104

74 Second Derivative Utilize difference equations: First derivative: Second derivative f = f(x+ 1) f(x) x 2 f = f(x+ 1)+f(x 1) 2f(x) x2 February 17, / 104

75 Derivative Observations First-order derivatives generate thick edges Second-order derivatives Have stronger response to details Produce a double response at step changes Order of response strength. Point, line, step. Second-order derivative is therefore preferred for enhancement Use isotropic (rotation invariant) formulation February 17, / 104

76 Partial Derivatives and the Gradient Derivative of a multidimensional function taken along one of its coordinates partial derivative; I I (x,y) and x y (x,y) The gradient function is I(x,y)= [ I ] x (x,y) I y (x,y) The magnitude of the gradient, I (x,y)= ( I x (x,y))2 +( I y (x,y))2 February 17, / 104

77 Derivative Filters The gradient approximation of the horizontal derivatives is the coefficient matrix: H D x =[ ]=0.5 [ 1 0 1] where the coefficients 0.5 and +0.5 correspond to the image elements I(x 1,y) and I(x+ 1,y). Vertical component of the gradient Hy D = = February 17, / 104

78 Partial Derivatives of a Two Dimensional Function (a) Synthetic image,(b) first derivatives in the horizontal direction I/ x. (c) vertical direction I/ y; (d) magnitude of the gradient I (d). February 17, / 104

79 Edge Operators-Various Approaches Differ in the type of filter used for estimating the gradient components and how they are combined. Roberts Operator: Simplest and oldest edge finder. Employs 2 small filters 2 2 for estimating the directional gradient along the image diagonals: ( ) ( ) Hx R 0 1 = H R y = 0 1 February 17, / 104

80 Edge Operators-Various Approaches Prewitt and Sobel Operators: Classic methods that differ in the filters they use: Prewitt: Computes an average gradient across 3 neighboring lines or columns respectively Hx P = Hy P = Sobel: The smoothing part assigns higher weight to the current center line and column, respectively: Hx S = Hy S = February 17, / 104

81 Estimates for Local Gradient Components Prewitt: Sobel: I(x,y) 1 ( (I H P x )(x,y) 6 (I Hy P )(x,y) I(x,y) 1 ( (I H S x )(x,y) 8 (I Hy S )(x,y) ) ) February 17, / 104

82 Edge Strength and Orientation Denote the scaled filter results as: D x (x,y)=h x I and D y (x,y)=h y I In both cases, the local edge strength E(x,y) is defined as the gradient magnitude (D x (x,y)) 2 +(D y (x,y)) 2 an the local edge orientation angle φ(x,y) is φ(x,y)=arctan( D y(x,y) D x (x,y) ) An improved version of the Sobel filter: Hx S = Hy S = February 17, / 104

83 Edge Operators Process of gradient-based edge extraction. Linear filters H x and H y produce two gradient images, D x and D y. They are used to compute the edge strength E and orientation φ for each image position (x,y). February 17, / 104

84 Laplacian Edge Operator Problem with edge operators based on first derivatives; edges are as wide as the underlying intensity transition. Alternative makes use of the second derivatives. Laplacian is the simplest isotropic derivative: 2 f = 2 f x f y 2 2 f [f(x+ 1,y)+f(x 1,y)+f(x,y+ 1)+f(x,y 1)] 4f(x,y) Isotropic to 90 rotations Add diagonal derivatives to make it 45 isotropic February 17, / 104

85 Mask Implementations and Enhancement Similar definition produces a sign change Enhancement adds (subtracts) derivative and observed image { f(x,y) g(x,y)= 2 f(x,y) if the center is negative f(x,y)+ 2 f(x,y) if the center is positive February 17, / 104

86 Example Top Left: image of the north pole of the moon. Top Right: Laplacian-filtered image. Bottom Left: Laplacian image scaled for display purposes. Bottom Right: Image enhancend February 17, / 104

87 Example Top Left: Composite Laplacian mask. Top Right:scanning electron microscope image Bottom Left: filtered image with first Laplacian mask. Bottom Right: filtered image with second Laplacian mask February 17, / 104

88 February 17, / 104

89 Unsharp Masking Unsharp masking process: f i (x,y)=f(x,y) f(x,y) Originally a darkroom process combining a blurred negative and positive film Generalization: high-boost filtering f hb (x,y) = Af(x,y) f(x,y) f hb (x,y) = (A 1)f(x,y) f s (x,y) Using the Laplacian to obtain the sharp image fs(x,y) { Af(x,y) f hb = 2 f(x,y) if the center is negative Af(x,y)+ 2 f(x,y) if the center is positive February 17, / 104

90 February 17, / 104

91 Edge Sharpening with second derivative Original intensity function f(x), first derivative f (x), second derivative f (x), and sharpened intensity function f(x)=f(x) w f (x) February 17, / 104

92 Results of Laplace filter H L (a) Image I(a), (b) second partial derivative 2 I/ 2 u in the horizontal direction, (c) second partial derivative 2 I/ 2 v in the vertical direction, and (d) Laplace filter 2 I(u,v). February 17, / 104

93 Edge Sharpening with the Laplace filter Original image and marked line (a, b), result of Laplace filter H L (c, d), and sharpened image (e, f). February 17, / 104

94 Unsharp Mask (a,b) Original image and detail,(c) intensity profile of line; USM filtering with Gaussian smoothing σ = 2.5 (d - f) and 10.0 (g - i). Sharpening factor is 1.0 (100%). February 17, / 104

95 Mask Implementation Standard Laplacian sharpening: A = 1. Increasing A, reduces the level of sharpening Scales (brightens) original image February 17, / 104

96 High-Boost Example Top Left: original image. Top Right: Laplacian of the original image with mask(left) in previous image (A = 0). Bottom Left: Laplacian of the original image with mask(right) in previous image (A = 1). Bottom Right: Laplacian of the original image with mask(right) in previous image (A=1.7). February 17, / 104

97 Edges and Contours Edges and contours, are of high importance for the visual perception and interpretation of images. Edges: local intensity changes distinctly along a particular orientation. The amount of change with respect to spatial distance is the first derivative of a function. February 17, / 104

98 Edge Detection February 17, / 104

99 Edge Detection The derivative of a digital image can be found by applying the linear filter. February 17, / 104

100 Edge Detection The edges are then thinned by applying a threshold. February 17, / 104

101 Sobel Example Left: Optical image of contact lens. Right: Sobel gradient. Common application: edge detection Threshold sobel output Binary edge mask February 17, / 104

102 Comparison of Edge Operators February 17, / 104

103 Composite Operator Example I Sharpen: add original and Laplacian Identify edges: sobel operator February 17, / 104

104 Composite Operator Example II Thicken identified edges Smooth Sobel output Form mask Product of smoothed Sobel and sharpened image Sharpened image Addition of original and mask. Only sharpen edges Final display Apply power law transformation February 17, / 104

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