# CSCI 5980 Assignment #5 Bundle Adjustment solved

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2 Data Capture and Overview
Figure 1: You will capture your cellphone images to reconstruct camera pose and 3D
points.
In this assignment, you will use your cellphone images (more than 5) to reconstruct
3D camera poses and points with full bundle adjustment. Make sure you have enough
baseline (translation) between images for well conditioned fundamental matrix while
retaining enough number of correspondences between image. Avoid a scene dominated
by a planar surface, i.e., the images need to contain many 3D objects as shown in
Figure 1.
2
CSCI 5980: Assignment #5
You will write a full pipeline of the structure from motion algorithm including matching, camera pose estimation using fundamental matrix, PnP, triangulation, and bundle
adjustment. A nonlinear optimization is always followed by the initial estimate by
linear least squares solution. The pipeline is described in Algorithm 1.
Algorithm 1 Structure from Motion
1: [Mx, My] = GetMatches(I1, · · · , IN )
2: Normalize coordinate in Mx and My, i.e., x = K−1x.
3: Select two images Ii1 and Ii2 for the initial pair reconstruction.
4: [R, C, X] = CameraPoseEstimation([Mx(:,i1) My(:,i1)], [Mx(:,i2) My(:,i2)])
5: P = {P1,P2} where Pi1 =

