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inpaint_nans.m
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inpaint_nans.m
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function B=inpaint_nans(A,method)
% Author: John D'Errico
% e-mail address: woodchips@rochester.rr.com
% Release: 2
% Release date: 4/15/06
% INPAINT_NANS: in-paints over nans in an array
% usage: B=INPAINT_NANS(A) % default method
% usage: B=INPAINT_NANS(A,method) % specify method used
%
% Solves approximation to one of several pdes to
% interpolate and extrapolate holes in an array
%
% arguments (input):
% A - nxm array with some NaNs to be filled in
%
% method - (OPTIONAL) scalar numeric flag - specifies
% which approach (or physical metaphor to use
% for the interpolation.) All methods are capable
% of extrapolation, some are better than others.
% There are also speed differences, as well as
% accuracy differences for smooth surfaces.
%
% methods {0,1,2} use a simple plate metaphor.
% method 3 uses a better plate equation,
% but may be much slower and uses
% more memory.
% method 4 uses a spring metaphor.
% method 5 is an 8 neighbor average, with no
% rationale behind it compared to the
% other methods. I do not recommend
% its use.
%
% method == 0 --> (DEFAULT) see method 1, but
% this method does not build as large of a
% linear system in the case of only a few
% NaNs in a large array.
% Extrapolation behavior is linear.
%
% method == 1 --> simple approach, applies del^2
% over the entire array, then drops those parts
% of the array which do not have any contact with
% NaNs. Uses a least squares approach, but it
% does not modify known values.
% In the case of small arrays, this method is
% quite fast as it does very little extra work.
% Extrapolation behavior is linear.
%
% method == 2 --> uses del^2, but solving a direct
% linear system of equations for nan elements.
% This method will be the fastest possible for
% large systems since it uses the sparsest
% possible system of equations. Not a least
% squares approach, so it may be least robust
% to noise on the boundaries of any holes.
% This method will also be least able to
% interpolate accurately for smooth surfaces.
% Extrapolation behavior is linear.
%
% method == 3 --+ See method 0, but uses del^4 for
% the interpolating operator. This may result
% in more accurate interpolations, at some cost
% in speed.
%
% method == 4 --+ Uses a spring metaphor. Assumes
% springs (with a nominal length of zero)
% connect each node with every neighbor
% (horizontally, vertically and diagonally)
% Since each node tries to be like its neighbors,
% extrapolation is as a constant function where
% this is consistent with the neighboring nodes.
%
% method == 5 --+ See method 2, but use an average
% of the 8 nearest neighbors to any element.
% This method is NOT recommended for use.
%
%
% arguments (output):
% B - nxm array with NaNs replaced
%
%
% Example:
% [x,y] = meshgrid(0:.01:1);
% z0 = exp(x+y);
% znan = z0;
% znan(20:50,40:70) = NaN;
% znan(30:90,5:10) = NaN;
% znan(70:75,40:90) = NaN;
%
% z = inpaint_nans(znan);
%
%
% See also: griddata, interp1
%
% Author: John D'Errico
% e-mail address: woodchips@rochester.rr.com
% Release: 2
% Release date: 4/15/06
% I always need to know which elements are NaN,
% and what size the array is for any method
[n,m]=size(A);
A=A(:);
nm=n*m;
k=isnan(A(:));
% list the nodes which are known, and which will
% be interpolated
nan_list=find(k);
known_list=find(~k);
% how many nans overall
nan_count=length(nan_list);
% convert NaN indices to (r,c) form
% nan_list==find(k) are the unrolled (linear) indices
% (row,column) form
[nr,nc]=ind2sub([n,m],nan_list);
% both forms of index in one array:
% column 1 == unrolled index
% column 2 == row index
% column 3 == column index
nan_list=[nan_list,nr,nc];
% supply default method
if (nargin<2) || isempty(method)
method = 0;
elseif ~ismember(method,0:5)
error 'If supplied, method must be one of: {0,1,2,3,4,5}.'
end
% for different methods
switch method
case 0
% The same as method == 1, except only work on those
% elements which are NaN, or at least touch a NaN.
