Computer Vision Advanced Edge Detectors Prof. Flvio Cardeal - PowerPoint PPT Presentation

Computer Vision Advanced Edge Detectors Prof. ¡Flávio ¡Cardeal ¡– ¡DECOM ¡/ ¡CEFET-­‑MG ¡ cardeal@decom.cefetmg.br ¡ ¡

Abstract • This lecture discusses advanced edge detectors that combine multiple approaches into a single algorithm. 2 ¡

LoG and DoG • The Laplacian of Gaussian (LoG) and the difference of Gaussians (DoG) are very important basic image transforms. • They are applied in several different domains in computer vision. 3 ¡

LoG Edge Detector • Applying the Laplacian for a Gauss-filtered image can be done in one step of convolution, based on the theorem: ∇ 2 (G σ ∗ I) = I ∗ ∇ 2 G σ Note that for calculating the Laplacian of a Gauss-filtered image, we only have to perform one convolution with . ∇ 2 G σ 4 ¡

LoG Edge Detector • The filter kernel for is not limited to be a 3 x 3 ∇ 2 G σ kernel as shown below: • In fact, because the Gauss function is continuous, we can calculate its exact Laplacian. How? 5 ¡

LoG Edge Detector x • For the first partial derivative with respect to , we = = obtain that − x 2 + y 2 2 σ 2 · e − y 2 π s · e − x 2 � � 1 = 1 2 σ 2 G σ (x,y) = 2 πσ 2 exp 2 σ 2 ∂ G σ x 2 πσ 4 e − (x 2 + y 2 )/ 2 σ 2 ∂ x (x,y) = − 6 ¡

LoG Edge Detector x y • We then repeat the derivative for and and obtain the LoG as follows: � x 2 + y 2 − 2 σ 2 � 1 e − (x 2 + y 2 )/ 2 σ 2 ∇ 2 G σ (x,y) = 2 πσ 4 σ 2 • The LoG is also known as the Mexican hat function. In fact, it is an “ inverted Mexican hat ”. The zero- crossings define the edges. 7 ¡

LoG Edge Detector Source: ¡R. ¡KleKe ¡ 8 ¡

Sampling the LoG Kernel • Let’s sample this Laplacian into a (2 k + 1) × (2 k + 1) filter kernel for an appropriate value of . k • But what is an appropriate value for ? k • We start with estimating the standard deviation σ for the given class of input images, and an appropriate value of follows from this. k 9 ¡

Sampling the LoG Kernel • By doing and, for example, , we ∇ 2 G σ ( x , y ) = 0 y = 0 have both zero-crossings in the x axis as the roots x 2 = 2 σ 2 of , that is: x 1 = − σ 2 x 2 = + σ 2 • Consider a parameter given by: w w = x 1 − x 2 = 2 σ 2 10 ¡

Parameter w Source: ¡R. ¡KleKe ¡ 11 ¡

Sampling the LoG Kernel • For representing the Mexican hat function properly by samples, it is proposed to use a window size of . 3 w × 3 w = 6 σ 2 × 6 σ 2 • In conclusion, we have that: √ √ 2 σ � × ceil � 6 2 k + 1 × 2 k + 1 = ceil � 6 2 σ � Smallest integer equal to or larger than the argument 12 ¡

Sampling the LoG Kernel • The value of needs to be estimated for the given σ image data. • Smoothing an image with a very “narrow” (i.e. ) σ < 1 Gauss function does not make much sense. • So, let us consider . The smallest kernel ( , σ ≥ 1 σ = 1 thus 3 = 8.485) will be of size 9 x 9 (i.e., = 4). w k 13 ¡

LoG Scale Space σ = 0.5 σ = 1 Consider again this Gaussian σ = 2 σ = 4 scale space with six layers. Source: ¡R. ¡KleKe ¡ σ = 8 σ = 16 14 ¡

LoG Scale Space σ = 0.5 σ = 1 So, here we have the resulting images σ = 2 σ = 4 after computing the Laplacians of those six layers. Source: ¡R. ¡KleKe ¡ σ = 8 σ = 16 15 ¡

LoG Scale Space • This is an example of a LoG scale space. • As in a Gaussian scale space, each layer is defined by the scale , the used standard deviation in the σ Gauss function. • We can generate subsequent layers when starting at an initial scale and using subsequent scales σ a n ⋅ σ for and . a > 1 n = 0,1,..., m 16 ¡

Difference of Gaussians (DoG) • The difference of Gaussians (DoG) operator is a common approximation of the LoG operator, justified by reduced run time. • Consider again the equation below, which defines a centered (i.e. zero-mean) Gauss function : G σ − x 2 + y 2 � � 1 2 πσ 2 exp G σ (x,y) = 2 σ 2 17 ¡

Difference of Gaussians (DoG) • The DoG is defined by an initial scale and a σ scaling factor as follows: a > 1 D σ ,a (x,y) = L(x,y, σ ) − L(x,y,a σ ) • So, it is the difference between a blurred copy of image and an even more blurred copy of . I I 18 ¡

