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## Smoothing Techniques in Image Processing

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**Smoothing Techniques in Image Processing**Prof. PhD. Vasile Gui Polytechnic University of Timisoara**Content**• Introduction • Brief review of linear operators • Linear image smoothing techniques • Nonlinear image smoothing techniques**Introduction**Why do we need image smoothing? What is “image” and what is “noise”? • Frequency spectrum • Statistical properties**Brief Review of Linear Operators[Pratt 1991]**• Generalized 2D linear operator • Separable linear operator: • Space invariant operator: Convolution sum**h22**h20 h21 L-1 h12 h11 h10 2 L-1 h02 h01 h00 h22 h20 h21 H 2 h12 h11 h10 h02 h01 h00 L-1 h22 G h20 h21 2 L-1 h12 h11 h10 2 h02 h01 h00 N-L+1 F N N+L-1 h22 h20 h21 h12 h11 h10 h22 h20 h21 h02 h12 h01 h00 h11 h10 h02 h01 h00 Brief Review of Linear Operators • Geometrical interpretation of 2D convolution**Brief Review of Linear Operators**• Matrix formulation • Unitary transform • Separable transform • Transform is invertible**Brief Review of Linear Operators**• Basis vectors are orthogonal • Inner product and energy are conserved • Unitary transform: a rotation in N-dimensional vector space**x2**2 1 x1 Brief Review of Linear Operators • 2D vector space interpretation of a unitary transform**Brief Review of Linear Operators**• DFT unitary matrix**Brief Review of Linear Operators**• 1D DFT • 2D DFT**Brief Review of Linear Operators**• Circular convolution theorem • Periodicity of f,h and g with N are assumed**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator**m,n** + L pixels m,n+1 Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator • Separable • Can be computed recursively, resulting in roughly 4 operations per pixel**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator Optimality properties • Signal and additive white noise • Noise variance is reduced N times**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator • Unknown constant signal plus noise • Minimize MSE of the estimation g:**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator • i.i.d. Gaussian signal with unknown mean. • Given the observed samples, maximize • Optimal solution: arithmetic mean**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator • Frequency response: Non-monotonically decreasing with frequency**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator • An example**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator • Image smoothed with 33, 55, 99 and 11 11 box filters**Linear Image smoothing techniquesBox filters. Arithmetic**mean LL operator Lena image filtered with 5x5 box filter Original Lena image**Linear Image smoothing techniquesBinomial filters [Jahne**1995] • Computes a weighted average of pixels in the window • Less blurring, less noise cleaning for the same size • The family of binomial filters can be defined recursively • The coefficients can be found from (1+x)n**Linear Image smoothing techniquesBinomial filters. 1D**versions As size increases, the shape of the filter is closer to a Gaussian one**Linear Image smoothing techniquesBinomial filters. 2D**versions**Linear Image smoothing techniquesBinomial filters. Frequency**response Monotonically decreasing with frequency**Linear Image smoothing techniquesBinomial filters. Example**Original Lena image Lena image filtered with binomial 5x5 kernel Lena image filtered with box filter 5x5**Step edge**Result of size L box filter Size L binomial L Linear Image smoothing techniquesBinomial and box filters. Edge blurring comparison • Linear filters have to compromise smoothing with edge blurring**f(1)**f1 f(2) f2 . . . f(m) . . . Order samples select f(N) fN median Nonlinear image smoothingThe median filter [Pratt 1991] • Block diagram N is odd**3**7 8 2 3 7 4 6 7 2, 3, 3, 4, 6, 7, 7, 7, 8 6 Nonlinear image smoothingThe median filter • Numerical example**Nonlinear image smoothingThe median filter**• Nonlinearity median{f1 + f2} median{ f1} + median{ f2}. However: median{ c f} = c median{ f }, median{ c + f} = c + median{ f}. • The filter selects a sample from the window, does not average • Edges are better preserved than with liner filters • Best suited for “salt and pepper” noise**Nonlinear image smoothingThe median filter**Noisy image 5x5 median filtered 5x5 box filter**Nonlinear image smoothingThe median filter**• Optimality • Grey level plateau plus noise. Minimize sum of absolute differences: • Result: • If 51% of samples are correct and 49% outliers, the median still finds the right level!