def _annotate_kpts(kpts, sel_fx, draw_ell, draw_pts, color=None, nRandKpts=None, rect=False):
#print('[viz] _annotate_kpts()')
if color is None:
color = 'distinct' if sel_fx is None else df2.ORANGE
ell_args = {
'ell': draw_ell,
'pts': draw_pts,
'rect': rect,
'ell_alpha': .4,
'ell_linewidth': 2,
'ell_color': color,
}
if draw_ell and nRandKpts is not None:
# show a random sample of kpts
nkpts1 = len(kpts)
fxs1 = np.arange(nkpts1)
size = nRandKpts
replace = False
p = np.ones(nkpts1)
p = p / p.sum()
fxs_randsamp = np.random.choice(fxs1, size, replace, p)
kpts = kpts[fxs_randsamp]
# TODO Fix this. This should not set the xlabel
df2.set_xlabel('displaying %r/%r keypoints' % (nRandKpts, nkpts1))
elif draw_ell or draw_pts:
# draw all keypoints
df2.draw_kpts2(kpts, **ell_args)
if sel_fx is not None:
# Draw selected keypoint
sel_kpts = kpts[sel_fx:sel_fx + 1]
df2.draw_kpts2(sel_kpts, ell_color=df2.BLUE, arrow=True, rect=True)
def draw_keypoint_patch(rchip, kp, sift=None, warped=False, **kwargs):
#print('--------------------')
#print('[extract] Draw Patch')
if warped:
wpatch, wkp = get_warped_patch(rchip, kp)
patch = wpatch
subkp = wkp
else:
patch, subkp = get_patch(rchip, kp)
#print('[extract] kp = '+str(kp))
#print('[extract] subkp = '+str(subkp))
#print('[extract] patch.shape = %r' % (patch.shape,))
color = (0, 0, 1)
fig, ax = df2.imshow(patch, **kwargs)
df2.draw_kpts2([subkp], ell_color=color, pts=True)
if not sift is None:
df2.draw_sift(sift, [subkp])
return ax
def draw_keypoint_patch(rchip, kp, sift=None, warped=False, **kwargs):
#print('--------------------')
#print('[extract] Draw Patch')
if warped:
wpatch, wkp = get_warped_patch(rchip, kp)
patch = wpatch
subkp = wkp
else:
patch, subkp = get_patch(rchip, kp)
#print('[extract] kp = '+str(kp))
#print('[extract] subkp = '+str(subkp))
#print('[extract] patch.shape = %r' % (patch.shape,))
color = (0, 0, 1)
fig, ax = df2.imshow(patch, **kwargs)
df2.draw_kpts2([subkp], ell_color=color, pts=True)
if not sift is None:
df2.draw_sift(sift, [subkp])
return ax
def _annotate_kpts(kpts,
sel_fx,
draw_ell,
draw_pts,
color=None,
nRandKpts=None,
rect=False):
#print('[viz] _annotate_kpts()')
if color is None:
color = 'distinct' if sel_fx is None else df2.ORANGE
ell_args = {
'ell': draw_ell,
'pts': draw_pts,
'rect': rect,
'ell_alpha': .4,
'ell_linewidth': 2,
'ell_color': color,
}
if draw_ell and nRandKpts is not None:
# show a random sample of kpts
nkpts1 = len(kpts)
fxs1 = np.arange(nkpts1)
size = nRandKpts
replace = False
p = np.ones(nkpts1)
p = p / p.sum()
fxs_randsamp = np.random.choice(fxs1, size, replace, p)
kpts = kpts[fxs_randsamp]
# TODO Fix this. This should not set the xlabel
df2.set_xlabel('displaying %r/%r keypoints' % (nRandKpts, nkpts1))
elif draw_ell or draw_pts:
# draw all keypoints
df2.draw_kpts2(kpts, **ell_args)
if sel_fx is not None:
# Draw selected keypoint
sel_kpts = kpts[sel_fx:sel_fx + 1]
df2.draw_kpts2(sel_kpts, ell_color=df2.BLUE, arrow=True, rect=True)
def test(hs, qcx):
fx2_scale = sv2.keypoint_scale(hs.feats.cx2_kpts[qcx])
fx = fx2_scale.argsort()[::-1][40]
rchip = hs.get_chip(qcx)
kp = hs.feats.cx2_kpts[qcx][fx]
# Show full image and keypoint
df2.figure(fignum=9000, doclf=True)
df2.imshow(rchip, plotnum=(1,3,1))
df2.draw_kpts2([kp], ell_color=(1,0,0), pts=True)
# Show cropped image and keypoint
patch, subkp = get_patch(rchip, kp)
df2.imshow(patch, plotnum=(1,3,2))
df2.draw_kpts2([subkp], ell_color=(1,0,0), pts=True)
# Show warped image and keypoint
wpatch, wkp = get_warped_patch(rchip, kp)
df2.imshow(wpatch, plotnum=(1,3,3))
