Załącznik 4
Kod w MATLABie obliczający L2 projekcję bitmapy przy zastosowaniu izogeometrycznej metody elementów skończonych
(zob. rozdział Implementacja w MATLABie algorytmu solwera zmienno-kierunkowego dla problemu projekcji bitmapy )
% This is a very fast implementation of bitmap projection with direction splitting.
% It caches basis functions values for integration points prior to main integration loop.
%
% How to use
%
% bitmap(filename as a string, number of elements along x axis, polynomial order alog x axis, number of elements along y axis, polynomial order alog y axis)
%
% Examples
%
% bitmap("mp.JPG", 10,1,10,1)
% bitmap("basket.JPG", 20,3,20,3)
function bitmap_splitting(filename,elementsx,px,elementsy,py)
tic;
% subroutine calculating number of basis functions
compute_nr_basis_functions = @(knot_vector,p) size(knot_vector, 2) - p - 1;
% create knot_vectors along x any y axis
knot_vectorx = simple_knot(elementsx,px);
knot_vectory = simple_knot(elementsy,py);
% read image
X = imread(filename);
% extract R, G and B components of the image
R = X(:,:,1);
G = X(:,:,2);
B = X(:,:,3);
% read size of image
ix = size(X,1);
iy = size(X,2);
% compute number of degrees of freedom
nx = number_of_dofs(knot_vectorx,px);
ny = number_of_dofs(knot_vectory,py);
% initiate matrices for further computations
Ax = sparse(nx,nx);
Ay = sparse(ny,ny);
FRx = zeros(nx,ny);
FGx = zeros(nx,ny);
FBx = zeros(nx,ny);
init = toc
tic;
% initiate matrices for precached basis function values at given points
splinex = zeros(elementsx,nx,px+1);
spliney = zeros(elementsy,ny,py+1);
% precache values of basis functions in integration points
for ex = 1:elementsx;
% range of nonzero functions over element
[xl,xh] = dofs_on_element(knot_vectorx,px,ex);
% range of element (left and right edge over x axis)
[ex_bound_l,ex_bound_h] = element_boundary(knot_vectorx,px,ex);
% quadrature points over element (over x axis)
qpx = quad_points(ex_bound_l,ex_bound_h,px+1);
% quadrature weights over element (over x axis)
qwx = quad_weights(ex_bound_l,ex_bound_h,px+1);
% loop over nonzero functions over element
for bi = xl:xh
% loop over quadrature points
for iqx = 1:size(qpx,2)
splinex(ex,bi,iqx)= compute_spline(knot_vectorx,px,bi,qpx(iqx));
end
end
end
for ey = 1:elementsy;
% range of nonzero functions over element
[yl,yh] = dofs_on_element(knot_vectory,py,ey);
% range of element (left and right edge over y axis)
[ey_bound_l,ey_bound_h] = element_boundary(knot_vectory,py,ey);
% quadrature points over element (over y axis)
qpy = quad_points(ey_bound_l,ey_bound_h,py+1);
% quadrature weights over element (over y axis)
qwy = quad_weights(ey_bound_l,ey_bound_h,py+1);
% loop over nonzero functions over element
for bi = yl:yh
% loop over quadrature points
for iqy = 1:size(qpy,2)
spliney(ey,bi,iqy)= compute_spline(knot_vectory,py,bi,qpy(iqy));
end
end
end
init_splines=toc
tic;
% integral B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
