ITK Filters. Thresholding Edge Detection Gradients Second Order Derivatives Neighborhood Filters Smoothing Filters Distance Map Image Transforms

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1 ITK Filters Thresholding Edge Detection Gradients Second Order Derivatives Neighborhood Filters Smoothing Filters Distance Map Image Transforms ITCS 6010:Biomedical Imaging and Visualization 1 ITK Filters: Overview

2 Thresholding itk::binarythresholdimagefilter ITCS 6010:Biomedical Imaging and Visualization 2 ITK Filters: Overview

3 Thresholding itk::thresholdimagefilter ITCS 6010:Biomedical Imaging and Visualization 3 ITK Filters: Overview

4 Canny Edge Detection Performs non-maximum suppression and hysteresis thresholding Supply low and high thresholds and parameters for Gaussian Smoothing (itk:discretegaussianimagefilter) ITCS 6010:Biomedical Imaging and Visualization 4 ITK Filters: Overview

5 Gradient Filters itk::gradientmagnitudefilter (without smoothing) Uses Central Difference operator. itk::gradientmagnituderecursivegaussianimagefilter smoothing) (with Approximates the convolution with the derivative of Gaussian with IIR filters Separable in each dimension itk::derivativeimagefilter: smoothing. Computes image derivatives without itk::recursivegaussianimagefilter: Computes 0th, first or second partial derivatives with smoothing (can get all second derivatives - Hessian matrix). ITCS 6010:Biomedical Imaging and Visualization 5 ITK Filters: Overview

6 Neighborhood Filters itk::meanimagefilter (average of a local neighborhood) itk::medianimagefilter (meadian of a local neighborhood) ITCS 6010:Biomedical Imaging and Visualization 6 ITK Filters: Overview

7 Smoothing Filters Blurring: itk::discretegaussianimagefilter Parameters: variance, max kernel size Also itk::binomialblurimagefilter iterative, approximates a Gaussian with large number of iterations itk::recursivegaussianimagefilter: uses IIR with approximation of convolution with Gaussian and its derivatives ITCS 6010:Biomedical Imaging and Visualization 7 ITK Filters: Overview

8 Smoothing Filters ITCS 6010:Biomedical Imaging and Visualization 8 ITK Filters: Overview

9 Nonlinear Image Filtering with Partial Differential Equations Motivation: Preserve boundaries while smoothing noisy parts of image/volume Linear filters (eg., Gaussian blurring) tends to smooth out edges, that complicates segmentation and image analysis tasks Non-linear filters fall under 3 classes: Heuristics - specific rules on pixels to achieve a particular result Statistics - filters designed on statistics, eg., median filter Partial Differential Equations (PDEs) - solutions to PDEs, whose input is some value (from image, for instance) ITCS 6010:Biomedical Imaging and Visualization 9 ITK Filters: Overview

10 Gaussian Blurring and Heat Equation Gaussian - low pass filter, useful in denoising, as noise has more energy at higher frequencies; Given an image f(x, y) in a domain D R 2, thus f : D R, define the Gaussian and its Fourier Transform as g(x, y) = 1 x2 +y 2 2πσ 2 e 2σ 2, G(u, v) = F{g} = e (x2 +y 2 )σ 2 2 Gaussian has nice properties: smooth, no ringing and can be applied repeatedly with the same filter Relationship to PDEs is through the diffusion equation: f t = 2 f 2 x + 2 f 2 y =.c f where the solution, f(x, y, t) is a sequence of images starting with f(x, y, 0) that evolves with time, t. is the vector of partial derivatives, and c is a conductance term ITCS 6010:Biomedical Imaging and Visualization 10 ITK Filters: Overview

11 Gaussian Blurring and Heat Equation(contd) f t = 2 f 2 x + 2 f 2 y =.c f An example of an initial value problem, with the initial image f(x, y, 0) = I(x, y) As this PDE is linear and stationary (shift-invariant), in the frequency domain, this is f t = (u2 + v 2 )F where F{ f/ x} = juf, j 2 = 1 This is an ODE, and has the solution F (u, v, t) = e (u2 +v 2 )t F (u, v, 0) ITCS 6010:Biomedical Imaging and Visualization 11 ITK Filters: Overview

12 Gaussian Blurring and Heat Equation (contd) F (u, v, t) = e (u2 +v 2 )t F (u, v, 0) This becomes a convolution in the spatial domain (Inverse FT) f(x, y, t) = 1 4πt e (x2 +y 2 )/4t f(x, y, 0) Thus, the solution to the diffusion equation is equivalent to convolving the initial conditions (eg., input image) with a Gaussian with σ = (2t). ITCS 6010:Biomedical Imaging and Visualization 12 ITK Filters: Overview

13 Solving the Heat Equation: Boundary Conditions Preferable to solve the diffusion equation due to boundary conditions (above relationship holds only for proper boundary conditions). By defining boundary conditions appropriate to images, a more useful solution without artifacts can be obtained. For second order PDEs, we use the Von Neumann conditions, by specifying value of derivatives of the solution across the boundary. If n represents a direction orthogonal to the boundary, choose f/ n = 0 for all points on boundaries, i.e. zero flow (diffusion) across the boundaries. ITCS 6010:Biomedical Imaging and Visualization 13 ITK Filters: Overview

14 Anisotropic Diffusion In image analysis, need to preserve edges (high frequencies) and reduce noise artifiacts (also high frequencies) Filtering with the diffusion equation, we can impact edge smoothing through a variable conductance term: f t =.c f The diffusion equation models the flow of material or energy from areas of high concentration to those of low concentration, controlled by the conductance c(x, y). Idea: Lower the conductance near interesting image features, like edges, to reduce blurring. ITCS 6010:Biomedical Imaging and Visualization 14 ITK Filters: Overview

15 Anisotropic Diffusion Perona and Malik proposed a non-linear form for c, anisotropic diffusion that depends on f: termed f t =.c( f ) f with c( f ) = e f 2 2K 2 where K is a free parameter that controls the degree to which the gradient of f controls conductance. ITCS 6010:Biomedical Imaging and Visualization 15 ITK Filters: Overview

16 Anisotropic Diffusion:Examples (Left to Right) (a) Original (b) Image with Gaussian noise, σ = 20 (c) Isotropic diffusion, MSE = 43.8 (d) Anisotropic Diffusion, MSE = 39. ITCS 6010:Biomedical Imaging and Visualization 16 ITK Filters: Overview

17 Edge Preserving Smoothing: ITK Filters itk::gradientanisotropicdiffusionfilter: n-dimensional version Uses the classic Perona-Malik formulation for scalar-valued images with conductance: C( x) = e ( U(x) K ) 2 More robust method used for estimating gradient magnitude. itk::curvatureanisotropicfilter : Uses a modified curvature diffusion equation, preserves finer details of structures. itk::curvatureflowimagefilter : based on evolving level sets under the control of a diffusion equation, with speed proportional to curvature. Analogous filters for vector images (eg., RGB images) ITCS 6010:Biomedical Imaging and Visualization 17 ITK Filters: Overview