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% vq_binary_switch.m
% David Rowe Sep 2021
%
% Experiments in making VQs robust to bit errors, this is an Octave
% implementation of [1].
%
% [1] Pseudo Gray Coding, Zeger & Gersho 1990
1;
% returns indexes of hamming distance 1 neighbours
function index_neighbours = distance_one_neighbours(N,k)
log2N = log2(N);
index_neighbours = [];
for b=0:log2N-1
index_neighbour = bitxor(k-1,2.^b) + 1;
index_neighbours = [index_neighbours index_neighbour];
end
end
% equation (33) of [1], for hamming distance 1
function c = cost_of_distance_one(vq, prob, k, verbose=0)
[N K] = size(vq);
log2N = log2(N);
c = 0;
for b=0:log2N-1
index_neighbour = bitxor(k-1,2.^b) + 1;
diff = vq(k,:) - vq(index_neighbour, :);
dist = sum(diff*diff');
c += prob(k)*dist;
if verbose
printf("k: %d b: %d index_neighbour: %d dist: %f prob: %f c: %f \n", k, b, index_neighbour, dist, prob(k), c);
end
end
endfunction
% equation (39) of [1]
function d = distortion_of_current_mapping(vq, prob, verbose=0)
[N K] = size(vq);
d = 0;
for k=1:N
c = cost_of_distance_one(vq, prob, k);
d += c;
if verbose
printf("k: %2d c: %f d: %f\n", k, c, d);
end
end
endfunction
function [vq distortion] = binary_switching(vq, prob, max_iteration, fast_en=1)
[N K] = size(vq);
iteration = 0;
i = 1;
finished = 0;
switches = 0;
distortion0 = distortion_of_current_mapping(vq, prob)
while !finished
% generate a list A(i) of which vectors have the largest cost of bit errors
c = zeros(1,N);
for k=1:N
c(k) = cost_of_distance_one(vq, prob, k);
end
[tmp A] = sort(c,"descend");
% Try switching each vector with A(i)
best_delta = 0;
for j=2:N
% we can't switch with ourself
if j != A(i)
if fast_en
delta = -cost_of_distance_one(vq, prob, A(i)) - cost_of_distance_one(vq, prob, j);
n1 = [distance_one_neighbours(N,A(i)) distance_one_neighbours(N,j)];
n1(n1 == A(i)) = [];
n1(n1 == j) = [];
for l=1:length(n1)
delta -= cost_of_distance_one(vq, prob, n1(l));
end
else
distortion1 = distortion_of_current_mapping(vq, prob);
end
% switch vq entries A(i) and j
tmp = vq(A(i),:);
vq(A(i),:) = vq(j,:);
vq(j,:) = tmp;
if fast_en
delta += cost_of_distance_one(vq, prob, A(i)) + cost_of_distance_one(vq, prob, j);
for l=1:length(n1)
delta += cost_of_distance_one(vq, prob, n1(l));
end
else
distortion2 = distortion_of_current_mapping(vq, prob);
delta = distortion2 - distortion1;
end
if delta < 0
if abs(delta) > best_delta
best_delta = abs(delta);
best_j = j;
end
end
% unswitch
tmp = vq(A(i),:);
vq(A(i),:) = vq(j,:);
vq(j,:) = tmp;
end
end % next j
% printf("best_delta: %f best_j: %d\n", best_delta, best_j);
if best_delta == 0
% Hmm, no improvement, lets try the next vector in the sorted cost list
if i == N
finished = 1;
else
i++;
end
else
% OK keep the switch that minimised the distortion
tmp = vq(A(i),:);
vq(A(i),:) = vq(best_j,:);
vq(best_j,:) = tmp;
switches++;
% set up for next iteration
iteration++;
distortion = distortion_of_current_mapping(vq, prob);
printf("it: %3d dist: %f %3.2f i: %3d sw: %3d\n", iteration, distortion,
distortion/distortion0, i, switches);
if iteration >= max_iteration, finished = 1, end
i = 1;
end
end
endfunction
% return indexes of hamming distance one vectors
function ind = neighbour_indexes(vq, k)
[N K] = size(vq);
log2N = log2(N);
ind = [];
for b=0:log2N-1
index_neighbour = bitxor(k-1,2.^b) + 1;
ind = [ind index_neighbour];
end
endfunction
function test_binary_switch
vq1 = [1 1; -1 1; -1 -1; 1 -1];
%f=fopen("vq1.f32","wb"); fwrite(f, vq1, 'float32'); fclose(f);
[vq2 distortion] = binary_switching(vq1, ones(1,4), 10);
% algorithm should put hamming distance 1 neighbours in adjacent quadrants
distance_to_closest_neighbours = 2;
% there are two hamming distance 1 neighbours
target_distortion = 2^2*distance_to_closest_neighbours*length(vq1);
assert(target_distortion == distortion);
printf("test_binary_switch OK!\n");
endfunction
function test_fast
N=16; % Number of VQ codebook vectors
K=2; % Vector length
Ntrain=10000;
training_data = randn(Ntrain,K);
[idx vq1] = kmeans(training_data, N);
f=fopen("vq1.f32","wb");
for r=1:rows(vq1)
fwrite(f,vq1(r,:),"float32");
end
fclose(f);
[vq2 distortion] = binary_switching(vq1, [1 ones(1,N-1)], 1000, fast_en = 0);
[vq3 distortion] = binary_switching(vq1, [1 ones(1,N-1)], 1000, fast_en = 1);
assert(vq2 == vq3);
printf("test_fast OK!\n");
endfunction
function demo
N=16; % Number of VQ codebook vectors
K=2; % Vector length
Ntrain=10000;
training_data = randn(Ntrain,K);
[idx vq1] = kmeans(training_data, N);
[vq2 distortion] = binary_switching(vq1, [1 ones(1,N-1)], 1000, 1);
figure(1); clf; plot(training_data(:,1), training_data(:,2),'+');
hold on;
plot(vq1(:,1), vq1(:,2),'og','linewidth', 2);
plot(vq2(:,1), vq2(:,2),'or','linewidth', 2);
% plot hamming distance 1 neighbours
k = 1;
ind = neighbour_indexes(vq2, k);
for i=1:length(ind)
plot([vq2(k,1) vq2(ind(i),1)],[vq2(k,2) vq2(ind(i),2)],'r-','linewidth', 2);
end
hold off;
endfunction
pkg load statistics
%test_binary_switch;
test_fast;
%demo
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