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Claims  |
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We claim:
1. A method for creating a panoramic image from a plurality of rectilinear
images obtained from a camera, the method comprising:
estimating a projective transformation for a pairwise registration process
between at least two overlapping rectilinear images, said projective
transformation relating the images and yielding at least one Hessian
matrix describing a registration error landscape in the neighborhood of
the projective transformation;
determining a vector of internal parameters and a vector of external
parameters from the at least one Hessian matrix to minimize image
discrepancy in an overlapping region between the at least two images; and
blending the at least two images to provide a smooth transition between the
at least two images.
2. The method according to claim 1, wherein determining the vectors
comprises determining the vectors to minimize image discrepancy in the
overlapping region during a process of global optimization.
3. The method according to claim 1, wherein the projective transformation
comprises a 3.times.3 matrix having nine projective parameters.
4. The method according to claim 1, wherein the pairwise registration
process includes a local registration error function between the at least
two images,
##EQU7##
5. The method according to claim 4, wherein the at least one Hessian
matrix, C.sub.ij, between the at least two images is determined from the
local registration a error function,
e.sub.ij (M.sub.ij).apprxeq.e.sub.ij.sup.0 +(M.sub.ij
-M.sub.ij.sup.0).sup.T C.sub.ij (M.sub.ij -M.sub.ij.sup.0).
6. The method according to 5,
wherein determining the vectors comprises determining the vectors to
minimize image discrepancy in the overlapping region during a process of
global optimization,
and wherein a global error function that is optimized during the global
optimization process is defined by a function of the sum of the products
of overlap area between the at least two images and the local registration
error function.
7. The method according to claim 1, wherein the vector of internal
parameters comprises a representation of focal length.
8. The method according to claim 7, wherein the vector of internal
parameters further comprises a representation of image center position.
9. The method according to claim 8, wherein the vector of internal
parameters further comprises a representation of pixel aspect ratio.
10. The method according to claim 9, wherein the vector of internal
parameters further comprises a representation of skew.
11. The method according to claim 1, wherein the vector of external
parameters comprises a representation of camera orientation with respect
to a common frame of reference.
12. The method according to claim 11, wherein the representation of camera
orientation comprises Euler angles including pan, tilt, and roll.
13. The method according to claim 11, wherein the representation of camera
orientation comprises Euler angles including roll, pitch, and yaw.
14. The method according to claim 11, wherein the representation of camera
orientation comprises unit quaternions.
15. The method according to claim 1, wherein blending the at least two
images comprises:
for signals in the overlapping region having a frequency higher than a
threshold frequency, blending the signals over a first blend region range;
and
for signals in the overlapping region having a frequency lower than the
threshold frequency, blending the signals over a blend region range wider
than the first blend region range.
16. A system for creating a panoramic image from a plurality of rectilinear
images obtained from a camera, the system comprising:
a processor for estimating a projective transformation during a pairwise
registration process between at least two overlapping rectilinear images
obtained from a camera, said projective transformation relating the images
and yielding at least one Hessian matrix describing a registration error
landscape in the neighborhood of the projective transformation;
a global optimizer for determining a vector of internal parameters and a
vector of external parameters from the at least one Hessian matrix to
minimize image discrepancy in an overlapping region between the at least
two overlapping rectilinear images; and
a blender for blending the at least two images to provide a smooth
transition between the at least two images.
17. The system according to claim 16, wherein the projective transformation
comprises a 3.times.3 matrix having nine projective parameters.
18. The system according to claim 16, wherein the pairwise registration
process includes a local registration error function between the at least
two images.
##EQU8##
19. The system according to claim 18, wherein the at least one Hessian
matrix, C.sub.ij, between the at least two images is determined from the
local registration error function,
e.sub.ij (M.sub.ij).apprxeq.e.sub.ij.sup.0 +(M.sub.ij
-M.sub.ij.sup.0).sup.T C.sub.ij (M.sub.ij.sup.0).
20. The system according to claim 19, wherein a global error function that
is optimized by the global optimizer is defined by a function of the
product of overlap area between the at least two images and the local
registration error function.
21. The system according to claim 16, wherein the vector of internal
parameters comprises a representation of focal length.
22. The system according to claim 21, wherein the vector of internal
parameters further comprises a representation of image center position.
23. The system according to claim 22, wherein the vector of internal
parameters further comprises a representation of pixel aspect ratio.
