Signal duplicates/repeats in spatio-temporal filtering

I'm trying to implement spatio-temporal filtering in order to seperate moving objects (microbubbles in ultrasound localization microscopy) in an image series to enhance localization. While my implementation already manages to distort objects moving at a different velocity or in a different direction, duplicates of my objects are appearing after I filtered the image series.

I don't quite understand where these objects come from. I included a minimal example to replicate the output. I noticed that a lower value for sigma_t reduces the effect. However, this also reduces attenuation of objects moving at different velocities.

My questions are:

1. Why do these objects appear after filtering the image?
2. How can I remove them?

n_frames = 100;

ims = zeros(400,400,n_frames);

for ii=1:n_frames
    ims(200,ii,ii) = 1;
end

dx = 1;
dz = 1;
fs = 1;

vel = 1;   % filtering velocity
phi = 90;   % filtering angle

sigma_t = 100;  % gaussian filter sigma

[zDim,xDim,Nframes] = size(ims);
% fourier transform
IMS = fftshift(fftn(ims));

% spatial sampling frequencies
Kx = 2*pi/dx;
Kz = 2*pi/dz;
% step size for spatial frequencies
dkx = Kx/xDim;
dkz = Kz/zDim;
% temporal step size
dt = 1/fs;

Omega = 2*pi/dt;
% frequency step size
domega = Omega/Nframes;

% null-centered frequency vectors
kx = [-xDim/2:xDim/2-1]*dkx;
kz = [-zDim/2:zDim/2-1]*dkz;
omegas = [-Nframes/2:Nframes/2-1]*domega;

[kxx,kzz] = meshgrid(kx,kz);
% kx in the first layer, kz in the second
k_grid(:,:,1) = kxx;
k_grid(:,:,2) = kzz;
% velocity vector with x- and z-components
vx = vel * sind(phi);
vz = vel * cosd(phi);
vvect = [vx vz];
% vx in the first layer, vz in the second layer; same dimensions like
% spatial frequencies
vvect = permute(vvect,[3,1,2]);
v_grid = repmat(vvect,size(k_grid,1),size(k_grid,2)); 

nvect_matrix = sum(k_grid.*v_grid,3);

% adding time frequencies
nvect_array = repmat(nvect_matrix,1,1,Nframes);

% fitting time frequencies to image size
omegas_new(1,1,:) = omegas;
omegas_array = repmat(omegas_new,size(k_grid,1),size(k_grid,2));

% filter transfer function
W = exp(-(omegas_array + nvect_array).^2 * sigma_t^2 / 2);
       
% filtering by multiplication in frequency domain 
filtered_IMS = IMS .* W;
        
% inverse fourier transform
filtered_ims = abs(ifftn(ifftshift(filtered_IMS)));

Answer 1

[Gyan Vaibhav]

Hi Sebastian,

I understand that you are trying to implement spatio-temporal filtering to separate moving objects.

The appearance of duplicate objects after spatio-temporal filtering can be attributed to the side lobes of the filter's frequency response.

Here are a few possible reasons:

1. Filter Side Lobes: As mentioned above, the filter has side lobes that can let through unwanted frequencies, leading to artifacts.
2. Aliasing: If the sampling rate (fs) is not high enough to capture the motion properly, aliasing can occur, which might create duplicates or ghost images of the moving objects.

Here are a few possible ways to fix:

1. Adjust the Filter: You can try adjusting the filter's parameters, such as “sigma_t”, to find a balance between removing unwanted objects and retaining the desired ones.
2. Increase Sampling Rate: If aliasing is a problem, increasing the sampling rate (“fs”) can help capture the motion more accurately and reduce the chances of aliasing artifacts.
3. Bandpass Filtering: Implement a bandpass filter that only allows a narrow range of frequencies corresponding to the expected velocities of the microbubbles.

Hope this helps.

Thanks

Gyan