**Test images**Results of 33 pixel median filter Nonlinear image smoothingThe median filter • Caution: points, thin lines and corners are erased by the median filter**Results of the 9 pixel cross shaped window median filter**Nonlinear image smoothingThe median filter • Cross shaped window can correct some of the problems**Nonlinear image smoothingThe median filter**• Implementing the median filter • Sorting needs O(N2) comparisons • Bubble sort • Quick sort • Huang algorithm (based on histogram) • VLSI median**x1**x(1) x2 x(2) x3 x(3) a Min(a,b) x4 x(4) b Max(a,b) x5 x(5) x6 x(6) x7 x(7) Nonlinear image smoothingThe median filter • VLSI median block diagram**Nonlinear image smoothingThe median filter**• Color median filter • There is no natural ordering in 3D (RGB) color space • Separate filtering on R,G and B components does not guarantee that the median selects a true sample from the input window • Vector median filter, defined as the sample minimizing the sum of absolute deviations from all the samples • Computing the vector median is very time consuming, although several fast algorithms exist**Nonlinear image smoothingThe median filter**• Example of color median filtering • 5x5 pixels window • Up: original image • Down: filtered image**Nonlinear image smoothingThe weighted median filter**• The basic idea is to give higher weight to some samples, according to their position with respect to the center of the window • Each sample is given a weight according to its spatial position in the window. • Weights are defined by a weighting mask • Weighted samples are ordered as usually • The weighted median is the sample in the ordered array such that neither all smaller samples nor all higher samples can cumulate more than 50% of weights. • If weights are integers, they specify how many times a sample is replicated in the ordered array**1**2 1 2 2 3 1 2 1 3 7 8 2 3 7 6 4 6 7 2, 2, 3, 3, 3, 3, 4, 6, 6, 7, 7, 7, 7, 7, 8 Nonlinear image smoothingThe weighted median filter • Numerical example for the weighted median filter**R3**R2 R4 m5 R1 mi = median( Ri ), i =1,2,3,4. Result = median( m1,m2,m3,m4,m5 ) Nonlinear image smoothingThe multi-stage median filter • Better detail preservation**weights**f(1) f1 a1 f(2) f2 a2 y . . . . . . ordering sum f(N) fN aN y = a1f(1) + a2f(2)+...+ aNf(N) Nonlinear image smoothingRank-order filters (L filters) • Block diagram**a1=1**. . . Min aN=1 . . . Max am=1 . . . . . . Median a1=.5 aN=.5 . . . Mid range Nonlinear image smoothingRank-order filters (L filters) • Some examples Percentile filters (rank selection): 0%, 50% 100% Particular cases of grey level morphological filters**Nonlinear image smoothingRank-order filters (L filters)**• Optimality Minkovski distance Case r = 1: best estimator ismedian. Caser = 2, best estimator is arithmetic mean. Caser , best estimator is mid-range.**Nonlinear image smoothingRank-order filters (L filters)**• Alpha-trimmed mean filter • = (2Q + 1) / N, defines de degree of averaging • = 1 corresponds to arithmetic mean = 1 / N corresponds to the median filter Properties: in between mean and median**Nonlinear image smoothingRank-order filters (L filters)**Median of absolute differences trimmed mean • Better smoothing than the median filter and good edge preservation M2 is a robust estimator of the variance**Nonlinear image smoothingRank-order filters (L filters)**• k nearest neighbour (kNN) median filter • Median of the k grey values nearest by rank to the central pixel • Aim: same as above**reg 2**reg 1 reg 4 reg 3 reg 6 reg 5 reg 8 reg 7 Nonlinear image smoothingSelected area filtering [Nagao 1980] • Kuwahara type filtering • Form regions Ri • Compute mean, mi and variance, Si for each region. • Result = mi : mi < mj for all j different from i, i,e. themean of the most homogeneous region • Matsujama & Nagao**Nonlinear image smoothingConditional mean**• Pixels in a neighbourhood are averaged only if they differ from the central pixel by less than a given threshold: L is a spacescale parameter and th is a range scale parameter**Nonlinear image smoothingConditional mean**• Example with L=3, th=32**Nonlinear image smoothingBilateral filter [Tomasi 1998]**• Space and range are treated in a similar way • Space and range similarity is required for the averaged pixels • Tomasi and Manduchi [1998] introduced soft weights to penalize the space and range dissimilarity. s() and r() are space and range similarity functions (Gaussian functions of the Euclidian distance between their arguments).