% (i,k=1,...,Nx; j,l=1,...,Ny)px
% loop over elements over x axis
for ex = 1:elementsx;
% range of nonzero functions over element
[xl,xh] = dofs_on_element(knot_vectorx,px,ex);
% range of element (left and right edge over x axis)
[ex_bound_l,ex_bound_h] = element_boundary(knot_vectorx,px,ex);
% Jacobian = size of element
J = ex_bound_h - ex_bound_l;
% quadrature points over element (over x axis)
qpx = quad_points(ex_bound_l,ex_bound_h,px+1);
% quadrature weights over element (over x axis)
qwx = quad_weights(ex_bound_l,ex_bound_h,px+1);
% loop over nonzero functions over element
for bi = xl:xh
for bk = xl:xh
% loop over quadrature points
for iqx = 1:size(qpx,2)
% B^x_k(x)
funk = splinex(ex,bk,iqx);
% B^x_i(x)
funi = splinex(ex,bi,iqx);
% B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
fun = funi*funk;
% integral z B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
% (i,k=1,...,Nx; j,l=1,...,Ny)
int = fun*qwx(iqx)*J;
if (int~=0)
Ax(bi,bk) = Ax(bi,bk) + int;
end
end
end
end
end
lhsx=toc
% integral B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
% (i,k=1,...,Nx; j,l=1,...,Ny)
% loop over elements on y axis
for ey = 1:elementsy
% range of nonzero functions over element
[yl,yh] = dofs_on_element(knot_vectory,py,ey);
% range of element (left and right edge over y axis)
[ey_bound_l,ey_bound_h] = element_boundary(knot_vectory,py,ey);
% Jacobian = size of element
J = ey_bound_h - ey_bound_l;
% quadrature points over element (over y axis)
qpy = quad_points(ey_bound_l,ey_bound_h,py+1);
% quadrature weights over element (over y axis)
qwy = quad_weights(ey_bound_l,ey_bound_h,py+1);
% loop over nonzero functions over element
for bj = yl:yh
for bl = yl:yh
% loop over quadrature points
for iqy = 1:size(qpy,2)
% B^y_l(y)
funl = spliney(ey,bl,iqy);
% B^y_j(y)
funj = spliney(ey,bj,iqy);
% B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
fun = funj*funl;
% integral z B^x_i(x) B^y_j(y) B^x_k(x) B^y_l(y)
% (i,k=1,...,Nx; j,l=1,...,Ny)
int = fun*qwy(iqy)*J;
if (int~=0)
Ay(bj,bl) = Ay(bj,bl) + int;
end
end
end
end
end
lhsy=toc
tic;
% Integral BITMAP(x,y) B^x_k(x) B^y_l(y)
% loop over elements on x axis
for ex = 1:elementsx;
% range of nonzero functions over element
[xl,xh] = dofs_on_element(knot_vectorx,px,ex);
% range of element (left and right edge over x axis)
[ex_bound_l,ex_bound_h] = element_boundary(knot_vectorx,px,ex);
% loop over elements on y axis
for ey = 1:elementsy
% range of nonzero functions over element
[yl,yh] = dofs_on_element(knot_vectory,py,ey);
% range of element (left and right edge over y axis)
[ey_bound_l,ey_bound_h] = element_boundary(knot_vectory,py,ey);
% Jacobian = size of element
Jx = ex_bound_h - ex_bound_l;
Jy = ey_bound_h - ey_bound_l;
J = Jx * Jy;
% quadrature points over element (over x axis)
qpx = quad_points(ex_bound_l,ex_bound_h,px+1);
% quadrature points over element (over y axis)
qpy = quad_points(ey_bound_l,ey_bound_h,py+1);
