This tour explores image segmentation using parametric active contours. $\newcommand{\dotp}[2]{\langle #1, #2 \rangle}$ $\newcommand{\qandq}{\quad\text{and}\quad}$ $\newcommand{\qwhereq}{\quad\text{where}\quad}$ $\newcommand{\qifq}{ \quad \text{if} \quad }$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\pa}[1]{\left(#1\right)}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Nn}{\mathcal{N}}$ $\newcommand{\Bb}{\mathcal{B}}$ $\newcommand{\EE}{\mathbb{E}}$ $\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\abs}[1]{\left|#1\right|}$ $\newcommand{\choice}[1]{ \left\{ \begin{array}{l} #1 \end{array} \right. }$ $\newcommand{\al}{\alpha}$ $\newcommand{\la}{\lambda}$ $\newcommand{\ga}{\gamma}$ $\newcommand{\Ga}{\Gamma}$ $\newcommand{\La}{\Lambda}$ $\newcommand{\si}{\sigma}$ $\newcommand{\Si}{\Sigma}$ $\newcommand{\be}{\beta}$ $\newcommand{\de}{\delta}$ $\newcommand{\De}{\Delta}$ $\newcommand{\phi}{\varphi}$ $\newcommand{\th}{\theta}$ $\newcommand{\om}{\omega}$ $\newcommand{\Om}{\Omega}$
Important: Please read the installation page for details about how to install the toolboxes.
using PyPlot
using NtToolBox
using Autoreload
WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\Autoreload\src\constants.jl:1 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\Autoreload\src\constants.jl:1 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\Autoreload\src\constants.jl:7 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\Autoreload\src\constants.jl:7 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\Autoreload\src\constants.jl:12
In this tours, the active contours are represented using parametric curve $ \ga : [0,1] \rightarrow \RR^2 $.
This curve is discretized using a piewise linear curve with $p$ segments, and is stored as a complex vector of points in the plane $\ga \in \CC^p$.
Initial polygon.
gamma0 = [.78, .14, .42, .18, .32, .16, .75, .83, .57, .68, .46, .40, .72, .79, .91, .90] + 1im*[.87, .82, .75, .63, .34, .17, .08, .46, .50, .25, .27, .57, .73, .57, .75, .79]
WARNING: Base.UTF8String is deprecated, use String
16-element Array{Complex{Float64},1}: 0.78+0.87im 0.14+0.82im 0.42+0.75im 0.18+0.63im 0.32+0.34im 0.16+0.17im 0.75+0.08im 0.83+0.46im 0.57+0.5im 0.68+0.25im 0.46+0.27im 0.4+0.57im 0.72+0.73im 0.79+0.57im 0.91+0.75im 0.9+0.79im
instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31 WARNING: Base.UTF8String is deprecated, use String instead. likely near C:\Users\Ayman\.julia\v0.5\IJulia\src\kernel.jl:31
Display the initial curve.
periodize = gamma -> [gamma; [gamma[1]]]
function cplot(gamma, s = "b", lw = 1)
plot(real(periodize(gamma)), imag(periodize(gamma)), s, linewidth = lw)
axis("equal")
axis("off")
end
cplot(gamma0, "b.-")
(0.1015,0.9485,0.040499999999999994,0.9095)
Number of points of the discrete curve.
p = 256
256
Shortcut to re-sample a curve according to arc length.
using Dierckx
curvabs = gamma -> [0; cumsum(1e-5 .+ abs(gamma[1:end - 1] - gamma[2:end]) )]
resample11 = (gamma, d) -> evaluate(Spline1D(d./d[end], real(gamma), k=1), (0:p-1)./p)
resample12 = (gamma, d) -> evaluate(Spline1D(d./d[end], imag(gamma), k=1), (0:p-1)./p)
resample = gamma -> resample11( [gamma; gamma[1]], curvabs( [gamma; gamma[1]] ) ) + 1im*resample12( [gamma; gamma[1]], curvabs( [gamma; gamma[1]] ) )
(::#9) (generic function with 1 method)
Initial curve $ \ga_1(t)$.
gamma1 = resample(gamma0)
256-element Array{Complex{Float64},1}: 0.78+0.87im 0.761553+0.868559im 0.743106+0.867118im 0.724659+0.865676im 0.706212+0.864235im 0.687764+0.862794im 0.669317+0.861353im 0.65087+0.859912im 0.632423+0.858471im 0.613976+0.857029im 0.595529+0.855588im 0.577082+0.854147im 0.558635+0.852706im ⋮ 0.889715+0.719573im 0.899979+0.734968im 0.909894+0.750424im 0.905407+0.768371im 0.900921+0.786317im 0.887764+0.798157im 0.872369+0.808421im 0.856974+0.818684im 0.841579+0.828947im 0.826185+0.83921im 0.81079+0.849474im 0.795395+0.859737im
Display the initial curve.
cplot(gamma1, "k")
(0.10727700934498144,0.9481138973125205,0.040705710689687556,0.909490204252872)
Shortcut for forward and backward finite differences.
