Fluid Dynamics¶

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This numerical tour explores fluid dynamics for image generation.

In [2]:

using PyPlot
using NtToolBox
using Interpolations

Velocity Flow Field¶

A velocity flow is simply a 2-D vector field $V = (V_i)_{i=1}^N \in \RR^{n \times n \times 2}$ where $V_i \in \RR^2$ is one of the $N=n \times n$ vectors at a position indexed by $i$.

It can be generated as a realization of Gaussian process. The blurring creates correlations in the flow.

In [2]:

n = 128
V = perform_blurring(randn(n,n,2),[40], "per");

Subsampling display operator.

In [3]:

myplot = V-> plot_vf(V[1:6:n,1:6:n, :])

Out[3]:

(::#1) (generic function with 1 method)

We can display the vector field using arrow.

In [4]:

figure(figsize = (7,7))
myplot(V)

We can renormalize the flow, which enhances the singularities. It defines $\tilde V$ as $\tilde V_i = V_i/\norm{V_i}$.

In [5]:

Normalize = V -> V ./ repeat( max(1e-9,sqrt.(sum(V.^2, 3))) , outer=(1,1,2));

Display.

In [6]:

figure(figsize = (7,7))
myplot(Normalize(V))

Incompressible Flows¶

An incompressible flow has a vanishing divergence. The set of vector incompressible flow defines a sub-space of $\RR^{n \times n \times 2}$ $$ \Ii = \enscond{V}{ \text{div}(V)=0 } \qwhereq \text{div}(V) = \pd{V}{x_1} + \pd{V}{x_2} \in \RR^{n \times n}. $$ Here $\pd{}{x_s}$ for $s=1,2$ are finite differences approximation of the horizontal and vertical derivative operators (we suppose here periodic boundary conditions).

The orthogonal projection $U = \text{Proj}_{\Ii}(V)$ on $\Ii$ is computed by solving a Poisson equation $$ U = V-\nabla A \qwhereq \Delta A = \text{div}(V). $$

This is especially simple for periodic boundary conditions since $A$ can be computed over the Fourier domain as $$ \forall \om \neq 0, \quad \hat A(\om) = \frac{\hat Y(\om)}{\mu(\om)} \qwhereq Y = \text{div}(V) \qandq \mu(\om_1,\om_2) = -4 \sin(\om_1 \pi / n)^2 -4 \sin(\om_2 \pi / n)^2 $$ and $\hat A(0)=0$.

Compute the kernel $\mu(\om)$.

In [7]:

Y, X = meshgrid(0:n-1,0:n-1)
mu = sin.(X*pi/n).^2
mu = -4*(mu + mu')
mu[1,1] = 1;

Computation of $A$.

In [8]:

A = V -> real.(ifft(fft(Div(V[:,:,1], V[:,:,2], "per"))./mu))

Out[8]:

(::#5) (generic function with 1 method)

Projection on incompressible flows.

In [9]:

ProjI = V -> V - Grad(A(V), "per")

Out[9]:

(::#7) (generic function with 1 method)

Display $U=\text{Proj}_{\Ii}(V)$.

In [10]:

U = ProjI(V)

figure(figsize=(7,7))
myplot(U)

Display $W=U-V$ the irrotational component of $V$.

In [11]:

figure(figsize=(7,7))
myplot(V-U)

Note that the decomposition $V=U+W$ is called the Hoge decomposition of the vector field.

Image Advection Along the Flow¶

A flow defines a warping operator that transport the content of an image along the streaming of the flow.

We load an image $f$.

In [12]:

f = load_image("NtToolBox/src/data/lena.png", 2*n)
f = f[n-Base.div(n,2):n+Base.div(n,2)-1, n-Base.div(n,2):n+Base.div(n,2)-1];

Given some vector field $U$, the warping operator $f_1 = \Ww_U(f)$ along the flow is defined $$ f_1(x) = f(x+U(x)) $$ i.e. it advects the values of $f$ by the vector field $U$ to obtain the values of $f_1$.

We define $U$ as a scaled normalized incompressible flow.

In [13]:

U = Normalize(ProjI(V));

Helper function: enforce periodicity.

In [14]:

periodic = P -> cat(3, mod(P[:,:,1]-1,n)+1, mod(P[:,:,2]-1,n)+1 );

Helper function: extend an image by 1 pixel to avoid boundary problems.

In [15]:

extend1 = f -> [f f[:,1]]
extend = f -> extend1(extend1(f)')';

Helper function: bilinear interpolation on a grid.

In [16]:

f1 = extend(f)
itp = interpolate((1:size(f1,1), 1:size(f1,2)), f1, Gridded(Linear()))
size(itp)

Out[16]:

(129,129)

In [17]:

function myinterp(P1,f1,Pi)
    itp = interpolate((1:size(f1,1), 1:size(f1,2)), f1, Gridded(Linear()))
    itpd = zeros(size(Pi)[1:2])
    for i in 1:size(Pi,1)
        for j in 1:size(Pi,2)
        itpd[i,j] = itp[Pi[i,j,1], Pi[i,j,2]]
        end
    end
    return itpd
end;

First we compute the initial and wraped grids.

In [18]:

(Y,X) = meshgrid(1:n,1:n)
P = cat(3, X,Y)
(Y1,X1) = meshgrid(1:n+1,1:n+1)
P1 = cat(3, X1,Y1);

Defines the warping operator $\Ww_U$.

In [19]:

W =  (f, U) -> myinterp(P1, extend(f), periodic(P - U));

Display a warped image $\Ww_{\rho U}(f)$ for some scaling $\rho$.

In [20]:

rho = 2
figure(figsize = (5,5))
imageplot(W(f, rho*U))

Exercise 1

Display $\Ww_{\rho U}(f)$ for various values of $\rho$.

In [21]:

include("NtSolutions/graphics_5_fluids/exo1.jl")