% quadrature weights over element (over x axis)
qwx = quad_weights(ex_bound_l,ex_bound_h,px+1);
% quadrature weights over element (over y axis)
qwy = quad_weights(ey_bound_l,ey_bound_h,py+1);
% loop over nonzero functions over element
for bk = xl:xh
for bl = yl:yh
% loop over quadrature points
for iqx = 1:size(qpx,2)
for iqy = 1:size(qpy,2)
% B^x_k(x)
funk = splinex(ex,bk,iqx);
% B^y_l(y)
funl = spliney(ey,bl,iqy);
% integral BITMAP(x,y) B^x_k(x) B^y_l(y) over RGB components
intR = funk*funl*qwx(iqx)*qwy(iqy)*J*bitmp(R,qpx(iqx),qpy(iqy));
intG = funk*funl*qwx(iqx)*qwy(iqy)*J*bitmp(G,qpx(iqx),qpy(iqy));
intB = funk*funl*qwx(iqx)*qwy(iqy)*J*bitmp(B,qpx(iqx),qpy(iqy));
FRx(bk,bl) = FRx(bk,bl) + intR;
FGx(bk,bl) = FGx(bk,bl) + intG;
FBx(bk,bl) = FBx(bk,bl) + intB;
end
end
end
end
end
end
rhs=toc
tic;
% solve one direction
[RRx,GGx,BBx]=solve_direction(Ax,FRx,FGx,FBx);
factorx=toc
tic;
% transpose matrices to solve over the other direction
FRy = transpose(RRx);
FGy = transpose(GGx);
FBy = transpose(BBx);
reorder=toc
tic
% solve second direction
[RRy,GGy,BBy]=solve_direction(Ay,FRy,FGy,FBy);
factory=toc
tic;
% transpose matrices back
RR = transpose(RRy);
GG = transpose(GGy);
BB = transpose(BBy);
% reconstruction of image
% set zero to reconstructed image matrices
R1 = zeros(ix,iy);
G1 = zeros(ix,iy);
B1 = zeros(ix,iy);
funx_tab = zeros(nx,ix);
funy_tab = zeros(ny,iy);
% precache basis functions values
% loop over basis functions
for bi = 1:nx
% loop over nonzero pixels over given function
for i=xx(knot_vectorx(bi)):xx(knot_vectorx(bi+px+1))
% scale coordinates [1-width] -> [0-1]
ii = (i-1)/(ix-1);
% B^x_i(x)
funx_tab(bi,i) = compute_spline(knot_vectorx,px,bi,ii);
end
end
% loop over basis functions
for bj = 1:ny
% loop over nonzero pixels over given function
for j=yy(knot_vectory(bj)):yy(knot_vectory(bj+py+1))
% scale coordinates [1-height] -> [0-1]
jj = (j-1)/(iy-1);
% B^y_j(y)
funy_tab(bj,j) = compute_spline(knot_vectory,py,bj,jj);
end
end
preprocess=toc
tic;
% image reconstruction
% loop over functions
for bi = 1:nx
for bj = 1:ny
% loop over nonzero pixels over given function
for i=xx(knot_vectorx(bi)):xx(knot_vectorx(bi+px+1))
for j=yy(knot_vectory(bj)):yy(knot_vectory(bj+py+1))
% B^x_i(x)
funi = funx_tab(bi,i);
% B^y_j(y)
funj = funy_tab(bj,j);
ff = funi*funj;
R1(i,j) = R1(i,j) + floor(ff*RR(bi,bj));
G1(i,j) = G1(i,j) + floor(ff*GG(bi,bj));
B1(i,j) = B1(i,j) + floor(ff*BB(bi,bj));
end
end
end
end
RGB=X;
RGB(:,:,1) = R1;
RGB(:,:,2) = G1;
RGB(:,:,3) = B1;
rebuild=toc
imshow(RGB);
% Subroutine to solve one direction as 1D problem with multiple RHS
function [RR,GG,BB]=solve_direction(A,FR,FG,FB)
% compute LU factorization of A matrix
[L,U,P,Q] = lu(A);
Q1=Q';
RR = zeros(size(FR,1),size(FR,2));
GG = zeros(size(FG,1),size(FG,2));
BB = zeros(size(FB,1),size(FB,2));
% loop over multiple RHS and color components
for i=1:size(FR,2)
RR(:,i)=solveRHS(L,U,P,Q1,FR(:,i));
GG(:,i)=solveRHS(L,U,P,Q1,FG(:,i));