% horizontal and vertical neighbors only
talks_to = [-1 0;0 -1;1 0;0 1];
neighbors_list=identify_neighbors(n,m,nan_list,talks_to);
% list of all nodes we have identified
all_list=[nan_list;neighbors_list];
% generate sparse array with second partials on row
% variable for each element in either list, but only
% for those nodes which have a row index > 1 or < n
L = find((all_list(:,2) > 1) & (all_list(:,2) < n));
nl=length(L);
if nl>0
fda=sparse(repmat(all_list(L,1),1,3), ...
repmat(all_list(L,1),1,3)+repmat([-1 0 1],nl,1), ...
repmat([1 -2 1],nl,1),nm,nm);
else
fda=spalloc(n*m,n*m,size(all_list,1)*5);
end
% 2nd partials on column index
L = find((all_list(:,3) > 1) & (all_list(:,3) < m));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(all_list(L,1),1,3), ...
repmat(all_list(L,1),1,3)+repmat([-n 0 n],nl,1), ...
repmat([1 -2 1],nl,1),nm,nm);
end
% eliminate knowns
rhs=-fda(:,known_list)*double(A(known_list));
k=find(any(fda(:,nan_list(:,1)),2));
% and solve...
B=A;
B(nan_list(:,1))=fda(k,nan_list(:,1))\rhs(k);
case 1
% least squares approach with del^2. Build system
% for every array element as an unknown, and then
% eliminate those which are knowns.
% Build sparse matrix approximating del^2 for
% every element in A.
% Compute finite difference for second partials
% on row variable first
[i,j]=ndgrid(2:(n-1),1:m);
ind=i(:)+(j(:)-1)*n;
np=(n-2)*m;
fda=sparse(repmat(ind,1,3),[ind-1,ind,ind+1], ...
repmat([1 -2 1],np,1),n*m,n*m);
% now second partials on column variable
[i,j]=ndgrid(1:n,2:(m-1));
ind=i(:)+(j(:)-1)*n;
np=n*(m-2);
fda=fda+sparse(repmat(ind,1,3),[ind-n,ind,ind+n], ...
repmat([1 -2 1],np,1),nm,nm);
% eliminate knowns
rhs=-fda(:,known_list)*double(A(known_list));
k=find(any(fda(:,nan_list),2));
% and solve...
B=A;
B(nan_list(:,1))=fda(k,nan_list(:,1))\rhs(k);
case 2
% Direct solve for del^2 BVP across holes
% generate sparse array with second partials on row
% variable for each nan element, only for those nodes
% which have a row index > 1 or < n
L = find((nan_list(:,2) > 1) & (nan_list(:,2) < n));
nl=length(L);
if nl>0
fda=sparse(repmat(nan_list(L,1),1,3), ...
repmat(nan_list(L,1),1,3)+repmat([-1 0 1],nl,1), ...
repmat([1 -2 1],nl,1),n*m,n*m);
else
fda=spalloc(n*m,n*m,size(nan_list,1)*5);
end
% 2nd partials on column index
L = find((nan_list(:,3) > 1) & (nan_list(:,3) < m));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,3), ...
repmat(nan_list(L,1),1,3)+repmat([-n 0 n],nl,1), ...
repmat([1 -2 1],nl,1),n*m,n*m);
end
% fix boundary conditions at extreme corners
% of the array in case there were nans there
if ismember(1,nan_list(:,1))
fda(1,[1 2 n+1])=[-2 1 1];
end
if ismember(n,nan_list(:,1))
fda(n,[n, n-1,n+n])=[-2 1 1];
end
if ismember(nm-n+1,nan_list(:,1))
fda(nm-n+1,[nm-n+1,nm-n+2,nm-n])=[-2 1 1];
end
if ismember(nm,nan_list(:,1))
fda(nm,[nm,nm-1,nm-n])=[-2 1 1];
end
% eliminate knowns
rhs=-fda(:,known_list)*double(A(known_list));
% and solve...