df2.draw_kpts2([wkp], ell_color=(1,0,0), pts=True)
#
df2.set_figtitle('warp test')
def _kpts_helper(kpts_, color, alpha, label):
# helper function taking into acount phantom labels
df2.draw_kpts2(kpts_, ell_color=color, ell_alpha=alpha)
df2.phantom_legend_label(label, color)
def _kpts_helper(kpts_, color, alpha, label):
# helper function taking into acount phantom labels
df2.draw_kpts2(kpts_, ell_color=color, ell_alpha=alpha)
df2.phantom_legend_label(label, color)
def in_depth_ellipse2x2(rchip, kp):
#-----------------------
# SETUP
#-----------------------
from hotspotter import draw_func2 as df2
np.set_printoptions(precision=8)
tau = 2 * np.pi
df2.reset()
df2.figure(9003, docla=True, doclf=True)
ax = df2.gca()
ax.invert_yaxis()
def _plotpts(data, px, color=df2.BLUE, label=''):
#df2.figure(9003, docla=True, pnum=(1, 1, px))
df2.plot2(data.T[0], data.T[1], '.', '', color=color, label=label)
df2.update()
def _plotarrow(x, y, dx, dy, color=df2.BLUE, label=''):
ax = df2.gca()
arrowargs = dict(head_width=.5, length_includes_head=True, label=label)
arrow = df2.FancyArrow(x, y, dx, dy, **arrowargs)
arrow.set_edgecolor(color)
arrow.set_facecolor(color)
ax.add_patch(arrow)
df2.update()
def _2x2_eig(M2x2):
(evals, evecs) = np.linalg.eig(M2x2)
l1, l2 = evals
v1, v2 = evecs
return l1, l2, v1, v2
#-----------------------
# INPUT
#-----------------------
# We will call perdoch's invA = invV
print('--------------------------------')
print('Let V = Perdoch.A')
print('Let Z = Perdoch.E')
print('--------------------------------')
print('Input from Perdoch\'s detector: ')
# We are given the keypoint in invA format
(x, y, ia11, ia21, ia22), ia12 = kp, 0
invV = np.array([[ia11, ia12], [ia21, ia22]])
V = np.linalg.inv(invV)
# <HACK>
#invV = V / np.linalg.det(V)
#V = np.linalg.inv(V)
# </HACK>
Z = (V.T).dot(V)
print('invV is a transform from points on a unit-circle to the ellipse')
helpers.horiz_print('invV = ', invV)
print('--------------------------------')
print('V is a transformation from points on the ellipse to a unit circle')
helpers.horiz_print('V = ', V)
print('--------------------------------')
print('Points on a matrix satisfy (x).T.dot(Z).dot(x) = 1')
print('where Z = (V.T).dot(V)')
helpers.horiz_print('Z = ', Z)
# Define points on a unit circle
theta_list = np.linspace(0, tau, 50)
cicrle_pts = np.array([(np.cos(t), np.sin(t)) for t in theta_list])
# Transform those points to the ellipse using invV
ellipse_pts1 = invV.dot(cicrle_pts.T).T
# Transform those points to the ellipse using V
ellipse_pts2 = V.dot(cicrle_pts.T).T
#Lets check our assertion: (x_).T.dot(Z).dot(x_) = 1
checks1 = [x_.T.dot(Z).dot(x_) for x_ in ellipse_pts1]
checks2 = [x_.T.dot(Z).dot(x_) for x_ in ellipse_pts2]
assert all([abs(1 - check) < 1E-11 for check in checks1])
#assert all([abs(1 - check) < 1E-11 for check in checks2])
print('... all of our plotted points satisfy this')
#=======================
# THE CONIC SECTION
#=======================
# All of this was from the Perdoch paper, now lets move into conic sections
# We will use the notation from wikipedia
# http://en.wikipedia.org/wiki/Conic_section
# http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
#-----------------------
# MATRIX REPRESENTATION
#-----------------------
# The matrix representation of a conic is:
(A, B2, B2_, C) = Z.flatten()
(D, E, F) = (0, 0, 1)
B = B2 * 2
assert B2 == B2_, 'matrix should by symmetric'
print('--------------------------------')
print('Now, using wikipedia\' matrix representation of a conic.')