I3 0

, Pi2 = R

I3 −C

6: R = {i1, i2}
7: while |R| < N do 8: i = GetBestFrame(Mx, My, R); 9: [Ri , Ci ] = PnP RANSAC([Mx(:,i) My(:,i)], X) 10: [Ri , Ci ] = PnP Nonlinear(Ri Ci , [Mx(:,i) My(:,i)], X) 11: Pi = Ri I3 −Ci 12: for f = 1 : |R| do 13: U = FindUnreconstructedPoints(X, Rf , i, Mx, My) 14: for j = 1 : |U| do 15: u = [Mx(Uj , i), My(Uj , i)] and v = [Mx(Uj , Rf ), My(Uj , Rf )] 16: x = LinearTriangulation(u, Pi , v, PRf ) 17: x = NonlinearTriangulation(X, u, Ri , Ci , v, RRf , CRf ) 18: X = X ∪ x 19: end for 20: end for 21: P = P ∪ Pi and R = R ∪ i. 22: [P, X] = BundleAdjustment(P, X, R, Mx, My) 23: end while 3 CSCI 5980: Assignment #5 Bundle Adjustment 3 Matching Given a set of images, I1, · · · , IN , you will find matches across all images where N is the number of images similar to HW #4. Pick a reference image, Iref , and match with other images using SIFT features from VLFeat, i.e., Iref ↔ I1, · · · , Iref ↔ IN (no need to match Iref ↔ Iref ). Your matches are outlier free, i.e., bidirectional knn match → ratio test → inliers from the fundamental matrix based RANSAC. Based on the matches, you will build a measurement matrix, Mx and My: [Mx, My] = GetMatches(I1, · · · , IN ) Mx: F×N matrix storing x coordinate of correspondences My: F×N matrix storing y coordinate of correspondences The f th feature point in image Ii corresponds to a point in image Ij . The x and y coordinates of the correspondence is stored at (f, i) and (f, j) elements in Mx and My, respectively. If (f, i) does not correspond to any point in image Ik, you set -1 to indicate no match as shown in Figure 2. Important: For better numerical stability, you can transform the measurements to the normalized coordinate by multiplying K−1 , i.e., x = K−1x where x is 2D measured points in homogeneous coordinate. You can run structure from motion in the normalized coordinate by factoring out K. When visualizing projection in the image, the coordinate needs to be transformed back to original coordinate by multiplying K. F N f ,i x f , j f x i j Mx N f ,i y f , j f y i j My ( xf ,i , y f ,i) ↔ ( xf , j , y f , j) ↔ ( xf ,k , y f ,k ) −1 k −1 k Figure 2: The f th feature point in image Ii corresponds to a point in image Ij . The x and y coordinates of the correspondence is stored at (f, i) and (f, j) elements in Mx and My, respectively. If (f, i) does not correspond to any point in image Ik, you set -1 to indicate no match. 4 CSCI 5980: Assignment #5 Bundle Adjustment 4 Camera Pose Estimation You will write a camera pose estimation code that takes correspondences between two images, Ii1 and Ii2 where i1 and i2 are the indices of the initial images to reconstruct selected manually. [R, C, X] = CameraPoseEstimation(u1, u2) R and C: the relative transformation of the i2 image u1 and u2: 2D-2D correspondences As studied in HW #4, you will compute: 1. Fundamental matrix via RANSAC on correspondences, Mx(:,i1), My(:,i2) 2. Essential matrix from the fundamental matrix 3. Four configurations of camera poses given the essential matrix 4. Disambiguation via chierality (using 3D point linear triangulation): X = LinearTriangulation(u, Pi , v, Pj ) Write-up: (a) Inlier matches Top view Oblique view (b) 3D camera pose Figure 3: Camera pose estimation. (1) Visualize inlier matches as shown in Figure 3(a). (2) Visualize camera pose and 3D reconstructed points in 3D as shown in Figure 3(b). 5 CSCI 5980: Assignment #5 Bundle Adjustment 5 Nonlinear 3D Point Refinement You will write a nonlinear triangulation code. Given the linear estimate for the point triangulation, X, you will refine the 3D point X = X Y Z T to minimize geometric error (reprojection error) via iterative nonlinear least squares estimation, ∆X = ∂f(X) ∂X T ∂f(X) ∂X !−1 ∂f(X) ∂X T (b − f(X)). (1) Write-up: (1) Derive the point Jacobian, i.e., ∂f(X)j ∂X and write the following code. df dX = JacobianX(K, R, C, X) (2) Write a code to refine the 3D point by minimizing the reprojection error and visualize reprojection error reduction similar to Figure 5. X = NonlinearTriangulation(X, u1, R1, C1, u2, R2, C2) Algorithm 2 Nonlinear Point Refinement 1: b = u T 1 u T 2 T 2: for j = 1 : nIters do 3: Build point Jacobian, ∂f(X)j ∂X . 4: Compute f(X). 5: ∆X =  ∂f(X) ∂X T ∂f(X) ∂X + λI −1 ∂f(X) ∂X T (b − f(X)) 6: X = X + ∆X 7: end for 6 CSCI 5980: Assignment #5 Bundle Adjustment 6 Camera Registration You will register an additional image, Ij using 2D-3D correspondences. Write-up: (1) (3D-2D correspondences) Given 3D triangulated points, find 2D-3D matches, X ↔ u. (2) (Perspective-n-Point algorithm) Write a code that computes 3D camera pose from 3D-2D correspondences: [R, C] = LinearPnP(u, X) X: n × 3 matrix containing n 3D reconstructed points u: n × 2 matrix containing n 2D points in the additional image I3 R and C: rotation and translation for the additional image. Hint: After the linear solve, rectify the rotation matrix such that det(R) = 1 and scale C according to the rectification. 7 CSCI 5980: Assignment #5 Bundle Adjustment (3) (RANSAC PnP) Write a RANSAC algorithm for the camera pose registration (PnP) given n matches using the following pseudo code: Algorithm 3 PnP RANSAC 1: nInliers ← 0 2: for i = 1 : M do 3: Choose 6 correspondences, Xr and ur, randomly from X and u. 4: [Rr, tr] = LinearPnP(ur, Xr) 5: Compute the number of inliers, nr, with respect to Rr, tr. 6: if nr > nInliers then
7: nInliers ← nr
8: R = Rr and t = tr
9: end if
10: end for
Visualize 3D registered pose as shown in Figure 4.
-40
-20
0
20
(a) Front view
0
10
20
30
40
50
60
0 -80 -60 -40 -20 0 20 40
(b) Top view
(4) (Reprojection) Visualize measurement and reprojection to verify the solution.
8
CSCI 5980: Assignment #5
7 Nonlinear Camera Refinement
Given the initial estimate Ri and ti
, you will refine the camera pose to minimize
geometric error (reprojection error) via iterative nonlinear least squares estimation,
∆p =

