Edge detection Nicolas ROUGON ARTEMIS Department - PowerPoint PPT Presentation

Gaussian filter ■ Discrete approximations of the Gaussian kernel > Real-coefficient filters ● Sampled Gaussian kernel over a [- M , M ] window Usual choice: M = C σ + 1 with C  [3,6] ► significant errors for large windows ● Discrete Gaussian kernel ► Scale-space theory Solution of the discrete heat equation ● Recursive Gaussian filters Optimal Infinite Impulse Response (IIR) approximations of G σ ► Canny filtering with given orders Deriche | Young | Triggs | Farnebäck Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gaussian filter ■ Discrete approximations of the Gaussian kernel > Integer-coefficient filters ● Binomial filters 1  n 1 1 B n = n 2 ● Optimal integer approximations over fixed-size windows  M = 2 (a  0.4)  M = 1 1 1 1 2 1 1-2a 1 a 1 1-2a B 2 =  4 4 3a Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gaussian filter original Gaussian Gaussian 3x3 | σ =1 5x5 | σ =1 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Median filter ■ Order-statistics filtering Denote the k th -smallest pixel value in the neighborhood of x  Ω ● Rank filter of rank k k 1 minimum median maximum erosion median filter dilation  odd ► Erosion/dilation provide the basic operators of mathematical morphology ( a.k.a. image algebra) Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Median filter ■ Performances ● Efficient denoising  optimal for moderate impulse noise («salt-and-pepper» noise) # noisy pixels < 20%  edge preservation + contrast enhancement ● Fine details are smoothed indistinguishable from noise ● Computational bottleneck = local pixel sorting ► quasilinear sorting algorithms with O( ) complexity Heapsort | Block sort | … Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Median filter original median median 3x3 5x5 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Adaptive median filter ■ Key ideas ● Median filtering is restricted to pixels corrupted by impulse noise. Other pixels are unfiltered > detail preservation + filtering bias reduction ● A local impulse noise test based on order-statistics in pixel neighborhood is first applied A candidate noisy pixel  differs from most of its neighbors  can be distinguished from similar neighboring pixels ● Adaptation to image local scale is performed by letting neighborhood size vary within a predefined range Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Adaptive median filter ■ Algorithm Denote | | the minimum | median | maximum pixel value in the r -size neighborhood of x  Ω ( 1  r  r max ) ● Step #1 - Impulse noise test for ( r =1; r  r max ; r ++) { // grow until noise is detectable if ( < < ) // most neighbors differ from noise goto Step #2; } // > proceed with filtering return L ( x ) // noise is undetectable > no filtering ● Step #2 - Denoising if ( < L ( x ) < ) // pixel differs from impulse noise return L ( x ) // > no filtering else return // else median filtering IMA 4103 - Nicolas ROUGON Institut Mines-Télécom

Adaptive median filter ■ Performances median adaptive original 3x3 median ● Improved denoising for impulse noise > 20% ● Improved edge contrast enhancement ● Fine detail preservation Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Nonlinear smoothing filters ■ PDE-based filters ● Iterative filters generating increasingly smoothed images defined as solutions of nonlinear diffusion PDEs anisotropic diffusion | geometric heat equations | … ■ Morphological filters ● Algebraic filters with monotonicity + idempotence properties derived by combining morphological erosion and dilation opening | closing | alternating sequential filters | … ■ Patch-based filters ● Weighted mean filters with weights depending on the similarity between patches around current and any other pixels non- local (NL) means | … Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Linear vs. nonlinear smoothing original median ● Nonlinear filters allow 3x3 for performing distinct intra- / inter-region smoothing ► reduced bias ► better discontinuity mean Gaussian preservation + 3x3 3x3 | σ =1 potential contrast enhancement > sharper details Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Edge detection methods ■ Detecting edges from low-order image derivatives D 1 L =  L D 0 L = L D 2 L = Hessian( L ) image derivative       ■ ■ ■ ■ ■ ■ companion L step roof slope edge models edge edge edge x    impractical class of gradient-based Laplacian-based for discrete signals methods |  L |  L = Trace(D 2 L ) edge map − edge criterion local maximum zero-crossing − Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based methods Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based edge detection ■ Luminance gradient ● Amplitude = contrast ● Phase = orientation ► level line density ► level line normal along normal L n n Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based edge detection d n ■ Directional derivative ● Assessing luminance variations in arbitrary directions requires to define directional differential operators ● Directional derivative (or Lie derivative) along a unit vector d  R n  Consistency with Cartesian derivatives  Consistency with curvilinear derivatives Viewing d as the tangent vector along a curve x ( s ) with arclength s Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based edge detection d n ■ Directional derivative ● Contrast = luminance variation along level line normal Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based edge detection d n ■ Edge point An edge point x is a local directional maximum of contrast in some An edge point x is a local directional maximum of contrast direction d ( x ) which locally defines the edge normal The direction d defines the edge normal ● Detecting edge points requires solving for 2 unknowns i.e. location x and orientation d ► under-constrained problem ● Local edge properties consist of − directional contrast  d L − location x − orientation d geometry photometry IMA 4103 - Nicolas ROUGON Institut Mines-Télécom

