Quantification of fibre width in biological images Lake Como School ”Computational methods for inverse problems in imaging” Mathilde Galinier PhD student at the Department of Physics, Informatics and Mathematics Universit` a degli studi di Modena e Reggio Emilia Wednesday, May 23rd 2018 1/14 Mathilde Galinier Quantification of fibre width in biological images
Introduction Aim : Enabling a better understanding of microbial ecosystems in order to improve waste-water treatment methods. Project based on the analysis of sewage pictures. Algorithm associating Canny edge detector-related methods and statistical analysis. Figure: Raw image u , taken with a fluorescence microscope. 2/14 Mathilde Galinier Quantification of fibre width in biological images
Step 1 : Convolution with Gaussian filters Image convolved by two Gaussian filters : � u 1 = K σ ∗ u u 2 = K βσ ∗ u , β < 1 with − k 2 + l 2 1 � � K σ ( k , l ) = 2 πσ 2 exp 2 σ 2 For example, a N × M Gaussian filter K σ applied to the image u provides : N − 1 M − 1 � � u 1 ij = K σ ( k , l ) u ( i − k , j − l ) k =0 l =0 which can be computed thanks to the Fourier transform : u 1 = F − 1 ( F ( K σ ) . F ( u )) 3/14 Mathilde Galinier Quantification of fibre width in biological images
Step 2 : Computation of the gradient Detection of the edges of an image f ( x , y ) based on the computation of the gradient. The vector : � ∂ f � � g x � ∂ x ∇ f ( x , y ) = ( x , y ) = ( x , y ) ∂ f g y ∂ y points in the direction of the greater rate of change of f at ( x , y ). Norm of ∇ f ( x , y ) : �∇ f ( x , y ) � 2 2 = g 2 x + g 2 y Direction of ∇ f ( x , y ) : θ ( x , y ) = arctan ( g y ) g x 4/14 Mathilde Galinier Quantification of fibre width in biological images
Step 2 : Computation of the gradient In our case, the matrix of interest is written, for each pixel : � G xx � C = 1 G xy � � ∇ u 1 ∇ u T 2 + ∇ u 2 ∇ u T = 1 G xy G yy 2 where : G xx = ∂ x u 1 .∂ x u 2 G yy = ∂ y u 1 .∂ y u 2 G xx = 1 2 ( ∂ x u 1 .∂ y u 2 + ∂ y u 1 .∂ x u 2 ) Topological gradient : largest eigenvalue of C , noted λ 1 . 5/14 Mathilde Galinier Quantification of fibre width in biological images
Step 2 : Computation of the gradient In our case, the matrix of interest is written, for each pixel : � G xx � C = 1 G xy � � ∇ u 1 ∇ u T 2 + ∇ u 2 ∇ u T = 1 G xy G yy 2 where : G xx = ∂ x u 1 .∂ x u 2 G yy = ∂ y u 1 .∂ y u 2 G xx = 1 2 ( ∂ x u 1 .∂ y u 2 + ∂ y u 1 .∂ x u 2 ) Topological gradient : largest eigenvalue of C , noted λ 1 . Remark : In the case u 1 = u 2 : ∂ x u 2 � � ∂ x u 1 ∂ y u 1 1 λ 1 = ∂ x u 2 1 + ∂ y u 2 1 = �∇ u 1 � 2 C = and ∂ y u 2 2 ∂ x u 1 ∂ y u 1 1 5/14 Mathilde Galinier Quantification of fibre width in biological images
Step 2 : Computation of the gradient Figure: Raw image u . 6/14 Mathilde Galinier Quantification of fibre width in biological images
Step 2 : Computation of the gradient Figure: Topological gradient of u . 7/14 Mathilde Galinier Quantification of fibre width in biological images
Step 3 : Non local maxima suppression 1. Find the 2 closest pixels along the edge normal. 2. Retain the pixel with maximum magnitude value. Topological gradient of u . 8/14 Mathilde Galinier Quantification of fibre width in biological images
Step 3 : Non local maxima suppression 1. Find the 2 closest pixels along the edge normal. 2. Retain the pixel with maximum magnitude value. Non local maxima suppression 9/14 Mathilde Galinier Quantification of fibre width in biological images
Step 4 : Hysteresis thresholding Keep pixels with a low intensity only if they are connected to a ’strong’ pixel. Binary image after hysteresis thresholding 10/14 Mathilde Galinier Quantification of fibre width in biological images
Step 5 : Computation of fiber width For each non-zero pixel, the algorithm searches for another edge in the direction of the edge normal. Binary image after hysteresis thresholding 11/14 Mathilde Galinier Quantification of fibre width in biological images
Statistical analysis Gauss-Newton algorithm for the fitting of the cumulative distribution functions : p � G ( p ) � 2 L 2 = � F p 1 , ··· , p K − F data � 2 min L 2 In red : Data cumulative distribution Histogram of fiber widths. Abscissa : fiber function. In blue : Fitting with lognormal width (pixels) ; Ordinate : number of fibers by category. cumulative distribution function. 12/14 Mathilde Galinier Quantification of fibre width in biological images
Comparison of the cumulative distribution functions over several days Figure: Lognormal cumulative distribution functions of a sample, for days 1,2 and 14. 13/14 Mathilde Galinier Quantification of fibre width in biological images
References Samuel Amstutz and J´ erˆ ome Fehrenbach. Edge detection using topological gradients: A scale-space approach. Springer Science+Business Media , Jan 2015. John Canny. A computational approach to edge detection. IEEE Transactions on pattern analysis and machine intelligence , PAMI-8(6), Nov 1986. Nick Efford. Digital Image Processing: A Practical Introduction Using Java . Pearson Education, 2000. 14/14
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