Quantification of fibre width in biological images Lake Como School - - PowerPoint PPT Presentation

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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

Mathilde Galinier Quantification of fibre width in biological images

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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.

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Step 1 : Convolution with Gaussian filters

Image convolved by two Gaussian filters : u1 = Kσ ∗ u u2 = Kβσ ∗ u, β < 1 with Kσ(k, l) = 1 2πσ2 exp

• −k2 + l2

2σ2

• For example, a N × M Gaussian filter Kσ applied to the image u

provides : u1ij =

N−1

• k=0

M−1

• l=0

Kσ(k, l)u(i − k, j − l) which can be computed thanks to the Fourier transform : u1 = F−1 (F(Kσ) . F(u))

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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 (x, y) = ∂f

∂x ∂f ∂y

• (x, y) =

gx gy

• (x, 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 = g2 x + g2 y

Direction of ∇f (x, y) : θ(x, y) = arctan(gy gx )

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Step 2 : Computation of the gradient

In our case, the matrix of interest is written, for each pixel : C = 1 2

• ∇u1∇uT

2 + ∇u2∇uT 1

• =

Gxx Gxy Gxy Gyy

• where :

   Gxx = ∂xu1.∂xu2 Gyy = ∂yu1.∂yu2 Gxx = 1

2 (∂xu1.∂yu2 + ∂yu1.∂xu2)

Topological gradient : largest eigenvalue of C, noted λ1.

Mathilde Galinier Quantification of fibre width in biological images

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Step 2 : Computation of the gradient

In our case, the matrix of interest is written, for each pixel : C = 1 2

• ∇u1∇uT

2 + ∇u2∇uT 1

• =

Gxx Gxy Gxy Gyy

• where :

   Gxx = ∂xu1.∂xu2 Gyy = ∂yu1.∂yu2 Gxx = 1

2 (∂xu1.∂yu2 + ∂yu1.∂xu2)

Topological gradient : largest eigenvalue of C, noted λ1.

Remark : In the case u1 = u2 : C =

• ∂xu2

1

∂xu1∂yu1 ∂xu1∂yu1 ∂yu2

1

• and

λ1 = ∂xu2

1 + ∂yu2 1 = ∇u12 2

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Step 2 : Computation of the gradient

Figure: Raw image u.

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Step 2 : Computation of the gradient

Figure: Topological gradient of u.

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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.

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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

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Step 4 : Hysteresis thresholding

Keep pixels with a low intensity only if they are connected to a ’strong’ pixel.

Binary image after hysteresis thresholding

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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

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Statistical analysis

Gauss-Newton algorithm for the fitting of the cumulative distribution functions :

min

p G(p)2 L2 = Fp1,··· ,pK − Fdata2 L2

Histogram of fiber widths. Abscissa : fiber width (pixels) ; Ordinate : number of fibers by category. In red : Data cumulative distribution

• function. In blue : Fitting with lognormal

cumulative distribution function.

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Comparison of the cumulative distribution functions over several days

Figure: Lognormal cumulative distribution functions of a sample, for days 1,2 and 14.

Mathilde Galinier Quantification of fibre width in biological images

References

Samuel Amstutz and J´ erˆ

• me 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.

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