BwdDiff = c -> c - [c[end]; c[1:end - 1]]
FwdDiff = c -> [c[2:end]; c[1]] - c
dotp = (c1, c2) -> real(c1.*conj(c2))
(::#15) (generic function with 1 method)
The tangent to the curve is computed as $$ t_\ga(s) = \frac{\ga'(t)}{\norm{\ga'(t)}} $$ and the normal is $ n_\ga(t) = t_\ga(t)^\bot. $
Shortcut to compute the tangent and the normal to a curve.
normalize = v -> v./max(abs(v), 1e-10)
tangent = gamma -> normalize( FwdDiff(gamma) )
normal = gamma -> -1im*tangent(gamma)
(::#21) (generic function with 1 method)
Move the curve in the normal direction, by computing $ \ga_1(t) \pm \delta n_{\ga_1}(t) $.
delta = .03
gamma2 = gamma1 .+ delta .* normal(gamma1)
gamma3 = gamma1 .- delta .* normal(gamma1)
256-element Array{Complex{Float64},1}: 0.782337+0.840091im 0.76389+0.83865im 0.745442+0.837209im 0.726995+0.835768im 0.708548+0.834326im 0.690101+0.832885im 0.671654+0.831444im 0.653207+0.830003im 0.63476+0.828562im 0.616313+0.827121im 0.597866+0.825679im 0.579418+0.824238im 0.560971+0.822797im ⋮ 0.864754+0.736214im 0.874728+0.751167im 0.88079+0.743148im 0.876303+0.761095im 0.880853+0.764018im 0.871123+0.773196im 0.855728+0.783459im 0.840333+0.793722im 0.824938+0.803986im 0.809544+0.814249im 0.794149+0.824512im 0.778754+0.834775im
Display the curves.
cplot(gamma1, "k")
cplot(gamma2, "r--")
cplot(gamma3, "b--")
axis("tight")
axis("off")
(0.09818192340611892,0.979037189504653,0.011000256259768873,0.942237845572892)
A curve evolution is a series of curves $ s \mapsto \ga_s $ indexed by an evolution parameter $s \geq 0$. The intial curve $\ga_0$ for $s=0$ is evolved, usually by minizing some energy $E(\ga)$ in a gradient descent $$ \frac{\partial \ga_s}{\partial s} = \nabla E(\ga_s). $$
Note that the gradient of an energy is defined with respect to the curve-dependent inner product $$ \dotp{a}{b} = \int_0^1 \dotp{a(t)}{b(t)} \norm{\ga'(t)} d t. $$ The set of curves can thus be thought as being a Riemannian surface.
The simplest evolution is the mean curvature evolution. It corresponds to minimization of the curve length $$ E(\ga) = \int_0^1 \norm{\ga'(t)} d t $$
The gradient of the length is $$ \nabla E(\ga)(t) = -\kappa_\ga(t) n_\ga(t) $$ where $ \kappa_\ga $ is the curvature, defined as $$ \kappa_\ga(t) = \frac{1}{\norm{\ga'(t)}} \dotp{ t_\ga'(t) }{ n_\ga(t) } . $$
Shortcut for normal times curvature $ \kappa_\ga(t) n_\ga(t) $.
normalC = gamma -> BwdDiff(tangent(gamma)) ./ abs( FwdDiff(gamma) )
(::#23) (generic function with 1 method)
Time step for the evolution. It should be very small because we use an explicit time stepping and the curve has strong curvature.
dt = 0.001 / 100
1.0e-5
Number of iterations.
