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Updated scaling factors for proper geometric representation of uncertainty ellipses #1067

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merged 4 commits into from
Apr 12, 2022
Merged

Updated scaling factors for proper geometric representation of uncertainty ellipses #1067

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d9a00de
Modified the scaling values for plotting uncertainty ellipses to prop…
magicbycalvin Jan 25, 2022
1b81776
Changed all instances of the Sigma value, k, to 5 for plotting the co…
magicbycalvin Jan 28, 2022
e524e28
Updated comments to reflect standard deviations instead of variances …
magicbycalvin Jan 28, 2022
df7fb47
Merge branch 'fix/ellipses' into develop
dellaert Apr 12, 2022
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52 changes: 37 additions & 15 deletions python/gtsam/utils/plot.py
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Original file line number Diff line number Diff line change
Expand Up @@ -75,8 +75,11 @@ def plot_covariance_ellipse_3d(axes,
Plots a Gaussian as an uncertainty ellipse

Based on Maybeck Vol 1, page 366
k=2.296 corresponds to 1 std, 68.26% of all probability
k=11.82 corresponds to 3 std, 99.74% of all probability
For the 3D case:
k = 3.527 corresponds to 1 std, 68.26% of all probability
k = 14.157 corresponds to 3 std, 99.74% of all probability
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Do you have an online reference?

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@senselessDev senselessDev Jan 26, 2022 •
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With code from other comment:

pct_to_sigma(sigma_to_pct(1, 1), 3) ** 2 --> 3.5267403802617303
pct_to_sigma(sigma_to_pct(3, 1), 3) ** 2 --> 14.156413609126677

So seems about right if we want this behavior.

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I think these (3.5 and 14.1) should still be changed to standard deviation


We choose k = 5 which corresponds to 99.99846% of all probability in 3D

Args:
axes (matplotlib.axes.Axes): Matplotlib axes.
Expand All @@ -87,7 +90,8 @@ def plot_covariance_ellipse_3d(axes,
n: Defines the granularity of the ellipse. Higher values indicate finer ellipses.
alpha: Transparency value for the plotted surface in the range [0, 1].
"""
k = 11.82
# Sigma value corresponding to the covariance ellipse
k = 5
U, S, _ = np.linalg.svd(P)

radii = k * np.sqrt(S)
Expand All @@ -113,7 +117,16 @@ def plot_point2_on_axes(axes,
linespec: str,
P: Optional[np.ndarray] = None) -> None:
"""
Plot a 2D point on given axis `axes` with given `linespec`.
Plot a 2D point and its corresponding uncertainty ellipse on given axis
`axes` with given `linespec`.

Based on Stochastic Models, Estimation, and Control Vol 1 by Maybeck,
page 366
For the 2D case:
k = 2.296 corresponds to 1 std, 68.26% of all probability
k = 11.820 corresponds to 3 std, 99.74% of all probability
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@senselessDev senselessDev Jan 26, 2022 •
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pct_to_sigma(sigma_to_pct(1, 1), 2) ** 2 --> 2.295748928898636
pct_to_sigma(sigma_to_pct(3, 1), 2) ** 2 --> 11.829158081900795

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same


We choose k = 5 which corresponds to 99.99963% of all probability for 2D.

Args:
axes (matplotlib.axes.Axes): Matplotlib axes.
Expand All @@ -125,16 +138,15 @@ def plot_point2_on_axes(axes,
if P is not None:
w, v = np.linalg.eig(P)

# "Sigma" value for drawing the uncertainty ellipse. 5 sigma corresponds
# to a 99.9999% confidence, i.e. assuming the estimation has been
# computed properly, there is a 99.999% chance that the true position
# of the point will lie within the uncertainty ellipse.
k = 5.0
# Sigma value corresponding to the covariance ellipse
k = 5

angle = np.arctan2(v[1, 0], v[0, 0])
# We multiply k by 2 since k corresponds to the radius but Ellipse uses
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Wow. I guess we were looking at the wrong ellipses all that time!

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On the bright side, your numbers would have still been correct!

The requested changes have been made. Please let me know if you need anything else.

Thanks!

# the diameter.
e1 = patches.Ellipse(point,
np.sqrt(w[0] * k),
np.sqrt(w[1] * k),
np.sqrt(w[0]) * 2 * k,
np.sqrt(w[1]) * 2 * k,
np.rad2deg(angle),
fill=False)
axes.add_patch(e1)
Expand Down Expand Up @@ -178,6 +190,14 @@ def plot_pose2_on_axes(axes,
"""
Plot a 2D pose on given axis `axes` with given `axis_length`.

Based on Stochastic Models, Estimation, and Control Vol 1 by Maybeck,
page 366
For the 2D case:
k = 2.296 corresponds to 1 std, 68.26% of all probability
k = 11.820 corresponds to 3 std, 99.74% of all probability
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same?


We choose k = 5 which corresponds to 99.99963% of all probability for 2D.

Args:
axes (matplotlib.axes.Axes): Matplotlib axes.
pose: The pose to be plotted.
Expand Down Expand Up @@ -205,13 +225,15 @@ def plot_pose2_on_axes(axes,

w, v = np.linalg.eig(gPp)

# k = 2.296
k = 5.0
# Sigma value corresponding to the covariance ellipse
k = 5

angle = np.arctan2(v[1, 0], v[0, 0])
# We multiply k by 2 since k corresponds to the radius but Ellipse uses
# the diameter.
e1 = patches.Ellipse(origin,
np.sqrt(w[0] * k),
np.sqrt(w[1] * k),
np.sqrt(w[0]) * 2 * k,
np.sqrt(w[1]) * 2 * k,
np.rad2deg(angle),
fill=False)
axes.add_patch(e1)
Expand Down