18/10/2017. Image Enhancement in the Spatial Domain: Gray-level transforms. Image Enhancement in the Spatial Domain: Gray-level transforms

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1 Gray-level transforms Gray-level transforms Generic, possibly nonlinear, pointwise operator (intensity mapping, gray-level transformation): Basic gray-level transformations: Negative: s L 1 r Generic log: s c ln(1 r) Power law: s c r Copyright notice: Most images in this pacage are Gonzalez and Woods,, Prentice-Hall Gray-level transforms Gray-level transforms

2 Gamma correction Gamma correction displayed! 1) Monitor response can "compensate" for Weber-law sensitivity of HVS: dp = dl/l p = log(l) higher sensit. in dar areas dar transitions can be compressed with power law L = x^gamma (e.g..4) 1a)...provided quantization errors are not incurred ) Beware of nonlinearities that are already included in image data (e.g., JPEG) see also: Gamma correction Gray-level transforms "The fact that a CRT s transfer function is very nearly the inverse of the lightness sensitivity of vision is an amazing, and fortunate, coincidence!" (Charles Poynton) Modern displays replicate the CRT's luminance response. Rec. ITU-R BT.1886 (011) states that.4-power EOTF shall be standard for HD content creation. Consumer displays are expected to conform. OETF: Rec. BT.709 (production of HD video) transfer function CIE primaries: X,Y,Z the slope of the function is limited near zero (linear portion) in order to minimize quantization noise in the dar regions of the picture.

3 Gray-level transforms Piece-wise linear mapping Note: stretching is useless if the image has to be thresholded) Gray-level slicing Bit-plane slicing

4 Bit-plane slicing Bit-plane slicing 4, 8, 16 gray levels respectively Reconstruction: Sum_n [ bit-plane_n * ^(n-1) ] May be useful for data compression Histogram-based processing Histogram-based processing Histogram: normalized frequency ( of gray level values (x). (can be inverted and preserves gray-level ordering)

5 Histogram equalization Histogram equalization Let the gray levels in an image be represented as random variables r in the range (0,1), with a probability density function (pdf): p r (r) Let be a monotonic, invertible transformation on r; s T(r) All the pixels below the curve p r (r) in the interval ( r, r dr) are mapped to pixels below ps (s) in ( s, s ds) i.e., the two areas are equal: ps ( s) ds pr ( r) dr Now, let the transformation be the cumulative distribution function (cdf) of r r s T ( r) p ( w) dw r 0 It is monotonic and invertible (if the pdf is nonzero for all r) The derivative of this function is of course r Substituting in p s ( s) ds p r ( r) dr p s ( s) i.e. the transformed variable has an exactly uniform pdf. In a practical discrete case: ds / dr p ( r) 1 s T ( r ) pr ( rj ) n j / n i.e., mapping each gray level r into the s value given above yields a uniform pdf for the output image. In general, only an approximately uniform distribution will be obtained. Note: no parameters are needed; the processing is automatic and straightforward. j0 j0 Histogram equalization Histogram equalization Example (continuous case): p r ( r) r 0 r Equalization is obtained via the transformation: r s T ( r) 0 The transformed variable has a uniform pdf. Indeed: 1 ( w ) dw r r Example (discrete case): S. Das, IIT Madras, Course on Computer Vision S. Das, IIT Madras, Course on Computer Vision

6 Histogram equalization Histogram equalization S. Das, IIT Madras, Course on Computer Vision Histogram equalization Histogram specification Remember that the mapping yields a (approx.) uniformly distributed output. Another variable z, with a different, nown and desired pdf p z, will satisfy the same equation: substituting: s T ( r ) pr ( rj ) n j / n G( z ) pz ( z j ) s z G 1 j0 ( s ) G ( T ( r )) i.e., mapping each gray level r into the z value given above yields the desired histogram (pdf) for the output image. j0 1 j0

7 Histogram specification Histogram specification s : uniformly distributed image G: determined as cdf of the desired pdf p z z : image with desired histogram Example: Then determine T(r) and G(z) (cdf s of the histograms): S. Das, IIT Madras, Course on Computer Vision Histogram specification Histogram specification r T ( r ) G( z ) z G 1 ( T ( r )) T(r) G(z) S. Das, IIT Madras, Course on Computer Vision