B=A;
k=nan_list(:,1);
B(k)=fda(k,k)\rhs(k);
case 3
% The same as method == 0, except uses del^4 as the
% interpolating operator.
% del^4 template of neighbors
talks_to = [-2 0;-1 -1;-1 0;-1 1;0 -2;0 -1; ...
0 1;0 2;1 -1;1 0;1 1;2 0];
neighbors_list=identify_neighbors(n,m,nan_list,talks_to);
% list of all nodes we have identified
all_list=[nan_list;neighbors_list];
% generate sparse array with del^4, but only
% for those nodes which have a row & column index
% >= 3 or <= n-2
L = find( (all_list(:,2) >= 3) & ...
(all_list(:,2) <= (n-2)) & ...
(all_list(:,3) >= 3) & ...
(all_list(:,3) <= (m-2)));
nl=length(L);
if nl>0
% do the entire template at once
fda=sparse(repmat(all_list(L,1),1,13), ...
repmat(all_list(L,1),1,13) + ...
repmat([-2*n,-n-1,-n,-n+1,-2,-1,0,1,2,n-1,n,n+1,2*n],nl,1), ...
repmat([1 2 -8 2 1 -8 20 -8 1 2 -8 2 1],nl,1),nm,nm);
else
fda=spalloc(n*m,n*m,size(all_list,1)*5);
end
% on the boundaries, reduce the order around the edges
L = find((((all_list(:,2) == 2) | ...
(all_list(:,2) == (n-1))) & ...
(all_list(:,3) >= 2) & ...
(all_list(:,3) <= (m-1))) | ...
(((all_list(:,3) == 2) | ...
(all_list(:,3) == (m-1))) & ...
(all_list(:,2) >= 2) & ...
(all_list(:,2) <= (n-1))));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(all_list(L,1),1,5), ...
repmat(all_list(L,1),1,5) + ...
repmat([-n,-1,0,+1,n],nl,1), ...
repmat([1 1 -4 1 1],nl,1),nm,nm);
end
L = find( ((all_list(:,2) == 1) | ...
(all_list(:,2) == n)) & ...
(all_list(:,3) >= 2) & ...
(all_list(:,3) <= (m-1)));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(all_list(L,1),1,3), ...
repmat(all_list(L,1),1,3) + ...
repmat([-n,0,n],nl,1), ...
repmat([1 -2 1],nl,1),nm,nm);
end
L = find( ((all_list(:,3) == 1) | ...
(all_list(:,3) == m)) & ...
(all_list(:,2) >= 2) & ...
(all_list(:,2) <= (n-1)));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(all_list(L,1),1,3), ...
repmat(all_list(L,1),1,3) + ...
repmat([-1,0,1],nl,1), ...
repmat([1 -2 1],nl,1),nm,nm);
end
% eliminate knowns
rhs=-fda(:,known_list)*double(A(known_list));
k=find(any(fda(:,nan_list(:,1)),2));
% and solve...
B=A;
B(nan_list(:,1))=fda(k,nan_list(:,1))\rhs(k);
case 4
% Spring analogy
% interpolating operator.
% list of all springs between a node and a horizontal
% or vertical neighbor
hv_list=[-1 -1 0;1 1 0;-n 0 -1;n 0 1];
hv_springs=[];
for i=1:4
hvs=nan_list+repmat(hv_list(i,:),nan_count,1);
k=(hvs(:,2)>=1) & (hvs(:,2)<=n) & (hvs(:,3)>=1) & (hvs(:,3)<=m);
hv_springs=[hv_springs;[nan_list(k,1),hvs(k,1)]];
end
% delete replicate springs
hv_springs=unique(sort(hv_springs,2),'rows');
% build sparse matrix of connections, springs
% connecting diagonal neighbors are weaker than
% the horizontal and vertical springs
nhv=size(hv_springs,1);
springs=sparse(repmat((1:nhv)',1,2),hv_springs, ...
repmat([1 -1],nhv,1),nhv,nm);
% eliminate knowns
rhs=-springs(:,known_list)*double(A(known_list));
% and solve...
