LK金字塔光流法实现——英文
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Pyramidal Implementation of the Lucas Kanade Feature Tracker
Description of the algorithm
Jean-Yves Bouguet
Intel Corporation
Microprocessor Research Labs
jean-yves.bouguet@intel.com
1 Problem Statement
Let I and J be two 2D grayscaled images. The two quantities I(x) = I(x, y) and J(x) = J(x, y) are then the
grayscale value of the two images are the location x = [x y]
T
, where x and y are the two pixel coordinates of a generic
image point x. The image I will sometimes be referenced as the first image, and the image J as the second image.
For practical issues, the images I and J are discret function (or arrays), and the upper left corner pixel coordinate
vector is [0 0]
T
. Let n
x
and n
y
be the width and height of the two images. Then the lower right pixel coordinate
vector is [n
x
− 1 n
y
− 1]
T
.
Consider an image point u = [u
x
u
y
]
T
on the first image I. The goal of feature tracking is to find the location
v = u + d = [u
x
+ d
x
u
y
+ d
y
]
T
on the second image J such as I(u) and J(v) are “similar”. The vector d = [d
x
d
y
]
T
is the image velocity at x, also known as the optical flow at x. Because of the aperture problem, it is essential to
define the notion of similarity in a 2D neighborhood sense. Let ω
x
and ω
y
two integers. We define the image velocity
d as being the vector that minimizes the residual function defined as follows:
(d) = (d
x
, d
y
) =
u
x
+ω
x
X
x=u
x
−ω
x
u
y
+ω
y
X
y=u
y
−ω
y
(I(x, y) − J(x + d
x
, y + d
y
))
2
. (1)
Observe that following that defintion, the similarity function is measured on a image neighborhood of size (2ω
x
+
1) × (2ω
y
+ 1). This neighborhood will be also called integration window. Typical values for ω
x
and ω
y
are 2,3,4,5,6,7
pixels.
2 Description of the tracking algorithm
The two key components to any feature tracker are accuracy and robustness. The accuracy component relates to
the local sub-pixel accuracy attached to tracking. Intuitively, a small integration window would be preferable in order
not to “smooth out” the details contained in the images (i.e. small values of ω
x
and ω
y
). That is especially required
at occluding areas in the images where two patchs potentially move with very different velocities.
The robustness component relates to sensitivity of tracking with respect to changes of lighting, size of image
motion,... In particular, in oder to handle large motions, it is intuively preferable to pick a large integration window.
Indeed, considering only equation 1, it is preferable to have d
x
≤ ω
x
and d
y
≤ ω
y
(unless some prior matching
information is available). There is therefore a natural tradeoff between local accuracy and robustness when choosing
the integration window size. In provide to provide a solution to that problem, we propose a pyramidal implementation
of the classical Lucas-Kanade algorithm. An iterative implementation of the Lucas-Kanade optical flow computation
provides sufficient local tracking accuracy.
2.1 Image pyramid representation
Let us define the pyramid representsation of a generic image I of size n
x
× n
y
. Let I
0
= I be the “zero
th
” level
image. This image is essentially the highest resolution image (the raw image). The image width and height at that
level are defined as n
0
x
= n
x
and n
0
y
= n
y
. The pyramid representation is then built in a recursive fashion: compute
I
1
from I
0
, then compute I
2
from I
1
, and so on... Let L = 1, 2, . . . be a generic pyramidal level, and let I
L−1
be
the image at level L − 1. Denote n
L−1
x
and n
L−1
y
the width and height of I
L−1
. The image I
L−1
is then defined as
follows:
I
L
(x, y) =
1
4
I
L−1
(2x, 2y) +
1
8
I
L−1
(2x − 1, 2y) + I
L−1
(2x + 1, 2y) + I
L−1
(2x, 2y − 1) + I
L−1
(2x, 2y + 1)
+
1
16
I
L−1
(2x − 1, 2y − 1) + I
L−1
(2x + 1, 2y + 1) + I
L−1
(2x − 1, 2y + 1) + I
L−1
(2x + 1, 2y + 1)
.
(2)
1
资源评论
ForeverCust2013-12-05并不全面，不是我想要的代码。不过有点可以参考的东西。
watchit2008
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