Tmax = 3.0 / 100
niter = round(Tmax/dt)
3000.0
Initialize the curve for $s=0$.
gamma = gamma1
256-element Array{Complex{Float64},1}: 0.78+0.87im 0.761553+0.868559im 0.743106+0.867118im 0.724659+0.865676im 0.706212+0.864235im 0.687764+0.862794im 0.669317+0.861353im 0.65087+0.859912im 0.632423+0.858471im 0.613976+0.857029im 0.595529+0.855588im 0.577082+0.854147im 0.558635+0.852706im ⋮ 0.889715+0.719573im 0.899979+0.734968im 0.909894+0.750424im 0.905407+0.768371im 0.900921+0.786317im 0.887764+0.798157im 0.872369+0.808421im 0.856974+0.818684im 0.841579+0.828947im 0.826185+0.83921im 0.81079+0.849474im 0.795395+0.859737im
Evolution of the curve.
gamma = gamma .+ dt .* normalC(gamma)
256-element Array{Complex{Float64},1}: 0.779911+0.869658im 0.761553+0.868559im 0.743106+0.867118im 0.724659+0.865676im 0.706212+0.864235im 0.687764+0.862794im 0.669317+0.861353im 0.65087+0.859912im 0.632423+0.858471im 0.613976+0.857029im 0.595529+0.855588im 0.577082+0.854147im 0.558635+0.852706im ⋮ 0.889715+0.719573im 0.899971+0.734974im 0.909471+0.750493im 0.905407+0.768371im 0.900638+0.786147im 0.887716+0.798096im 0.872369+0.808421im 0.856974+0.818684im 0.841579+0.828947im 0.826185+0.83921im 0.81079+0.849474im 0.795395+0.859737im
To stabilize the evolution, it is important to re-sample the curve so that it is unit-speed parametrized. You do not need to do it every time step though (to speed up).
gamma = resample(gamma)
256-element Array{Complex{Float64},1}: 0.779911+0.869658im 0.76173+0.868569im 0.743572+0.867154im 0.725414+0.865735im 0.707255+0.864317im 0.689097+0.862898im 0.670939+0.86148im 0.652781+0.860061im 0.634623+0.858642im 0.616465+0.857224im 0.598307+0.855805im 0.580149+0.854387im 0.56199+0.852968im ⋮ 0.890755+0.721134im 0.900799+0.736326im 0.909116+0.752056im 0.905023+0.769802im 0.89969+0.787023im 0.886136+0.799159im 0.871021+0.80932im 0.855866+0.819423im 0.840712+0.829526im 0.825557+0.839629im 0.810403+0.849732im 0.795246+0.859832im
Exercise 1: Perform the curve evolution. You need to resample it a few times.
run -i nt_solutions/segmentation_2_snakes_param/exo1
Geodesic active contours minimize a weighted length $$ E(\ga) = \int_0^1 W(\ga(t)) \norm{\ga'(t)} d t, $$ where $W(x)>0$ is the geodesic metric, that should be small in areas where the image should be segmented.
Size of the image $n$.
n = 200
200
Create a synthetic weight $W(x)$.
nbumps = 40
theta = rand(nbumps,1).*(2*pi)
r = .6*n/2
a = [.62*n,.6*n]
x = round( a[1] + r*cos(theta) )
y = round( a[2] + r*sin(theta) )
W = zeros(n, n)
for i in 1:nbumps
W[Int(x[i]), Int(y[i])] = 1
end
W = gaussian_blur(W, 6.0)
W = rescale( -min(W, .05), .3,1)
200×200 Array{Float64,2}: 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ⋮ ⋮ ⋱ ⋮ 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Display the metric $W$.
imageplot(W)
Pre-compute the gradient $\nabla W(x)$ of the metric.
G = grad(W)
G = G[:,:,1] .+ 1im*G[:, :, 2]
200×200 Array{Complex{Float64},2}: 0.0+0.0im … 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im … 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im … 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im ⋮ ⋱ 8.25562e-13+8.98837e-13im 2.03948e-12+2.19447e-12im 3.72147e-13+3.99458e-13im 9.23484e-13+9.80549e-13im 1.63647e-13+1.73639e-13im … 4.09006e-13+4.29212e-13im 7.08322e-14+7.41629e-14im 1.77192e-13+1.83853e-13im 2.9754e-14+3.08642e-14im 7.52731e-14+7.74936e-14im 1.22125e-14+1.26565e-14im 3.13083e-14+3.19744e-14im 5.10703e-15+5.10703e-15im 1.28786e-14+1.28786e-14im 1.77636e-15+1.9984e-15im … 4.88498e-15+5.10703e-15im 8.88178e-16+8.88178e-16im 2.22045e-15+1.9984e-15im 4.44089e-16+4.44089e-16im 6.66134e-16+6.66134e-16im 0.0+0.0im 2.22045e-16+4.44089e-16im 0.0+0.0im 2.22045e-16+2.22045e-16im
Display the image of the magnitude $\norm{\nabla W(x)}$ of the gradient.
imageplot(abs(G))
Shortcut to evaluate the gradient and the potential along a curve.
