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SYDE 575: Introduction to Image Processing Spatial-Frequency Domain: Implementations Textbook: Chapter 4 Filtering in Spatial and SpatialFrequency Domains Basic spatial filtering is essentially 2D discrete convolution between an image f and filter function h g ( x, y ) = f ( x , y ) * h ( x , y ) Convolution in spatial domain becomes multiplication in frequency domain G(u , v ) = F (u , v )H (u , v ) Spatial-Frequency Implementations We will discuss these implementations: • Low pass filters: ideal, Butterworth, Gaussian • High pass filters: ideal, Butterworth, Gaussian • Edge enhancement: high boost filtering • HVS modelling: Difference of Gaussians (DoG), Gabor, Laplacian of Gaussian • Periodic noise filtering (Section 5.4) Blurring/Noise reduction Noise characterized by sharp transitions in image intensity Such transitions contribute significantly to high frequency components of Fourier transform Intuitively, attenuating certain high frequency components result in blurring and reduction of image noise Ideal LPF Cuts off all high-frequency components at a distance greater than a certain distance from origin (D0: cutoff frequency) ì1, if D(u,v) £ D0 H (u , v ) = í î0, if D(u,v) > D0 Visualization Source: Gonzalez and Woods Effect of Different Cutoff Frequencies Source: Gonzalez and Woods Effect of Different Cutoff Frequencies Source: Gonzalez and Woods Effect of Different Cutoff Frequencies As cutoff frequency decreases Image becomes more blurred Noise becomes more reduced Analogous to larger spatial filter sizes Noticeable ringing artifacts that increase as the amount of high frequency components removed is increased Why ringing? Why is there ringing? Ideal low-pass filter function is a rectangular function The inverse Fourier transform of a rectangular function is a sinc function Convolution of a sinc and a step function generates ringing on both sides of the edge Ringing Source: Gonzalez and Woods Butterworth LPF H (u , v ) = 1 1 + [ D(u , v ) / D0 ] 2n Transfer function does not have sharp discontinuity establishing cutoff between passed and filtered frequencies Cutoff frequency D0 defines point at which H(u,v)=0.5 Similar to exponential LPF Butterworth LPF Source: Gonzalez and Woods Spatial Representations Tradeoff between amount of smoothing and ringing Source: Gonzalez and Woods Butterworth LPFs of Different Orders Source: Gonzalez and Woods Gaussian LPF H (u , v ) = e -D 2 2 ( u, v)/ 2 D0 This is another form of a Gaussian filter, as used by Gonzalez & Woods (textbook) Transfer function is smooth, like Butterworth filter Gaussian in frequency domain remains a Gaussian in spatial domain Advantage: No ringing artifacts Gaussian LPF Source: Gonzalez and Woods Gaussian LPF Source: Gonzalez and Woods Spatial-Frequency High Pass Filters (HPFs) • HPFs are effectively the opposite of LPFs • High pass filtering in the spatial-frequency domain is related to low pass filtering HHP(u,v) = 1 – HLP(u,v) hHP(x,y) = d(x,y) – hLP (x,y) • Note: DC gain is zero for a HPF Impact of High Pass Filtering Edges and fine detail characterized by sharp transitions in image intensity Such transitions contribute significantly to high frequency components of Fourier transform Intuitively, attenuating low frequency components and preserving high frequency components will retain image intensity edges HPF Transfer Functions Ideal HPF ì0 H (u , v ) = í î1 Butterworth HPF H (u , v ) = if D(u , v ) £ D0 if D(u , v ) > D0 1 1 + [ D0 / D(u , v )] Gaussian HPF H (u , v ) = 1 - e - D2 ( u, v)/ 2 D02 2n HPF Transfer Functions Spatial Representations of HPFs Ideal HPF Filtering Butterworth HPF Filtering Gaussian HPF Filtering Observations of HPFs As with ideal LPF, ideal HPF shows significant ringing artifacts Second-order Butterworth HPF shows sharp edges with minor ringing artifacts Gaussian HPF shows good sharpness in edges with no ringing artifacts Spatial-Frequency Edge Enhancement • Edge enhancement can be performed directly in the spatial-frequency domain • Example: high boost filtering (unsharp masking) High frequency emphasis Advantageous to accentuate enhancements made by high-frequency components of image in certain situations (e.g., image visualization) Solution: multiply high-pass filter by a constant and add offset so zero frequency term not eliminated g(x,y) = f(x,y) + k gHPF(x,y) As discussed earlier, this is referred to as high-boost filtering High Boost Filtering In spatial domain: g(x,y) = f(x,y) + k gHPF(x,y) Impulse response: h(x,y) = d(x,y) + k hHPF(x,y) Transfer function in spatial-frequency domain: H(u,v) = 1 + k HHPF(u,v) or: H(u,v) = 