con = np.array(((' A', 'B / 2', 'D / 2'), ('B / 2', ' C', 'E / 2'),
('D / 2', 'E / 2', ' F')))
helpers.horiz_print('A matrix A_Q = ', con)
# A_Q is our conic section (aka ellipse matrix)
A_Q = np.array(((A, B / 2, D / 2), (B / 2, C, E / 2), (D / 2, E / 2, F)))
helpers.horiz_print('A_Q = ', A_Q)
#-----------------------
# DEGENERATE CONICS
#-----------------------
print('----------------------------------')
print('As long as det(A_Q) != it is not degenerate.')
print('If the conic is not degenerate, we can use the 2x2 minor: A_33')
print('det(A_Q) = %s' % str(np.linalg.det(A_Q)))
assert np.linalg.det(A_Q) != 0, 'degenerate conic'
A_33 = np.array(((A, B / 2), (B / 2, C)))
helpers.horiz_print('A_33 = ', A_33)
#-----------------------
# CONIC CLASSIFICATION
#-----------------------
print('----------------------------------')
print('The determinant of the minor classifies the type of conic it is')
print('(det == 0): parabola, (det < 0): hyperbola, (det > 0): ellipse')
print('det(A_33) = %s' % str(np.linalg.det(A_33)))
assert np.linalg.det(A_33) > 0, 'conic is not an ellipse'
print('... this is indeed an ellipse')
#-----------------------
# CONIC CENTER
#-----------------------
print('----------------------------------')
print('the centers of the ellipse are obtained by: ')
print('x_center = (B * E - (2 * C * D)) / (4 * A * C - B ** 2)')
print('y_center = (D * B - (2 * A * E)) / (4 * A * C - B ** 2)')
# Centers are obtained by solving for where the gradient of the quadratic
# becomes 0. Without going through the derivation the calculation is...
# These should be 0, 0 if we are at the origin, or our original x, y
# coordinate specified by the keypoints. I'm doing the calculation just for
# shits and giggles
x_center = (B * E - (2 * C * D)) / (4 * A * C - B**2)
y_center = (D * B - (2 * A * E)) / (4 * A * C - B**2)
helpers.horiz_print('x_center = ', x_center)
helpers.horiz_print('y_center = ', y_center)
#-----------------------
# MAJOR AND MINOR AXES
#-----------------------
# Now we are going to determine the major and minor axis
# of this beast. It just the center augmented by the eigenvecs
print('----------------------------------')
# The angle between the major axis and our x axis is:
l1, l2, v1, v2 = _2x2_eig(A_33)
x_axis = np.array([1, 0])
theta = np.arccos(x_axis.dot(v1))
# The eccentricity is determined by:
nu = 1
numer = 2 * np.sqrt((A - C)**2 + B**2)
denom = nu * (A + C) + np.sqrt((A - C)**2 + B**2)
eccentricity = np.sqrt(numer / denom)
from scipy.special import ellipeinc
#-----------------------
# DRAWING
#-----------------------
# Lets start off by drawing the ellipse that we are goign to work with
# Create unit circle sample
# Draw the keypoint using the tried and true df2
# Other things should subsiquently align
df2.draw_kpts2(np.array([(0, 0, ia11, ia21, ia22)]),
ell_linewidth=4,
ell_color=df2.DEEP_PINK,
ell_alpha=1,
arrow=True,
rect=True)
# Plot ellipse points
_plotpts(ellipse_pts1, 0, df2.YELLOW, label='invV.dot(cicrle_pts.T).T')