EvalG = gamma -> bilinear_interpolate(G, imag(gamma), real(gamma))
EvalW = gamma -> bilinear_interpolate(W, imag(gamma), real(gamma))
(::#27) (generic function with 1 method)
Create a circular curve $\ga_0$.
r = .98*n/2 # radius
p = 128 # number of points on the curve
theta = transpose( linspace(0, 2*pi, p + 1) )
theta = theta[1 : end]
gamma0 = n/2 * (1 + 1im) + r*(cos(theta) + 1im*sin(theta))
129-element Array{Complex{Float64},1}: 198.0+100.0im 197.882+104.809im 197.528+109.606im 196.939+114.38im 196.117+119.119im 195.063+123.812im 193.78+128.448im 192.271+133.015im 190.54+137.503im 188.591+141.9im 186.428+146.197im 184.057+150.382im 181.484+154.446im ⋮ 184.057+49.6179im 186.428+53.8031im 188.591+58.0996im 190.54+62.497im 192.271+66.9848im 193.78+71.5521im 195.063+76.1879im 196.117+80.8811im 196.939+85.6204im 197.528+90.3943im 197.882+95.1914im 198.0+100.0im
Initialize the curve at time $t=0$ with a circle.
gamma = gamma0
129-element Array{Complex{Float64},1}: 198.0+100.0im 197.882+104.809im 197.528+109.606im 196.939+114.38im 196.117+119.119im 195.063+123.812im 193.78+128.448im 192.271+133.015im 190.54+137.503im 188.591+141.9im 186.428+146.197im 184.057+150.382im 181.484+154.446im ⋮ 184.057+49.6179im 186.428+53.8031im 188.591+58.0996im 190.54+62.497im 192.271+66.9848im 193.78+71.5521im 195.063+76.1879im 196.117+80.8811im 196.939+85.6204im 197.528+90.3943im 197.882+95.1914im 198.0+100.0im
For this experiment, the time step should be larger, because the curve is in $[0,n-1] \times [0,n-1]$.
dt = 1
1
Number of iterations.
Tmax = 5000
niter = round(Tmax/ dt)
5000.0
Display the curve on the background.
lw = 2
clf
imageplot(transpose(W))
cplot(gamma, "r", lw)
(-0.5,199.5,199.5,-0.5)
The gradient of the energy is $$ \nabla E(\ga) = -W(\ga(t)) \kappa_\ga(t) n_\ga(t) + \dotp{\nabla W(\ga(t))}{ n_\ga(t) } n_\ga(t). $$
Pointwise innerproduct on the curve.
#dotp = lambda c1,c2: real(c1)*real(c2) + imag(c1)*imag(c2)
Evolution of the curve according to this gradient.
N = normal(gamma)
g = - EvalW(gamma) * normalC(gamma) + dotp(EvalG(gamma), N) * N # I'm not sure of this product. But when it is .* , we encounter a problem with function resample
gamma = gamma .- dt*g
129-element Array{Complex{Float64},1}: -8.60367e11-3.50375e13im -8.59819e11-3.50152e13im -8.59232e11-3.49913e13im -8.58198e11-3.49492e13im -8.56191e11-3.48674e13im -8.51551e11-3.46785e13im -8.44902e11-3.44077e13im -8.3394e11-3.39613e13im -8.12444e11-3.30859e13im -7.79551e11-3.17464e13im -7.41951e11-3.02151e13im -6.95247e11-2.83132e13im -6.64033e11-2.7042e13im ⋮ -6.93909e11-2.82587e13im -7.38248e11-3.00644e13im -7.83726e11-3.19164e13im -8.12467e11-3.30868e13im -8.31343e11-3.38555e13im -8.44715e11-3.44001e13im -8.52298e11-3.47089e13im -8.55557e11-3.48416e13im -8.58052e11-3.49432e13im -8.59335e11-3.49955e13im -8.60077e11-3.50257e13im -8.59972e11-3.50214e13im
To avoid the curve from being poorly sampled, it is important to re-sample it evenly.