∂f(p)
∂p
T
∂f(p)
∂p
!−1
∂f(p)
∂p
T
(b − f(p)),
f(p) =

u1/w1
v1/w1
.
.
.
un/wn
vn/wn

,

u
v
w

 = Ri

I3 −C


X
1

, b =

x1
y1
.
.
.
xn
yn

(2)
where p =

CT q
T
T
. C ∈ R3
is the camera optical center and q ∈ S
3
is the
quaternion representation of the camera rotation.
It is possible to minimize the overshooting by adding damping, λ as follows:
∆p =

∂f(p)
∂p
T
∂f(p)
∂p
+ λI
!−1
∂f(p)
∂p
T
(b − f(p)), (3)
where λ is the damping parameter. You can try λ ∈ [0, 10].
Note that the conversion between quaternion and rotation matrix is given as follows:
R =

1 − 2q
2
z − 2q
2
y −2qzqw + 2qyqx 2qyqw + 2qzqx
2qxqy + 2qwqz 1 − 2q
2
z − 2q
2
x
2qzqy − 2qxqw
2qxqz − 2qwqy 2qyqz + 2qwqx 1 − 2q
2
y − 2q
2
x

 ,
q =

qw
(R32 − R23)/(4qw)
(R13 − R31)/(4qw)
(R21 − R12)/(4qw)

, where qw =

1 + R11 + R22 + R33
2
, kqk = 1 (4)
9
CSCI 5980: Assignment #5
Write-up:
(1) Derive the quaternion Jacobian to rotation using Equation (4), i.e., ∂R
∂q
and write
the following code. Note: ignore the normalization kqk = 1.
dR dq = JacobianQ(q)
(2) Derive the rotation Jacobian to projection using Equation (2), i.e., ∂f(p)j
∂R where
f(p)j =

uj/wj vj/wj
T
and write the following code. Note: use vectorized form of
the rotation matrix.
df dR = JacobianR(R, C, X)
(3) Derive the expression of ∂f(p)j
∂q
using the chain rule.
(4) Derive the camera center Jacobian to projection using Equation (2), i.e., ∂f(p)j
∂C
and
write the following code.
df dC = JacobianC(R, C, X)
(5) Write a code to refine the camera pose by minimizing the reprojection error and
visualize reprojection error reduction as shown in Figure 5:
[R, C] = PnP Nonlinear(R C, u, X)
Algorithm 4 Nonlinear Camera Pose Refinement
1: p =

CT q
T
T
2: for j = 1 : nIters do
3: C = p1:3, R=Quaternion2Rotation(q), q = p4:7
4: Build camera pose Jacobian for all points, ∂f(p)j
∂p =
h
∂f(p)j
∂C
∂f(p)j
∂q
i
.
5: Compute f(p).
6: ∆p =

∂f(p)
∂p
T ∂f(p)
∂p + λI
−1
∂f(p)
∂p
T
(b − f(p)) using Equation (3).
7: p = p + ∆p
8: Normalize the quaternion scale, p4:7 = p4:7/kp4:7k.
9: end for
10
CSCI 5980: Assignment #5
Measurement
Linear estimate (reproj: 0.199104)
Nonlinear estimate (reproj: 0.119272)
(a) Reprojection error
Measurement
Linear estimate (reproj: 0.199104)
Nonlinear estimate (reproj: 0.119272)
(b) Close up
Figure 5: Nonlinear refinement reduces the reprojection error (0.19→0.11).
You will write a nonlinear refinement code that simultaneously optimizes camera poses
and 3D points using the sparse nature of the Jacobian matrix.
[P, X] = BundleAdjustient(P, X, R, Mx, My)
For example, consider 3 camera poses and 2 points. The Jacobian matrix can be written
as follows:
J =