# Plot ellipse axis
# !HELP! I DO NOT KNOW WHY I HAVE TO DIVIDE, SQUARE ROOT, AND NEGATE!!!
l1, l2, v1, v2 = _2x2_eig(A_33)
dx1, dy1 = (v1 / np.sqrt(l1))
dx2, dy2 = (v2 / np.sqrt(l2))
_plotarrow(0, 0, dx1, -dy1, color=df2.ORANGE, label='ellipse axis')
_plotarrow(0, 0, dx2, -dy2, color=df2.ORANGE)
# Plot ellipse orientation
orient_axis = invV.dot(np.eye(2))
dx1, dx2, dy1, dy2 = orient_axis.flatten()
_plotarrow(0, 0, dx1, dy1, color=df2.BLUE, label='ellipse rotation')
_plotarrow(0, 0, dx2, dy2, color=df2.BLUE)
df2.legend()
df2.dark_background()
df2.gca().invert_yaxis()
return locals()
def draw_one_kp(patch, kp, plotx, color, desc=None):
fig, ax = df2.imshow(patch, plotnum=(num_rows, num_cols, plotx))
df2.draw_kpts2([kp], ell_color=color, pts=True)
if not desc is None and not '--nodesc' in sys.argv:
df2.draw_sift(desc, [kp])
df2.draw_border(ax, color, 1)
def viz_top_features(hs, res, low, high, fignum=0, draw_chips=True):
from collections import defaultdict
qcx = res.qcx
cx2_nx = hs.tables.cx2_nx
top_patches_list = get_top_scoring_patches(hs, res, low, high)
num_rows = high-low
num_cols = 4
if params.__MATCH_TYPE__ == 'vsmany':
num_cols = 6
# Initialize Figure
fig = df2.figure(fignum+1, plotnum=(num_rows, num_cols,1))
cx2_rchip = defaultdict(int)
cx2_kplist = defaultdict(list)
def draw_one_kp(patch, kp, plotx, color, desc=None):
fig, ax = df2.imshow(patch, plotnum=(num_rows, num_cols, plotx))
df2.draw_kpts2([kp], ell_color=color, pts=True)
if not desc is None and not '--nodesc' in sys.argv:
df2.draw_sift(desc, [kp])
df2.draw_border(ax, color, 1)
for tx, (patches1, patches2, patchesN, cx, feat_score) in enumerate(top_patches_list):
(kp1, subkp1, wkp1, patch1, wpatch1, cx1, desc1) = patches1
(kp2, subkp2, wkp2, patch2, wpatch2, cx2, desc2) = patches2
# draw on table
# Draw Query Keypoint
plotx = (tx*num_cols)
draw_one_kp(patch1, subkp1, plotx+1, df2.GREEN, desc1)
qnx = cx2_nx[qcx]
df2.plt.gca().set_xlabel('qcx=%r; qnx=%r' % (qcx, qnx))
draw_one_kp(wpatch1, wkp1, plotx+2, df2.GREEN, desc1)
# Draw ith < k match
draw_one_kp(patch2, subkp2, plotx+3, df2.BLUE, desc2)
nx = cx2_nx[cx]
df2.plt.gca().set_xlabel('cx=%r; nx=%r' % (cx, nx))
draw_one_kp(wpatch2, wkp2, plotx+4, df2.BLUE, desc2)
df2.plt.gca().set_xlabel('score=%r' % (feat_score))
# Draw k+1th match
if params.__MATCH_TYPE__ == 'vsmany':
(kpN, subkpN, wkpN, patchN, wpatchN, cxN, descN) = patchesN
draw_one_kp(patchN, subkpN, plotx+5, df2.ORANGE, descN)
nxN = cx2_nx[qcx]
df2.plt.gca().set_xlabel('cxN=%r; nxN=%r' % (cxN, nxN))
draw_one_kp(wpatchN, wkpN, plotx+6, df2.ORANGE, descN)
# Get other info
cx2_rchip[cx] = hs.get_chip(cx)
cx2_kplist[qcx].append(kp1)
cx2_kplist[cx].append(kp2)
# Draw annotations on
df2.figure(plotnum=(num_rows, num_cols,1), title='Query Patch')
df2.figure(plotnum=(num_rows, num_cols,2), title='Query Patch')
df2.figure(plotnum=(num_rows, num_cols,3), title='Result Patch')
df2.figure(plotnum=(num_rows, num_cols,4), title='Result Warped')
if params.__MATCH_TYPE__ == 'vsmany':
df2.figure(plotnum=(num_rows, num_cols,5), title='Normalizer Patch')
df2.figure(plotnum=(num_rows, num_cols,6), title='Normalizer Warped')
df2.set_figtitle('Top '+str(low)+' to '+str(high)+' scoring matches')
if not draw_chips:
return
#
# Draw on full images
cx2_rchip[qcx] = hs.get_chip(qcx)
cx_keys = cx2_kplist.keys()
cx_vals = map(len, cx2_kplist.values())
cx_list = [x for (y,x) in sorted(zip(cx_vals, cx_keys))][::-1]
num_chips = len(cx_list)
pltnum_fn = lambda ix: (int(np.ceil(num_chips/2)), 2, ix)
plotnum = pltnum_fn(1)