8 Histogram specification Histogram specification distributions: original target obtained n p (z ) S. Das, IIT Madras, Course on Computer Vision Histogram specification Histogram specification

9 Local histogram modification Enhancement based on local statistics At each location the local histogram is computed, the required mapping is determined, and the pixel is mapped. (At the next step, just update the histogram) (CL)AHE paper m Local values can be estimated for different image statistics, and used to locally control a gray-level modification function. E.g.: local mean and variance in the neighborhood Sxy: Sxy s, tsxy r( s, t) p[ r( s, t)] Sxy Enhancement example: pixels in medium-variance, low-mean areas are scaled by a factor A>1: A f ( x, g( x, f ( x, if m Sxy s, tsxy [ r( s, t) m ] p[ r( s, t)] Sxy 0M f & 1D f Sxy D f otherwise M f and D f respectively are the global average and s.d. of the image; they are used to mae the operator more robust. Enhancement based on local statistics Enhancement based on local statistics

10 Enhancement based on local statistics Using multiple images: subtraction Using multiple images: averaging Using multiple images: averaging Assume an image is formed as: where n(x, is i.i.d. zero-mean noise. If we can average K acquisitions of the image, the variance of the noise is reduced by the factor K: K K 1 g( x, g( x, f ( x, n( x, 1 g ( x, f ( x, K 1 K 1 This approach is useful when the sensor noise is relatively high: poorly illuminated (static) scenes, astronomical images, n ( x, Fig.3.30 A) Ideal B) Noise added (s.d.=64) C) K=8 D) K=16

11 Local operators Local operators Generic, possibly nonlinear, neighborhood-based operator: g(x,=t[f(x,] The coefficients mas can be used in different ways, the simplest of which is linear filtering via the normalized convolution sum: g( x, a b sa tb a w( s, t) f ( x s, y t) b sa tb w( s, t) Note: if the output image is required to be the same size as the input image, the latter must be suitably padded. Local operators Local operators Matlab implementation using imfilter

12 18/10/017 Local operators Matlab: correlation or convolution Linear lowpass filters Both mass have power-of-two coefficients, which are simple to implement. In the second one even the sum of the coefficients is a power of two. Note: a uniform input image is not changed Note: for an image having amplitude 1 and frequency p in both hor. and vert. directions: Local operators Matlab: image padding + filtering [ lowpass, w = ones(31,31) ] -1 the output amplitude is respectively: G1 = ( 5x1 + 4x(-1) ) / 9 = 1/9 G = ( 8x1 + 8x(-1) ) / 16 = 0 Linear lowpass filters

13 Linear lowpass filters Linear lowpass filters Original 3x3 5x5 9x9 Linear lowpass filters Linear lowpass filters 15x15 35x35 A first elementary result in image segmentation!

14 Nonlinear "lowpass" filters Nonlinear "lowpass" filters Let Sxy be an mxn neighborhood of (x,; define the Median filter: fˆ( x, median{ g( s, t)} ( s, t) Sxy Sort the pixel values in Sxy and tae the one in position (mn+1)/ The filter can be iteratively applied to the data, possibly until convergence ("root signal") Note: mn should be odd; if it is even one can tae as output the average of the values in positions mn/ and mn/+1. The formal statistical properties of the filter change. [More about order statistics later, when dealing with image restoration] Linear highpass filters Linear highpass filters Define a 1-D digital derivative (other definitions are possible): First-order: f f ( x 1) f ( x) x Note: phase response is not zero Second-order: f x [ f ( x 1) f ( x)] [ f ( x) f ( x 1)] f ( x 1) f ( x 1) f ( x) Measuring the derivatives of a signal: -D case: Gradient: f f f, x y f f f ; x y tan 1 f f / y x f f Laplacian: f x y f ( x 1, f ( x 1, f ( x, f ( x, y 1) f ( x, y 1) f ( x,