Gradient-based edge detection d n ■ Gradient-based edge detection schemes 2 approaches ● Search for regular local maxima of |  L | ● Approximate locally edges as level lines ► Discrete estimators of  L ● Generate edge orientation hypotheses ► Discrete estimators ● Test for local directional maximum of  d L of |  L | in candidate directions Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Gradient-based edge detection ► ► Preprocessing Edge detection Postprocessing ● denoising ● artifacts removal ► Edge map Detection ● enhancement ● thinning estimation upper- ● … ● linking |  L | or |  d L | threshold ● Preprocessing can be built-in into edge detection > robust edge detectors ● Hyperparameter: contrast threshold ► critical impact ► A trade-off between saliency (  robustness) (  high value) vs . level of detail (  sensitivity) (  low value) must be set based on noise/ scene texture/ lighting conditions Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

L Contrast map ■ Properties ● L x  vertical + diagonal edges ● L y  horizontal + diagonal edges ● These maps complement/ reinforce when combined into the contrast map |  L | |  L | | L x | | L y | Sobel filter Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Contrast map natural ■ Gradient norms |  L | unit ball topology R 2 L 2 L 1 4-connectivity L  8-connectivity hybrid ● Isotropy ● Computational cost L  L 1 L  L 2 L 1 L 2 hybrid hybrid Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Finite differences ● Finite difference (FD) techniques allow for estimating an arbitrary order-derivative of a function as a linear combination of its values at given neighboring points ► A derivative is expressed as a discrete convolution against a kernel ( = differential filter )  Setting the kernel size results from on a trade-off between accuracy (  small) vs. robustness (  large) based on SNR and application-dependent computational constraints  FPGA / GPU implementations for real-time applications Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Finite differences ● FD estimates are derived analytically from Taylor expansions, or geometrically from linear fits of the function graph  A basic instance for 1 st -order derivatives consists in approximating tangents by chords L (x) x- d x x+ d x x Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