gamma = resample(gamma)
128-element Array{Complex{Float64},1}: -8.60367e11-3.50375e13im -8.53795e11-3.47699e13im -8.47223e11-3.45022e13im -8.40651e11-3.42346e13im -8.34079e11-3.3967e13im -8.27507e11-3.36993e13im -8.20935e11-3.34317e13im -8.14363e11-3.3164e13im -8.07791e11-3.28964e13im -8.01218e11-3.26287e13im -7.94646e11-3.23611e13im -7.88074e11-3.20935e13im -7.81502e11-3.18258e13im ⋮ -7.81713e11-3.18344e13im -7.88285e11-3.2102e13im -7.94857e11-3.23697e13im -8.01429e11-3.26373e13im -8.08001e11-3.2905e13im -8.14573e11-3.31726e13im -8.21145e11-3.34402e13im -8.27717e11-3.37079e13im -8.34289e11-3.39755e13im -8.40862e11-3.42432e13im -8.47434e11-3.45108e13im -8.54006e11-3.47784e13im
Exercise 2: Perform the curve evolution.
run -i nt_solutions/segmentation_2_snakes_param/exo2
One can use a gradient-based metric to perform edge detection in medical images.
Load an image $f$.
n = 256
name = "NtToolBox/src/data/cortex.png"
f = load_image(name, n)
256×256 Array{Float32,2}: 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋮ ⋱ ⋮ ⋮ 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Display.
imageplot(f)
An edge detector metric can be defined as a decreasing function of the gradient magnitude. $$ W(x) = \psi( d \star h_a(x) ) \qwhereq d(x) = \norm{\nabla f(x)}. $$ where $h_a$ is a blurring kernel of width $a>0$.
Compute the magnitude of the gradient.
G = grad(f)
d0 = sqrt(sum(G.^2, 3))
imageplot(d0[:, :, 1])
Blur it by $h_a$.
a = 2
d = gaussian_blur(d0, a)
imageplot(d[:, :, 1])
Compute a decreasing function of the gradient to define $W$.
d = min(d, .4)
W = rescale(-d, .8, 1)
256×256×1 Array{Float64,3}: [:, :, 1] = 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ⋮ ⋮ ⋱ ⋮ ⋮ 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 … 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Display it.
imageplot(W[:, :, 1])
Number of points.
p = 128
128
Exercise 3: Create an initial circle $\gamma_0$ of $p$ points. When plotting the image, you need to transpose it to have axis coherent with the cplot.
run -i nt_solutions/segmentation_2_snakes_param/exo3
Step size.
dt = 2
2
Number of iterations.
Tmax = 9000
niter = round(Tmax/ dt)
4500.0
Exercise 4: Perform the curve evolution.
run -i nt_solutions/segmentation_2_snakes_param/exo4
It is possible to perform the evolution of a non-closed curve by adding boundary constraint $$ \ga(0)=x_0 \qandq \ga(1)=x_1. $$
In this case, the algorithm find a local minimizer of the geodesic distance between the two points.
Note that a much more efficient way to solve this problem is to use the Fast Marching algorithm to find the global minimizer of the geodesic length.
Load an image $f$.
n = 256
f = load_image(name, n)
f = f[45:105, 60:120]
n = size(f)[1]
61
Display.
imageplot(f)
Exercise 5: Compute an edge attracting criterion $W(x)>0$, that is small in area of strong gradient.
run -i nt_solutions/segmentation_2_snakes_param/exo5
Start and end points $x_0$ and $x_1$.
x0 = 4 + 55j
x1 = 53 + 4j
Initial curve $\ga_0$.
p = 128
t = transpose(linspace(0, 1, p))
gamma0 = t*x1 + (1-t)*x0
Initialize the evolution.
gamma = gamma0
Display.
clf
imageplot(transpose(W))
cplot(gamma,'r', 2)
plot(real(gamma[0]), imag(gamma[0]), 'b.', markersize=20)
plot(real(gamma[-1]), imag(gamma[-1]), 'b.', markersize=20);
Re-sampling for non-periodic curves.
curvabs = lambda gamma: concatenate( ([0], cumsum( 1e-5 + abs(gamma[:-1:]-gamma[1::]) ) ) )
resample1 = lambda gamma,d: interpc(arange(0,p)/float(p-1), d/d[-1],gamma)
resample = lambda gamma: resample1( gamma, curvabs(gamma) )
Time step.
dt = 1/10
Number of iterations.
Tmax = 2000*4/ 7
niter = round(Tmax/ dt)
Exercise 6: Perform the curve evolution. Be careful to impose the boundary conditions at each step.
run -i nt_solutions/segmentation_2_snakes_param/exo6