Difference of Gaussians (DoG) • As for LoG, edges (following the step-edge model) are detected at zero-crossings. • Why DoG is an approximation of LoG? Because: ∇ 2 G σ (x,y) ≈ G a σ (x,y) − G σ (x,y) (a − 1 ) σ 2 with as a recommended parameter. a = 1.6 19 ¡

DoG Scale Space • Different scales produce layers in the DoG σ D σ , a scale space. • Let’s see the next figure for a comparison of three layers in the DoG scale space. • The scaling factor used is: . a = 1.6 20 ¡

DoG Scale Space Source: ¡R. ¡KleKe ¡ LoG DoG σ = 0.5 σ = 0.5 a = 1.6 21 ¡

DoG Scale Space Source: ¡R. ¡KleKe ¡ a = (1.6) 3 LoG DoG σ = 0.5 σ = 0.5 22 ¡

DoG Scale Space Source: ¡R. ¡KleKe ¡ a = (1.6) 5 LoG DoG σ = 0.5 σ = 0.5 23 ¡

Embedded Confidence • A confidence measure is quantified information derived from calculated data, to be used for deciding about the existence of a particular feature . • If the calculated data match the underlying model of the feature detector reasonably well, then this should correspond to high values of the measure . 24 ¡

The Meer-Georgescu Algorithm • The Meer-Georgescu algorithm detects edges while applying a confidence measure based on the assumption of the validity of the step-edge model. • Four parameters are considered in this method. 25 ¡

The Meer-Georgescu Algorithm • Specifically, for a gradient vector g ( p ) = ∇ I ( x , y ) at a pixel location , those parameters are: p = ( x , y ) g ( p ) = g ( p ) 2 1. The estimated gradient magnitude ; 2. The estimated gradient direction ; θ ( p ) 3. An edge confidence value ; η ( p ) 4. The percentile of the cumulative gradient ρ k magnitude distribution. 26 ¡

Computing the Gradient (2 k + 1) × (2 k + 1) • Let be a matrix representation of a A window centered at the current pixel location in p input image . I Τ • Moreover, let: be a (2 k + 1) × (2 k + 1) W = sd matrix of weights, obtained as the product of two vectors and . s = [ s 1 ,..., s 2 k + 1 ] d = [ d 1 ,..., d 2 k + 1 ] 27 ¡

Computing the Gradient • Vectors and meet the following requirements: d s 1. Both are unit vectors in the L1-norm, i.e. and ; d 1 + ... d 2 k + 1 = 1 s 1 + ... s 2 k + 1 = 1 2. is an asymmetric vector, i.e. , d d 1 = − d 2 k + 1 , … , . d 2 = − d 2 k d k + 1 = 0 s s 1 = s 2 k + 1 ≤ ... 3. is a symmetric vector, i. e. . ≤ s 2 = s 2 k ≤ .... ≤ s k + 1 28 ¡

Computing the Gradient • For example, the vectors and below define a 5 x s d 5 matrix : W d = [ − 0.125, -0.25, 0, 0.25, 0.125] Τ s = [0.0625, 0.25, 0.375, 0.25, 0.0625] Τ " % -0.0078 -0.0156 0 0.0156 0.0078 $ ' -0.0312 -0.0625 0 0.0625 0.0312 $ ' W = sd Τ = $ ' -0.0469 -0.0938 0 0.0938 0.0469 $ ' -0.0312 -0.0625 0 0.0625 0.0312 $ ' $ ' -0.0078 -0.0156 0 0.0156 0.0078 # & 29 ¡

Computing the Gradient • Let be the i- th row of matrix . By using: a i A sd ⊤ A � � d 1 = Tr ( WA ) = Tr 2 k + 1 � W ⊤ A = s ⊤ Ad = d ⊤ a i � � � � d 2 = Tr s i i = 1 we obtain the first two parameters used in the algorithm: � d 1 � � d 2 1 + d 2 θ (p) = arctan g(p) = 2 d 2 30 ¡

Generating the ρη Diagram • Let be a matrix representing (2 k + 1) × (2 k + 1) A ideal a template of an ideal step edge having the gradient direction . θ ( p ) A ideal • The value specifies the proposed Τ η ( p ) = Tr( A ideal A ) confidence measure. 31 ¡

Generating the ρη Diagram • The values in and are normalized such A ideal A that , with in case of a perfect η ( p ) = 1 0 ≤ η ( p ) ≤ 1 match with the ideal step edge. • Let be the ordered list of g 1 < ... < g k < ... < g N distinct (rounded) gradient-magnitudes in image , I with cumulative distribution values (probabilities): [ ] ρ k = Prob g ≤ g k for 1 ≤ k ≤ N . F 32 ¡ magnitude

Generating the ρη Diagram • For a given pixel in , assume that is the closest g k I real to its edge magnitude ; then we have the g ( p ) percentile . ρ ( p ) = ρ k • Altogether, for each pixel , we have a percentile p and a confidence between 0 and 1. ρ ( p ) η ( p ) 33 ¡