d y FD gradient filters d x ■ 1 st -order FD over 2D grids FD type expression kernel centered -1 0 1 left-sided -1 1 right-sided -1 1 ● Usual choice: unit pixel size ( d x = d y = 1) Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Roberts filter ● Lateral FD kernels respond at different (subpixel) locations  -1 ► location inconsistency  -1 1 ► ► in |  L | 1 ● 45 ° -rotation fuses their zero-crossings, yielding the Roberts filter -1 0 0 -1 D y = D x = 0 1 0 1 ► noise/texture-sensitive  small size kernels ► directional bias  tailored to diagonal edges ► interpolation  subpixel edge point location Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Robust gradient filters ● Gradient filters with enhanced noise robustness properties are derived by combining differentiation along a direction with smoothing in the orthogonal direction ● These separable filters are designed as tensor products of 1D differentiation (  ) and smoothing (S) kernels     T T D S S x x x y y     T T D S S y y x x y ► Discrete (1x n ) differentiation / smoothing kernels yield ( n x n ) gradient kernels Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Prewitt filter ● Combining 1 st -order centered FD gradient with mean filtering 1 unit  x  -1 0 1 S x = 1 1 1 kernel 3 yields the Prewitt filter -1 -1 -1 -1 0 1 1 1 -1 0 1 0 0 0 D x = D y = 3 3 -1 0 1 1 1 1 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Sobel filter ● FD gradient filters with smoother spectral responses are derived by using smoothing kernels with stronger continuity properties ● Switching from mean to Gaussian filtering 1 1 binomial ► 1 2 1 1 1 1 S x = kernel 3 4 yields the Sobel filter -1 -2 -1 -1 0 1 1 1 -2 0 2 0 0 0 D x = D y = 4 4 -1 0 1 1 2 1 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Frei-Chen filter ● Integer-coefficient FD gradient filters have directional bias  Roberts | Sobel  Prewitt ● Gradient filters with enhanced isotropy properties are derived by setting the Euclidean metric over the image grid 1 2 ► L  L 2 1 1 ● Reweighting the Sobel filter accordingly yields the Frei-Chen filter  real-coefficient filter ► higher computational cost / memory usage Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Frei-Chen filter ● Formally, the Frei Chen filter arises from a decomposition of (3x3) image patches into smooth (1D) + edge (4D) + line (4D) components. It is derived as a basis of the edge subspace - -1 0 1 -1 -1 2 1 1 - 0 0 0 D 0° = 0 D 90° = 2 2 2 2 2 2 -1 0 1 1 1 2 - 1 0 -1 0 2 2 1 1 -1 0 -1 0 1 D 45° = 1 D 135° = 2 2 2 2 - 2 -1 0 0 1 2 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Frei-Chen filter ● The companion edge map is defined as the fraction of the patch belonging to the edge subspace  2  2 using the Frobenius norm ( ) over the patch space A a ij ● Standard threshold values ≈ 95% Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Template matching ■ Principle Joint estimation of local image orientation d and contrast  d L via hypothesis testing ● Orientation sampling ► candidate directions d i  ● FD approximation of ► discrete templates D i d i ● Selection rule ► local estimates d ≈ d i* − orientation  d L ≈ |D i* L | − contrast ► Computational cost increasing with neighborhood size and angular resolution Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Template matching ■ Standard kernel bases ● Size: 3x3 ● Angular resolution: 45 ° ● Directional kernels D i x 45 ° ( i  [1..7]) are derived from D 0 ° via circular permutations of peripheral coefficients 90 ° 135 ° 45 ° 180 ° 0 ° 225 ° 315 ° 270 ° Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Template matching ■ Standard kernel bases ● Robinson ● Kirsch -1 0 1 -3 -3 5 1 1 -2 0 2 -3 0 5 D 0° = D 0° = 4 15 -1 0 1 -3 -3 5 ● Prewitt compass ● Many other bases − larger kernels -1 1 1 − finer angular resolution 1 -1 -2 1 D 0° = Nevatia-Babu | Zucker-Hummel 15 -1 1 1 Morgenthaler | … Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD gradient filters ■ Performances criterion kernel type kernel size (bias  ) Roberts Prewitt Sobel Kirsch  Accuracy trade-off Robustness Roberts Prewitt Kirsch Sobel  / noise, texture (variance  ) Roberts Kirsch Sobel  Complexity Prewitt Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Improving gradient-based edge map 3 steps with specific goals E raw E NMS E HT E Non-Maximum Hysteresis ► ► ► ► Linking Suppression Thresholding ● Remove artifacts ● Reconnect close ● Reconnect distant edge fragments edge fragments  noise points (= edgels)  thick edges Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Step #1: Non-maximum suppression > Removing edge artifacts ● Noise does not exhibit directional consistency ● Thick structures violate the local maximum contrast property ● Hence the idea of filtering both by checking that the definition of an edge point holds Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Step #1: Non-maximum suppression Algorithm For all pixel x in the raw edge map E raw ● Interpolate |  L | along  L at neighboring points in the 8-connected neighborhood of x x i , j ● Test if |  L ( x )| is a local maximum along  L If not, remove x from E raw  L ( x i , j ) ► Simultaneous edge thinning Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Interpolation ■ 1D interpolation ● 0 th -order: nearest-neighbor (NN) > piecewise-constant interpolant ● 1 st -order: linear L > piecewise-linear interpolant u ● Higher-order: polynomial i x i +1 spline | B-spline * | … > piecewise-smooth interpolant * continuous stitching properties Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Interpolation ■ 2D interpolation ● 0 th -order: nearest-neighbor (NN) x i , j x i +1, j u > piecewise-constant interpolant ● 1 st -order: bilinear v x x i , j +1 x i +1, j +1 > piecewise-linear interpolant ● Higher-order: polynomial spline | B-spline | … Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Step #2: Hysteresis thresholding > Reconnecting close edgels ● Short gaps between edgels in E NMS often correspond to weakly contrasted edge parts which are not detected due to a too high contrast threshold ● Lowering the threshold can lead to include irrelevant edges and spurious noise / texture pixels in the edge map ● The latter are usually not connected to shape edges. Hence the idea of filtering them using a connectivity constraint Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Step #2: Hysteresis thresholding Algorithm Given λ high > λ low > 0, add to the edge map E NMS any pixel x s.t. ● |  L ( x )| is a local directional maximum ● |  L ( x )| ≥ λ low ● x is connected to some y  E NMS s.t. |  L ( y )| ≥ λ high ● ( λ high , λ low ) are set empirically or derived from a noise estimate ● Typically: where k  [2,3] λ high = k λ low ● Very efficient in practice − improved noise robustness − 15-20 pixel gaps are filled compared to direct threshold Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing n ■ Step #3: Linking Key idea: from segment endpoint(s), propagate a front over Ω and move along its normal n until reconnection ● Front lines are generated as the level sets of some connection cost function V to edges ► defined over Ω | minimal over edges ● Shortest connecting path  starting from endpoint, iterate until reconnection ► requires interpolating  V over R n  unidirectional or bidirectional scheme Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Post-processing ■ Step #3: Linking ● Connection criteria scheme cost function V gradient-based closing with  structuring element size morphological distance-based digital distance to binary edge map E − V 1 image cost minimal path − V 2 edge cost methods − α smoothness term Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Laplacian-based methods Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Laplacian-based edge detection ■ Luminance Laplacian ● Isotropic operator ● 2 nd -order operator ► orientation-free ► contrast-free ► noise sensitive ● Laplacian zero-crossings (ZC) comprise  local directional maxima of |  L | ► shape edges  local minima of |  L | ► non generic in natural  constant luminance regions (= plateau) (noisy) images Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Laplacian-based edge detection ■ Laplacian edge map ● Since  is continuous, its ZC consist of closed * curves/surfaces i.e. geometric sets (as opposed to point sets) * except along image boundaries ● Salient edge points are filtered using a local contrast criterion  differential ► requires computing  L  statistical R 1 x based on luminance local variance in some neighborhood of x R 2 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Laplacian-based edge detection ► ► Preprocessing Edge detection Postprocessing ● denoising ● artifacts removal ► Edge map Detection ● enhancement ● thinning estimation zero- ● … ● linking  L crossings ● ZC detection is based on sign change (> parameter-free)  the 8-connected neighborhood is used (> the Jordan curve theorem holds)  ZC location is computed at subpixel scale via interpolation ● Since edges are defined as level lines, postprocessing is simplified for they are ensured to be thin and connected Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