1 + kHHPF(u,v) = 1 + k(1- HLPF(u,v)) = (1+k) - kHLPF(u,v) Recall: Set k=1 for unsharp masking Results Source: Gonzalez and Woods Examples of Frequency Domain Filtering Source: Gonzalez and Woods Human Visual System Models • For a generic spatial-frequency image enhancement filter, what should the transfer function look like? 1) DC gain is typically reduced so 0<H(u)<1 2) H(u) approaches zero as u increases 3) H(u) > 1 for frequency range where signal dominates • Sketch: Model of HVS • Light entering the eye is processed by two steps 1) Cornea/Lens H1(u): modelled as LPF e.g., Gaussian 2) Retina H2(u): modelled as edge enhancement e.g., 1-Laplacian Combined: H(u) = H1(u) H2(u) 2u2s2 2 -2p = (1+(2pu/a) ) e Sketch: Difference of Gaussians • There are a number of Gaussian-based functions that mimic lateral inhibition • Difference of Gaussians takes the difference of two Gaussians with different s 2 u2 s 2 2 u2 s 2 -2p -2p 1 2 H(u) = A e - Be With A>B and s1<< s2 Sketch in frequency and time domains • Can vary s1 and s2 to create filter bank with varying peak frequencies Gabor Filter • Gabor is a Gaussian band pass filter H(u) = (A/2) 2u 2s 2 -2p 1 e * [ d(u-up) + d(u+up)] • In time domain, a Gaussian-modulated sinusoid (real part of Gabor filter) 2 0.5 -0.5(x/s) h(x) = A/(s(2p) ) e cos(2pupx) Sketch in frequency and time • Similar shape as Difference of Gaussians, but with ringing • Note: complex form of filter used for texture feature extraction Laplacian of a Gaussian • Consider the Marr-Hildreth operator i.e., a Laplacian of a Gaussian 2u2s2 2u2s2 2 -2p 2 2 -2p H(u) = (-j2pu) e = 4p u e • Sketch in time and frequency domains • What is the impact of this filter? Why? Periodic Noise Reduction Typically occurs from electrical or electromechanical interference during image acquisition Spatially dependent noise Example: spatial sinusoidal noise Example Source: Gonzalez and Woods Observations Symmetric pairs of bright spots appear in the Fourier spectra Why? Fourier transform of cosine function is the sum of a pair of impulse functions cos(2pu0x) <-> 0.5[d(u + u0) + d(u – u0)] Intuitively, sinusoidal noise can be reduced by attenuating these bright spots Bandreject Filters Removes or attenuates a band of frequencies about the origin of the Fourier transform Sinusoidal noise may be reduced by filtering the band of frequencies upon which the bright spots associated with period noise appear Example: Ideal Bandreject Filters Ideal bandreject filter W ì if D(u , v ) < D0 ï1 2 ï W W ï H (u , v ) = í0 if D0 - £ D (u , v ) £ D 0 + 2 2 ï W ï if D (u , v ) > D0 + ï1 2 î Example Source: Gonzalez and Woods Notch Reject Filters Idea: Sinusoidal noise appears as bright spots in Fourier spectra Reject frequencies in predefined neighborhoods about a center frequency In this case, center notch reject filters around frequencies coinciding with the bright spots Some Notch Reject Filters Source: Gonzalez and Woods Example: Moire pattern reduction Source: Gonzalez and Woods Homomorphic Filtering Image can be modeled as a product of illumination (i) and reflectance (r) f ( x , y ) = i (x , y )r (x , y ) Unlike additive noise, can not operate on frequency components of illumination and reflectance separately Á[ f ( x, y )] ¹ Á[ i( x, y )] Á[ r ( x, y )] Homomorphic Filtering Idea: What if we take the logarithm of the image? ln f ( x, y ) = ln i ( x , y ) + ln r ( x , y ) Now the frequency components of i and r can be operated on separately Á[ ln f ( x, y )] = Á[ ln i ( x , y )] + Á[ ln r ( x , y )] Homomorphic Filtering Framework Source: Gonzalez and Woods Homomorphic Filtering: Image Enhancement Simultaneous dynamic range compression (reduce illumination variation) and contrast enhancement (increase reflectance variation) Illumination component characterized by slow spatial variations (low spatial frequencies) Reflectance component characterized by abrupt spatial variations (high spatial frequencies) Homomorphic Filtering: Image Enhancement Can be accomplished using a high frequency emphasis filter in log space DC gain of 0.5 (reduce illumination variations) High frequency gain of 2 (increase reflectance variations) Output of homomorphic filter g ( x , y ) » i ( x , y ) ( r ( x , y )) 2 Example Source: Gonzalez and Woods Homomorphic Filtering: Noise Reduction Multiplicative noise model f ( x , y ) = s (x , y )n (x , y ) signal noise Transforming into log space turns multiplicative noise to additive noise ln f ( x, y ) = ln s ( x , y ) + ln n (x , y ) Low-pass filtering can now be applied to reduce noise Example Original Multiplicative Noise Source: Jernigan, 2003 Example Homomorphic LPF Source: Jernigan, 2003