fig2 = df2.figure(fignum-1, plotnum=plotnum)
for ix, cx in enumerate(cx_list):
plotnum = pltnum_fn(ix+1)
fig2 = df2.figure(fignum-1, plotnum=plotnum)
rchip = cx2_rchip[cx]
fig, ax = df2.imshow(rchip)
title_pref = ''
color = df2.BLUE
if res.qcx == cx:
title_pref = 'q'
color = df2.GREEN
df2.draw_kpts2(cx2_kplist[cx], ell_color=color, ell_alpha=1)
df2.draw_border(ax, color, 2)
nx = cx2_nx[cx]
ax.set_title(title_pref+'cx = %r; nx=%r' % (cx, nx))
gname = hs.get_gname(cx)
ax.set_xlabel(gname)
def in_depth_ellipse2x2(rchip, kp):
#-----------------------
# SETUP
#-----------------------
from hotspotter import draw_func2 as df2
np.set_printoptions(precision=8)
tau = 2 * np.pi
df2.reset()
df2.figure(9003, docla=True, doclf=True)
ax = df2.gca()
ax.invert_yaxis()
def _plotpts(data, px, color=df2.BLUE, label=''):
#df2.figure(9003, docla=True, pnum=(1, 1, px))
df2.plot2(data.T[0], data.T[1], '.', '', color=color, label=label)
df2.update()
def _plotarrow(x, y, dx, dy, color=df2.BLUE, label=''):
ax = df2.gca()
arrowargs = dict(head_width=.5, length_includes_head=True, label=label)
arrow = df2.FancyArrow(x, y, dx, dy, **arrowargs)
arrow.set_edgecolor(color)
arrow.set_facecolor(color)
ax.add_patch(arrow)
df2.update()
def _2x2_eig(M2x2):
(evals, evecs) = np.linalg.eig(M2x2)
l1, l2 = evals
v1, v2 = evecs
return l1, l2, v1, v2
#-----------------------
# INPUT
#-----------------------
# We will call perdoch's invA = invV
print('--------------------------------')
print('Let V = Perdoch.A')
print('Let Z = Perdoch.E')
print('--------------------------------')
print('Input from Perdoch\'s detector: ')
# We are given the keypoint in invA format
(x, y, ia11, ia21, ia22), ia12 = kp, 0
invV = np.array([[ia11, ia12],
[ia21, ia22]])
V = np.linalg.inv(invV)
# <HACK>
#invV = V / np.linalg.det(V)
#V = np.linalg.inv(V)
# </HACK>
Z = (V.T).dot(V)
print('invV is a transform from points on a unit-circle to the ellipse')
helpers.horiz_print('invV = ', invV)
print('--------------------------------')
print('V is a transformation from points on the ellipse to a unit circle')
helpers.horiz_print('V = ', V)
print('--------------------------------')
print('Points on a matrix satisfy (x).T.dot(Z).dot(x) = 1')
print('where Z = (V.T).dot(V)')
helpers.horiz_print('Z = ', Z)
# Define points on a unit circle
theta_list = np.linspace(0, tau, 50)
cicrle_pts = np.array([(np.cos(t), np.sin(t)) for t in theta_list])
# Transform those points to the ellipse using invV
ellipse_pts1 = invV.dot(cicrle_pts.T).T
# Transform those points to the ellipse using V
ellipse_pts2 = V.dot(cicrle_pts.T).T
#Lets check our assertion: (x_).T.dot(Z).dot(x_) = 1
checks1 = [x_.T.dot(Z).dot(x_) for x_ in ellipse_pts1]
checks2 = [x_.T.dot(Z).dot(x_) for x_ in ellipse_pts2]
assert all([abs(1 - check) < 1E-11 for check in checks1])
#assert all([abs(1 - check) < 1E-11 for check in checks2])
print('... all of our plotted points satisfy this')
#=======================
# THE CONIC SECTION
#=======================
# All of this was from the Perdoch paper, now lets move into conic sections
# We will use the notation from wikipedia
# http://en.wikipedia.org/wiki/Conic_section
# http://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections
#-----------------------
# MATRIX REPRESENTATION
#-----------------------
# The matrix representation of a conic is:
(A, B2, B2_, C) = Z.flatten()
(D, E, F) = (0, 0, 1)
B = B2 * 2
assert B2 == B2_, 'matrix should by symmetric'
print('--------------------------------')
print('Now, using wikipedia\' matrix representation of a conic.')