d y FD Laplacian filters d x ■ 2 nd -order FD over 2D grids 2 nd -order FD estimates are derived by composing 1 st -order FD estimates FD type expression kernel centered 1 -2 1 ● Usual choice: unit pixel size ( d x = d y = 1) Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD Laplacian filters ■ Standard 2D Laplacian kernels ● 4-connected ● 8-connected 0 1 0 1 1 1  =  = 1 -4 1 1 -8 1 0 1 0 1 1 1  only separable (3x3) FD -1 2 -1 Laplacian kernel  = 2 -4 2 -1 2 -1  many other kernels Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

FD Laplacian filters ■ Standard 2D Laplacian kernels ● Small-size Laplacian kernels are noise-sensitive ► not suited to edge detection in natural images ● Estimating a reliable Laplacian 1 1 1 1 1 1 1 map requires large kernels 1 1 1 1 1 1 1 > from (7x7) to (11x11) 1 1 1 1 1 1 1 ► computational load  = 1 1 1 -48 1 1 1 boundary conditions 1 1 1 1 1 1 1 ● Many kernels are available 1 1 1 1 1 1 1 in the literature 1 1 1 1 1 1 1 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Regularized Laplacian filters Key idea: Improve Laplacian filter robustness w.r.t. noise by performing low-pass filtering prior to differentiation ● Linear low-pass filter σ K σ ● Convolution theorem   ► the Laplacian map ► the Laplacian operator is smoothed is smoothed ► not operative since ► regularized Laplacian operator L is noisy Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Regularized Laplacian filters ● The kernel defines a band-pass filter ● As for FD Laplacian filters, implementation in the spatial domain requires large discrete kernels ( e.g. obtained by sampling ) ► computational load | boundary conditions ► applicable only when σ is small ● Implementation in the spectral domain using FFT + precomputed Laplacian of kernel spectrum ► quasilinear complexity | truncation error-free Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Laplacian of Gaussian (LoG) filter ● Choosing K σ as a centered Gaussian kernel G σ with variance σ 2 yields the Laplacian of Gaussian (LoG) filter (a.k.a. Marr-Hildreth or Mexican Hat filter) ► The hyperparameter σ is set based on -  G σ noise/texture properties Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Laplacian of Gaussian (LoG) filter ● Strongly-smoothing ( G σ * L )  C  ( Ω )  σ > 0 ► well-posed differentiation: arbitrary-order derivatives are estimated in a robust way (= Gaussian derivatives) ● Separable ► computational efficiency ● Well-localized in both space and frequency TF ( G σ )  G 1/ σ ► good trade-off between accuracy vs. smoothing ● Delocalization artifacts (  with σ ) Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ LoG kernels ● For small values of σ , 0 0 3 2 2 2 3 0 0 LoG kernels are 0 2 3 5 5 5 3 2 0 readily derived by 3 3 5 3 0 3 5 3 3 sampling -12 -23 -12 2 5 3 2 3 5  σ = -23 -40 -23 2 5 0 0 5 2 ► computational load -12 -23 -12 2 5 3 3 5 2 boundary conditions 3 3 5 3 0 3 5 3 3 0 2 3 5 5 5 3 2 0 0 0 3 2 2 2 3 0 0 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