B=A;
B(nan_list(:,1))=springs(:,nan_list(:,1))\rhs;
case 5
% Average of 8 nearest neighbors
% generate sparse array to average 8 nearest neighbors
% for each nan element, be careful around edges
fda=spalloc(n*m,n*m,size(nan_list,1)*9);
% -1,-1
L = find((nan_list(:,2) > 1) & (nan_list(:,3) > 1));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([-n-1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% 0,-1
L = find(nan_list(:,3) > 1);
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([-n, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% +1,-1
L = find((nan_list(:,2) < n) & (nan_list(:,3) > 1));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([-n+1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% -1,0
L = find(nan_list(:,2) > 1);
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([-1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% +1,0
L = find(nan_list(:,2) < n);
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% -1,+1
L = find((nan_list(:,2) > 1) & (nan_list(:,3) < m));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([n-1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% 0,+1
L = find(nan_list(:,3) < m);
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([n, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% +1,+1
L = find((nan_list(:,2) < n) & (nan_list(:,3) < m));
nl=length(L);
if nl>0
fda=fda+sparse(repmat(nan_list(L,1),1,2), ...
repmat(nan_list(L,1),1,2)+repmat([n+1, 0],nl,1), ...
repmat([1 -1],nl,1),n*m,n*m);
end
% eliminate knowns
rhs=-fda(:,known_list)*double(A(known_list));
% and solve...
B=A;
k=nan_list(:,1);
B(k)=fda(k,k)\rhs(k);
end
% all done, make sure that B is the same shape as
% A was when we came in.
B=reshape(B,n,m);
% ====================================================
% end of main function
% ====================================================
% ====================================================
% begin subfunctions
% ====================================================
function neighbors_list=identify_neighbors(n,m,nan_list,talks_to)
% identify_neighbors: identifies all the neighbors of
% those nodes in nan_list, not including the nans
% themselves
%
% arguments (input):
% n,m - scalar - [n,m]=size(A), where A is the
% array to be interpolated
% nan_list - array - list of every nan element in A
% nan_list(i,1) == linear index of i'th nan element
% nan_list(i,2) == row index of i'th nan element
% nan_list(i,3) == column index of i'th nan element
% talks_to - px2 array - defines which nodes communicate
% with each other, i.e., which nodes are neighbors.
%
% talks_to(i,1) - defines the offset in the row
% dimension of a neighbor
% talks_to(i,2) - defines the offset in the column
% dimension of a neighbor
%
% For example, talks_to = [-1 0;0 -1;1 0;0 1]
% means that each node talks only to its immediate
% neighbors horizontally and vertically.
%
% arguments(output):
% neighbors_list - array - list of all neighbors of
% all the nodes in nan_list
if ~isempty(nan_list)
% use the definition of a neighbor in talks_to
nan_count=size(nan_list,1);
talk_count=size(talks_to,1);
nn=zeros(nan_count*talk_count,2);
j=[1,nan_count];
for i=1:talk_count
nn(j(1):j(2),:)=nan_list(:,2:3) + ...
repmat(talks_to(i,:),nan_count,1);
j=j+nan_count;
end
% drop those nodes which fall outside the bounds of the
% original array
L = (nn(:,1)<1)|(nn(:,1)>n)|(nn(:,2)<1)|(nn(:,2)>m);
nn(L,:)=[];
% form the same format 3 column array as nan_list
neighbors_list=[sub2ind([n,m],nn(:,1),nn(:,2)),nn];
% delete replicates in the neighbors list
neighbors_list=unique(neighbors_list,'rows');
% and delete those which are also in the list of NaNs.
neighbors_list=setdiff(neighbors_list,nan_list,'rows');
else
neighbors_list=[];
end