BB(:,i)=solveRHS(L,U,P,Q1,FB(:,i));
end
end
% Solves single RHS problem for predone LU factorization
function res=solveRHS(L,U,P,Q1,b)
y1 = L\(P*b);
y2=U\y1;
res=Q1\y2;
end
% Scales [0-1] back to pixel coordinates
function resx=xx(x)
resx = floor((ix-1)*x+1);
end
% Scales [0-1] back to pixel coordinates
function resy=yy(y)
resy = floor((iy-1)*y+1);
end
% Helper subroutine for integration over bitmap
function val=bitmp(M,x,y)
val = zeros(size(x));
for i=1:size(x,1)
for j=1:size(x,1)
val(i,j)=M(xx(x(1,i)),yy(y(1,j)));
end
end
end
% Subroutine computing order of polynomials
function p=compute_p(knot_vector)
% first entry in knot_vector
initial = knot_vector(1);
% lenght of knot_vector
kvsize = size(knot_vector,2);
p = 0;
% checking number of repetitions of first entry in knot_vector
while (p+2 <= kvsize) && (initial == knot_vector(p+2))
p = p+1;
end
return
end
% Subroutine checking sanity of knot_vector
function t=check_sanity(knot_vector,p)
initial = knot_vector(1);
kvsize = size(knot_vector,2);
t = true;
counter = 1;
% if number of repeated knots at the beginning of knot_vector doesn't match polynomial order
for i=1:p+1
if (initial ~= knot_vector(i))
% return FALSE
t = false;
return
end
end
% if there are too many repeated knots in the middle of knot_vector
for i=p+2:kvsize-p-1
if (initial == knot_vector(i))
counter = counter + 1;
if (counter > p)
% return FALSE
t = false;
return
end
else
initial = knot_vector(i);
counter = 1;
end
end
initial = knot_vector(kvsize);
% if number of repeated knots at the end of knot_vector doesn't match polynomial order
for i=kvsize-p:kvsize
if (initial ~= knot_vector(i))
% return FALSE
t = false;
return
end
end
% if subsequent element in knot_vector is smaller than previous one
for i=1:kvsize-1
if (knot_vector(i)>knot_vector(i+1))
% return FALSE
t = false;
return
end
end
return
end
% Subroutine computing basis functions according to recursive Cox-de-Boor formulae
function y=compute_spline(knot_vector,p,nr,x)
% function (x-x_i)/(x_{i-p}-x_i)
fC= @(x,a,b) (x-a)/(b-a);
% function (x_{i+p+1}-x)/(x_{i+p+1}-x_{i+1})
fD= @(x,c,d) (d-x)/(d-c);
% x_i
a = knot_vector(nr);
% x_{i-p}
b = knot_vector(nr+p);
% x_{i+1}
c = knot_vector(nr+1);
% x_{i+p+1}
d = knot_vector(nr+p+1);
% linear function for p=0
if (p==0)
y = 0 .* (x < a) + 1 .* (a <= x & x <= d) + 0 .* (x > d);
return
end
% B_{i,p-1}
lp = compute_spline(knot_vector,p-1,nr,x);
% B_{i+1,p-1}
rp = compute_spline(knot_vector,p-1,nr+1,x);
% (x-x_i)/(x_{i-p)-x_i)*B_{i,p-1}
if (a==b)
% if knots in knot_vector are repeated we have to include it in formula
y1 = 0 .* (x < a) + 1 .* (a <= x & x <= b) + 0 .* (x > b);
else
y1 = 0 .* (x < a) + fC(x,a,b) .* (a <= x & x <= b) + 0 .* (x > b);
end
% (x_{i+p+1}-x)/(x_{i+p+1)-x_{i+1})*B_{i+1,p-1}
if (c==d)
% if knots in knot_vector are repeated we have to include it in formula
y2 = 0 .* (x < c) + 1 .* (c < x & x <= d) + 0 .* (d < x);
else
y2 = 0 .* (x < c) + fD(x,c,d) .* (c < x & x <= d) + 0 .* (d < x);