∂f(p1,X1)
∂p1
02×7 02×7
∂f(p1,X1)
∂X1
02×3
02×7
∂f(p2,X1)
∂p2
02×7
∂f(p2,X1)
∂X1
02×3
02×7 02×7
∂f(p3,X1)
∂p3
∂f(p3,X1)
∂X1
02×3
∂f(p1,X2)
∂p1
02×7 02×7 02×3
∂f(p1,X2)
∂X2
02×7
∂f(p2,X2)
∂p2
02×7 02×3
∂f(p2,X2)
∂X2
02×7 02×7
∂f(p3,X2)
∂p3
02×3
∂f(p3,X2)
∂X2

=

Jp JX

(5)
where Jp and JX are the Jacobian for camera and point, respectively, and λ ∈ [0, 10].
The normal equation, J
TJ∆x = J
T
(b − f(x)) can be decomposed into:

A B
BT D
  ∆pb
∆Xb

=

ep
eX

, (6)
where
A = J
T
pJp + λI, B = J
T
pJX, D = J
T
XJX + λI
ep = J
T
p
(b − f(x)), eX = J
T
X(b − f(x))
where pb =

p
T
1
· · · p
T
I

and Xb =

XT
1
· · · XT
M

where I and M are the number
of images and points, respectively.
11
CSCI 5980: Assignment #5
The decomposed normal equation in Equation (6) allows us to efficiently compute the
inverse of J
TJ using Schur complement of D:
∆pb = (A − BD−1B
T
)
−1
(ep − BD−1
eX),
∆Xb = D−1
(eX − B
T∆pb)
where D is a block diagonal matrix whose inverse can be efficiently computed by inverting small block matrix:
D =

d1
.
.
.
dM

 , D−1 =

d
−1
1
.
.
.
d
−1
M

 (7)
The bundle adjustment algorithm is summarized in Algorithm 5. Note that not all
points are visible from cameras. You need to reason about the visibility, i.e., if the
point is not visible from the camera, the corresponding Jacobian and measurement
from J and b will be omitted, respectively.
12
CSCI 5980: Assignment #5
1: pb =

p
T
1
· · · p
T
I
T
and Xb =

XT
1
· · · XT
M

2: for iter = 1 : nIters do
3: Empty Jp, JX, b, f, Dinv.
4: for i = 1 : M do
5: d = 03×3
6: for j = 1 : I do
7: if the i
th point is visible from the j
th image then
8: J1 = 02×7I and J2 = 02×3M
9: J1(:, 7(j − 1) + 1 : 7j) = ∂f(pj ,Xi)
∂pj
10: J2(:, 3(i − 1) + 1 : 3i) = ∂f(pj ,Xi)
∂Xi
11: Jp =

J
T
p J
T
1
T
and JX =

J
T
X J
T
2
T
12: d = d +
∂f(pj ,Xi)
∂Xi
T ∂f(pj ,Xi)
∂Xi
13: b =

b
T u
T
ij
14: f =

f
T x
T
ij
where µ

xij
1

= Rj

I −Cj

15: end if
16: end for
17: d = d + λI
18: Dinv = blkdiag(Dinv, d
−1
)
19: end for
20: ep = J
T
p
(b − f)
21: eX = J
T
X(b − f)
22: A = J
T
pJp + λI, B = J
T
pJX, D−1 = Dinv
23: ∆pb = (A − BD−1BT
)
−1
(ep − BD−1eX)
24: Normalize quaternions.
25: ∆Xb = D−1
(eX − BT∆pb)
26: end for