15 Linear highpass filters Linear highpass filters NOTE: to any linear filter the superposition principle applies = Beware: all such definitions can be found in the literature = Image sharpening Image sharpening g( x, f ( x, f ( x, (Use + sign if mass in Fig.3.39 c or d are used) This is usually named Unsharp Masing : g( x, f ( x, f ( x, LP Note: a uniform input image is output unchanged (sum of the coeffs. is 1) Note: for an image having amplitude 1 and frequency p in both hor. and vert. directions: and if =1, the output amplitude is respectively: G 1 = 5x1 + (-4)x(-1) = 9 G = (9-4)x1 + (-4)x(-1) = 9

16 Image sharpening Image sharpening High-boost filtering Generalization of the sharpening filter (beware: the average gray level changes!): Image sharpening Image sharpening Estimation of the gradient (wrong, but useful) Roberts Sobel f f f x y

17 Image sharpening Image sharpening f = b. e h = T(g) (b) a b c d Original (NMR) Laplacian(a) a+b Sobel(a) e = LP(d) g = a + f z ( a c) / x Rational unsharp masing z ( d e) / y (b a c)( z ( )( 4 ) x 4z ) b d e zy z x y b' b 3zx 3zy A control term avoids noise amplification and excessive amplification of sharp and large edges. Pea amplification is 1 for z x =z y =1 pea position is controlled by Control term Absolute lum. difference Problem: image enhancement Poorly illuminated images Baclit objects Light sources within the image Solution: Retinex-based approaches improve the lightness in dar areas and emphasize the details Retinex-based algorithms A specific, simple version of a multi-faceted approach Retinex_at_50_McCann17.pdf Courtesy Dr. Stefano Marsi

18 Retinex-based algorithms Retinex-based algorithms Retinex = Retina + Cortex We perceive an object in the same way, independently of the illumination conditions A perceived image I can be considered as a product of the scene illumination L and the objects reflectance R I ( x, L( x, R( x, L I I R Through the estimation of the illumination it is possible to obtain an approximation of the reflectance. In Illumination Luminanc e estimation Estimation ~ L ~ R : * The components are separately processed and then recombined The Reflectance (details) is suitably emphasized The Illumination is improved in dar areas In the case of logarithmic sensors: The multiplication can be replaced with an addition bloc The division can be replaced with a subtraction bloc The Reflectance can be emphasized through a multiplicative coefficient Out Estimation of the illumination Estimation of the illumination Illumination characteristics Illumination typically changes very smoothly between contiguous pixels Abrupt transitions can also appear in presence of light sources in the image in presence of different illumination systems (e.g. baclight) The Illumination should be estimated using a narrow band (edge-preserving) lowpass filter In the literature these filters are realized: via very large mass via recursive filtering via multi-resolution decomposition We can use a Recursive Rational Filter (RRF) A simple first order IIR filter is y( n) (1 a) x( n) ay( n 1) a can be used to trim the bandwidth: -- a 1 : very narrow passband -- a = 0 : the filter is switched off

19 Estimation of the illumination Estimation of the illumination [ y( n 1, m) S y( n, m) v y( n, m 1) Sh] [( S S S 1 v h v S )(1 ) 1] x( n, m) h where A S h 1 x( n, m 1) log 1 x( n, m 1) A S v 1 x( n 1, m) log 1 x( n 1, m) Sh and Sv resp. are the horizontal and vertical uniformity sensors Note: filtering has to be applied forward and bacward to get zero-phase overall response Processing the illumination Processing the reflectance A modified gamma function is used To increase the lightness in dar zones To avoid dynamic compression in bright areas followed by a histogram stretching To better exploit the system dynamics The reflectance must be processed in the log domain R ~ log R 1 R exp ( L ~ ) 55 K L K L ~ ~ ( ) A sigmoid-lie function is used to: emphasize the details when these are poorly defined. limit the emphasis when the details are already well defined to avoid artifacts generation reduce the signal when it is extremely wea (the information is superseded by the noise)

20 Processing the illumination Processing example The adopted function is: 1 1 R K 1 exp( c( R1 ) R1 ) where c is a suitable coefficient which controls the slope of the sigmoid and creates the central dead zone Reflectance Estimation Processed Reflectance c(r 1 ) Original output R (R 1 ) Illumination Estimation Processed Illumination a comparison Noise control 1. Original. Histogram equalization 3. Gamma correction 4. Retinex-based method (a) (b) without (a) and with (b) the dead zone

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