con = np.array(((' A', 'B / 2', 'D / 2'),
('B / 2', ' C', 'E / 2'),
('D / 2', 'E / 2', ' F')))
helpers.horiz_print('A matrix A_Q = ', con)
# A_Q is our conic section (aka ellipse matrix)
A_Q = np.array((( A, B / 2, D / 2),
(B / 2, C, E / 2),
(D / 2, E / 2, F)))
helpers.horiz_print('A_Q = ', A_Q)
#-----------------------
# DEGENERATE CONICS
#-----------------------
print('----------------------------------')
print('As long as det(A_Q) != it is not degenerate.')
print('If the conic is not degenerate, we can use the 2x2 minor: A_33')
print('det(A_Q) = %s' % str(np.linalg.det(A_Q)))
assert np.linalg.det(A_Q) != 0, 'degenerate conic'
A_33 = np.array((( A, B / 2),
(B / 2, C)))
helpers.horiz_print('A_33 = ', A_33)
#-----------------------
# CONIC CLASSIFICATION
#-----------------------
print('----------------------------------')
print('The determinant of the minor classifies the type of conic it is')
print('(det == 0): parabola, (det < 0): hyperbola, (det > 0): ellipse')
print('det(A_33) = %s' % str(np.linalg.det(A_33)))
assert np.linalg.det(A_33) > 0, 'conic is not an ellipse'
print('... this is indeed an ellipse')
#-----------------------
# CONIC CENTER
#-----------------------
print('----------------------------------')
print('the centers of the ellipse are obtained by: ')
print('x_center = (B * E - (2 * C * D)) / (4 * A * C - B ** 2)')
print('y_center = (D * B - (2 * A * E)) / (4 * A * C - B ** 2)')
# Centers are obtained by solving for where the gradient of the quadratic
# becomes 0. Without going through the derivation the calculation is...
# These should be 0, 0 if we are at the origin, or our original x, y
# coordinate specified by the keypoints. I'm doing the calculation just for
# shits and giggles
x_center = (B * E - (2 * C * D)) / (4 * A * C - B ** 2)
y_center = (D * B - (2 * A * E)) / (4 * A * C - B ** 2)
helpers.horiz_print('x_center = ', x_center)
helpers.horiz_print('y_center = ', y_center)
#-----------------------
# MAJOR AND MINOR AXES
#-----------------------
# Now we are going to determine the major and minor axis
# of this beast. It just the center augmented by the eigenvecs
print('----------------------------------')
# The angle between the major axis and our x axis is:
l1, l2, v1, v2 = _2x2_eig(A_33)
x_axis = np.array([1, 0])
theta = np.arccos(x_axis.dot(v1))
# The eccentricity is determined by:
nu = 1
numer = 2 * np.sqrt((A - C) ** 2 + B ** 2)
denom = nu * (A + C) + np.sqrt((A - C) ** 2 + B ** 2)
eccentricity = np.sqrt(numer / denom)
from scipy.special import ellipeinc
#-----------------------
# DRAWING
#-----------------------
# Lets start off by drawing the ellipse that we are goign to work with
# Create unit circle sample
# Draw the keypoint using the tried and true df2
# Other things should subsiquently align
df2.draw_kpts2(np.array([(0, 0, ia11, ia21, ia22)]), ell_linewidth=4,
ell_color=df2.DEEP_PINK, ell_alpha=1, arrow=True, rect=True)
# Plot ellipse points
_plotpts(ellipse_pts1, 0, df2.YELLOW, label='invV.dot(cicrle_pts.T).T')
# Plot ellipse axis
# !HELP! I DO NOT KNOW WHY I HAVE TO DIVIDE, SQUARE ROOT, AND NEGATE!!!
l1, l2, v1, v2 = _2x2_eig(A_33)
dx1, dy1 = (v1 / np.sqrt(l1))
dx2, dy2 = (v2 / np.sqrt(l2))
_plotarrow(0, 0, dx1, -dy1, color=df2.ORANGE, label='ellipse axis')
_plotarrow(0, 0, dx2, -dy2, color=df2.ORANGE)
# Plot ellipse orientation
orient_axis = invV.dot(np.eye(2))
dx1, dx2, dy1, dy2 = orient_axis.flatten()
_plotarrow(0, 0, dx1, dy1, color=df2.BLUE, label='ellipse rotation')
_plotarrow(0, 0, dx2, dy2, color=df2.BLUE)
df2.legend()
df2.dark_background()
df2.gca().invert_yaxis()
return locals()