L Laplacian map ■ FD Laplacian vs. LoG |  L | |  L | 4-connected Laplacian LoG ( σ =1.2) Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Regularized Laplacian filters ● The kernel width σ acts as a scale parameter which allows for hierarchising image structure in terms of level of detail (LoD) 0 σ full LoD high LoD low LoD fine edge detail most salient edges Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Robust Laplacian filters ■ Difference of Gaussians (DoG) filter Given σ 1 > σ 2 > 0, the DoG filter is the linear filter with kernel ● accurately approximates the LoG kernel whenever ● For a given accuracy, the DoG kernel bandwidth is slightly larger than the LoG kernel bandwidth ● Biological consistency Retinal cell assemblies of mammals behave as DoG filter banks Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Image sharpening ● Blurring may occur during image acquisition ( e.g. defocusing), scanning or scaling ► low-pass filtering mostly noticeable along edges ● Image sharpening refers to techniques aiming at enhancing luminance transitions ► performed by amplifying high-frequencies ► reduces effects of blurring ● 2 main approaches to image sharpening  Laplacian-based  Unsharp masking Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Laplacian-based image sharpening Key idea: Use the Laplacian both to − detect edges (ZC) L − distinguish edge sides (sign) ● Edge enhancement is achieved by L x subtracting from the image a fraction of its Laplacian L xx ► a Laplacian kernel  induces a sharpening kernel Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Standard 2D Laplacian-based sharpening kernels ( b = 1) ● 4-connected ● 8-connected 0 -1 0 -1 -1 -1 K S = K S = -1 5 -1 -1 9 -1 0 -1 0 -1 -1 -1 1 -2 1 K S = -2 5 -2 1 -2 1 ● Robust sharpening kernels are build from LoG / DoG kernels Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Laplacian-based image sharpening original 4-connected Laplacian-based Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Laplacian-based image sharpening original 4-connected Laplacian-based Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Laplacian-based image sharpening 4-connected Laplacian-based 8-connected Laplacian-based Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Unsharp masking Key idea: Build a detail mask by subtracting from the image a smoothed version of itself L ● Edge enhancement is achieved by adding to the image a fraction K * L of the detail mask L – K * L ► called highboost filtering if b > 1 ► a smoothing kernel K induces a sharpening kernel Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Unsharp masking original highboost filtering Gaussian kernel ( σ = 2.0) | b = 1.5 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Image sharpening ■ Laplacian-based sharpening vs. Unsharp masking ● Unsharp masking relies on smoothing and does not involve differentiation ► intrinsically more robust to noise than Laplacian-based methods (especially when using small-size Laplacian kernels) ► the smoothing kernel bandwidth provides an additional scale hyperparameter allowing for finer performance control ● Using robust Laplacian kernels, Laplacian-based techniques tend to perform similarly to unsharp masking Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Canny-Deriche filtering Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Canny-Deriche filtering ■ Key ideas Canny-Deriche filtering provides a class of linear differential filters with arbitrary-order ● Optimized for a noisy edge model ● Based on quantitative performance criteria ► mathematical derivation ● Recursive implementation ► computational efficiency J. Canny - A computational approach to edge detection IEEE Transactions on Pattern Analysis and Machine Intelligence 8(6):679-698, Nov. 1986 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Canny-Deriche filtering ■ 1D edge model Noisy step edge (location x 0 | contrast ρ ) ρ with additive white Gaussian noise ● Detection using a linear filter with kernel K ► Optimal kernel? x 0 Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Canny-Deriche filtering ■ Performance criteria Expressed as functions of kernel K and its derivatives ● Good detection ► high specificity ● Good localization ► high accuracy ● Single response at edge points ► no ambiguity Institut Mines-Télécom IMA 4103 - Nicolas ROUGON

Canny-Deriche filtering ■ Optimal detection Low probability of false alarms (= no detection | wrong detection) ● SNR criterion ► to be maximized Institut Mines-Télécom IMA 4103 - Nicolas ROUGON