I suppose my solution has a weakness that should've been obvious to me. It fails if it has to do extrapolation to bridge a gap. In other words, if there is a gap at theta=0, there's nothing to suggest that the ends of the spline should meet. They'll just go off wherever the endslope takes them.
There's a workaround. It's not very elegant, but this is what I came up with in the time I had available at the moment.
load('x_and_y.mat')
x = a(:,1); y = a(:,2);
[th1,r1] = cart2pol(x,y);
b = [th1, r1];
b = rmoutliers(b,'percentiles',[0 97]);
th = b(:,1);
r = b(:,2);
[th, sortmap] = sort(th,'ascend');
r = r(sortmap);
% offset angles so that there isn't a gap at th=0
thoffset = pi/2;
th = mod(th+thoffset,2*pi)-pi; % shift angles
% one approach would be to fit a spline to both interpolate and denoise
newth = linspace(-pi,pi,1000)';
newth = mod(newth-thoffset,2*pi)-pi;
pp = fit(th,r,'smoothingspline','smoothingparam',1-1E-5); % feel free to adjust
newr = pp(newth);
% shift angles back
th = mod(th-thoffset,2*pi)-pi;
newth = mod(newth-thoffset,2*pi)-pi;
[newx, newy] = pol2cart(newth,newr);
subplot(2,1,1)
plot(th,r,'.','MarkerFaceColor',[0,0,1])
xlim([-4 4])
grid on
subplot(2,1,2)
plot(newth,newr,'.','MarkerFaceColor',[0,0,1])
xlim([-4 4])
grid on
figure
subplot(1,2,1)
plot(x,y,'.','MarkerFaceColor',[0,0,1])
xlim([-200 200])
ylim([-200 200])
grid on
axis equal
subplot(1,2,2)
plot(newx,newy,'.','MarkerFaceColor',[0,0,1])
xlim([-200 200])
ylim([-200 200])
grid on
axis equal
