Question

I have a list of points moving in two dimensions (x– and y-axis) represented as rows in an array. I might have N points – i.e., N rows:

1 t1 x1 y1
2 t2 x2 y2
. 
.
.
N tN xN yN

where ti, xi, and yi, is the time-index, x-coordinate, and the y-coordinate for point i. The time index-index ti is an integer from 1 to T. The number of points at each such possible time index can vary from 0 to N (still with only N points in total).

My goal is the filter out all the points that do not move in a certain way; or to keep only those that do. A point must move in a parabolic trajectory – with decreasing x– and y-coordinate (i.e., moving to the left and downwards only). Points with other dynamic behaviour must be removed.

Can I use a simple sorting mechanism on this array – and then analyse the order of the time-index? I have also considered the fact each point having the same time-index ti are physically distinct points, and so should be paired up with other points. The complexity of the problem grew – and now I turn to you.

NOTE: You can assume that the points are confined to a sub-region of the (x,y)-plane between two parabolic curves. These curves intersect only at only at one point: A point close to the origin of motion for any point.

More Information:

I have made some datafiles available:

• MATLAB datafile (1.17 kB)
• same data as CSV with semicolon as column separator (2.77 kB)

Necessary context:

The datafile hold one uint32 array with 176 rows and 5 columns. The columns are:

1. pixel x-coordinate in 175-by-175 lattice
2. pixel y-coordinate in 175-by-175 lattice
3. discrete theta angle-index
4. time index (from 1 to T = 10)
5. row index for this original sorting

The points “live” in a 175-by-175 pixel-lattice – and again inside the upper quadrant of a circle with radius 175. The points travel on the circle circumference in a counterclockwise rotation to a certain angle theta with horizontal, where they are thrown off into something close to a parabolic orbit. Column 3 holds a discrete index into a list with indices 1 to 45 from 0 to 90 degress (one index thus spans 2 degrees). The theta-angle was originally deduces solely from the points by setting up the trivial equations of motions and solving for the angle. This gives rise to a quasi-symmetric quartic which can be solved in close-form. The actual metric radius of the circle is 0.2 m and the pixel coordinate were converted from pixel-coordinate to metric using simple linear interpolation (but what we see here are the points in original pixel-space).

My problem is that some points are not behaving properly and since I need to statistics on the theta angle, I need to remove the points that certainly do NOT move in a parabolic trajoctory. These error are expected and fully natural, but still need to be filtered out.

MATLAB plot code:

% load data and setup variables:
load mat_points.mat;
num_r = 175;
num_T = 10;
num_gridN = 20;

% begin plotting:
figure(1000);
clf;
plot( ...
   num_r * cos(0:0.1:pi/2), ...
   num_r * sin(0:0.1:pi/2), ...
   'Color', 'k', ...
   'LineWidth', 2 ...
);
axis equal;
xlim([0 num_r]);
ylim([0 num_r]);
hold all;

% setup grid (yea... went crazy with one):
vec_tickValues = linspace(0, num_r, num_gridN);
cell_tickLabels = repmat({''}, size(vec_tickValues));
cell_tickLabels{1} = sprintf('%u', vec_tickValues(1));
cell_tickLabels{end} = sprintf('%u', vec_tickValues(end));
set(gca, 'XTick', vec_tickValues);
set(gca, 'XTickLabel', cell_tickLabels);
set(gca, 'YTick', vec_tickValues);
set(gca, 'YTickLabel', cell_tickLabels);
set(gca, 'GridLineStyle', '-');
grid on;

% plot points per timeindex (with increasing brightness):
vec_grayIndex = linspace(0,0.9,num_T);
for num_kt = 1:num_T
   vec_xCoords = mat_points((mat_points(:,4) == num_kt), 1);
   vec_yCoords = mat_points((mat_points(:,4) == num_kt), 2);
   plot(vec_xCoords, vec_yCoords, 'o', ...
      'MarkerEdgeColor', 'k', ...
      'MarkerFaceColor', vec_grayIndex(num_kt) * ones(1,3) ...
   );
end

Thanks 🙂

Answer 1

Why, it looks almost as if you’re simulating a radar tracking debris from the collision of two missiles…

Anyway, let’s coin a new term: object. Objects are moving along parabolae and at certain times they may emit flashes that appear as points. There are also other points which we are trying to filter out.

We will need some more information:

1. Can we assume that the objects obey the physics of things falling under gravity?
2. Must every object emit a point at every timestep during its lifetime?
3. Speaking of lifetime, do all objects begin at the same time? Can some expire before others?
4. How precise is the data? Is it exact? Is there a measure of error? To put it another way, do we understand how poorly the points from an object might fit a perfect parabola?