end
y = lp .* y1 + rp .* y2;
return
end
% Computes number of elements in given knot_vector
function n=number_of_elements(knot_vector,p)
initial = knot_vector(1);
kvsize = size(knot_vector,2);
n = 0;
for i=1:kvsize-1
if (knot_vector(i) ~= initial)
initial = knot_vector(i);
n = n+1;
end
end
end
% Creates simple knot_vector without repetitions in the middle
function knot=simple_knot(elems, p)
pad = ones(1, p);
knot = [0 * pad, 0:elems, elems * pad];
knot = knot/elems;
end
% Computes number of degrees of freedom over give knot_vector
function n=number_of_dofs(knot,p)
n = length(knot) - p - 1;
end
% Finds index of first knot in knot_vector related to give element
function first=first_dof_on_element(knot_vector,p,elem_number)
[l,h] = element_boundary(knot_vector,p,elem_number);
first = find(knot_vector==l, 1, 'last') - p;
end
% Finds lower and higher boundary of element
function [low,high]=element_boundary(knot_vector,p,elem_number)
initial = knot_vector(1);
kvsize = size(knot_vector,2);
k = 0;
low=0;
high=0;
for i=1:kvsize
if (knot_vector(i) ~= initial)
initial = knot_vector(i);
k = k+1;
end
if (k == elem_number)
low = knot_vector(i-1);
high = knot_vector(i);
return;
end
end
end
% Returns range (indexes) of nonzero functions over element on given knot_vector
function [low,high]=dofs_on_element(knot_vector,p,elem_number)
low = first_dof_on_element(knot_vector,p,elem_number);
% we expect exactly p+1 nonzero functions over element
high = low + p;
end
% Row vector of points of the k-point Gaussian quadrature on [a, b]
function xs=quad_points(a, b, k)
% mapping points
map = @(x) 0.5 * (a * (1 - x) + b * (x + 1));
switch (k)
case 1
xs = [0];
case 2
xs = [-1/sqrt(3), ...
1/sqrt(3)];
case 3
xs = [-sqrt(3/5), ...
0, ...
sqrt(3/5)];
case 4
xs = [-sqrt((3+2*sqrt(6/5))/7), ...
sqrt((3-2*sqrt(6/5))/7), ...
sqrt((3-2*sqrt(6/5))/7), ...
sqrt((3+2*sqrt(6/5))/7)];
case 5
xs = [-1/3*sqrt(5+2*sqrt(10/7)), ...
-1/3*sqrt(5-2*sqrt(10/7)), ...
0, ...
1/3*sqrt(5-2*sqrt(10/7)), ...
1/3*sqrt(5+2*sqrt(10/7))];
otherwise
xs = [-1/3*sqrt(5+2*sqrt(10/7)), ...
-1/3*sqrt(5-2*sqrt(10/7)), ...
0, ...
1/3*sqrt(5-2*sqrt(10/7)), ...
1/3*sqrt(5+2*sqrt(10/7))];
end
xs = map(xs);
end
% Row vector of weights of the k-point Gaussian quadrature on [a, b]
function ws=quad_weights(a, b, k)
switch (k)
case 1
ws = [2];
case 2
ws = [1, 1];
case 3
ws = [5/9, ...
8/9, ...
5/9];
case 4
ws = [(18-sqrt(30))/36, ...
(18+sqrt(30))/36, ...
(18+sqrt(30))/36, ...
(18-sqrt(30))/36];
case 5
ws = [(322-13.0*sqrt(70))/900, ...
(322+13.0*sqrt(70))/900, ...
128/225, ...
(322+13.0*sqrt(70))/900, ...
(322-13.0*sqrt(70))/900];
otherwise
ws = [(322-13.0*sqrt(70))/900, ...
(322+13.0*sqrt(70))/900, ...
128/225, ...
(322+13.0*sqrt(70))/900, ...
(322-13.0*sqrt(70))/900];
end
end
end
Listing 1 (Pobierz): MATLAB code of bitmap projection with alternating-directions solver.
Autorzy kodów w MATLABie: Marcin Łoś i Maciej Woźniak.