24. The system according to claim 23, wherein the vector of internal
parameters further comprises a representation of skew.
25. The system according to claim 16, wherein the vector of external
parameters comprises a representation of camera orientation with respect
to a common frame of reference.
26. The system according to claim 25, wherein the representation of camera
orientation comprises Euler angles including pan, tilt, and roll.
27. The system according to claim 25, wherein the representation of camera
orientation comprises Euler angles including roll, pitch, and yaw.
28. The system according to claim 25, wherein the representation of camera
orientation comprises unit quaternions.
29. The system according to claim 16, wherein:
for signals in the overlapping region having a frequency higher than a
threshold frequency, the blender blends the signals over a first blend
region range; and
for signals in the overlapping region having a frequency lower than the
threshold frequency, the blender blends the signals over a blend region
range wider than the first blend region range.
30. A method for creating a panoramic image from a plurality of rectilinear
images obtained from a camera, the method comprising:
estimating a projective transformation during a pairwise registration
process between at least two overlapping rectilinear images obtained from
a camera, said projective transformation providing at least one Hessian
matrix relating the at least two rectilinear images;
determining a vector of internal parameters and a vector of external
parameters from the at least one Hessian matrix to minimize image
discrepancy in an overlapping region between the at least two overlapping
rectilinear images;
blending the at least two images to provide a smooth transition between the
at least two images; and
mapping the blended images onto a three-dimensional geometry surface.
31. The method according to claim 30, wherein determining the vectors
comprises determining the vectors to minimize image discrepancy in the
overlapping region during a process of global optimization.
32. The method according to claim 30, wherein the three-dimensional
geometry surface is a sphere.
33. A computer program product for creating a panoramic image from a
plurality of rectilinear images obtained from a camera, the computer
program product comprising:
a computer-readable medium; and
computer program code, encoded on the medium, for:
estimating a projective transformation during a pairwise registration
process between at least two overlapping rectilinear images obtained from
a camera, said projective transformation relating the images and yielding
at least one Hessian matrix describing a registration error landscape in
the neighborhood of the projective transformation;
determining a vector of internal parameters and a vector of external
parameters from the at least one Hessian matrix to minimize image
discrepancy in an overlapping region between the at least two overlapping
rectilinear images; and
blending the at least two images to provide a smooth transition between the
at least two images.
34. The computer program product according to claim 33, wherein the
computer program code for determining the vectors comprises computer
program code for determining the vectors to minimize image discrepancy in
the overlapping region during a process of global optimization.
35. The computer program product according to claim 33, wherein the
projective transformation comprises a 3.times.3 matrix having nine
projective parameters.
36. The computer program product according to claim 33, wherein the
pairwise registration process includes a local registration error function
between the at least two images.
##EQU9##
37. The computer program product according to claim 36, wherein the at
least one Hessian matrix, C.sub.ij, between the at least two images is
determined from the local registration error function,
e.sub.ij (M.sub.ij).apprxeq.e.sub.ij.sup.0 +(M.sub.ij
-M.sub.ji.sup.0).sup.T C.sub.ij (M.sub.ij -M.sub.ij.sup.0).
38. The computer program product according to claim 37, wherein a global
error function that is optimized during the global optimization process is
defined by a function of the sum of the products of overlap area between
the at least two images and the local registration error function.
39. The computer program product according to claim 33, wherein the vector
of internal parameters comprises a representation of focal length.
40. The computer program product according to claim 39, wherein the vector
of internal parameters further comprises a representation of image center
position.
41. The computer program product according to claim 40, wherein the vector
of internal parameters further comprises a representation of pixel aspect
ratio.
42. The computer program product according to claim 41, wherein the vector
of internal parameters further comprises a representation of skew.
43. The computer program product according to claim 33, wherein the vector
of external parameters comprises a representation of camera orientation
with respect to a common frame of reference.
44. The computer program product according to claim 43, wherein the
representation of camera orientation comprises Euler angles including pan,
tilt and roll.
45. The computer program product according to claim 43, wherein the
representation of camera orientation comprises Euler angles including
roll, pitch, and yaw.
46. The computer program product according to claim 43, wherein the
representation of camera orientation comprises unit quarternions.
47. The computer program product according to claim 33, wherein the
computer program code for blending the at least two images comprises
computer program code for:
for signals in the overlapping region having a frequency higher than a
threshold frequency, blending the signals over a first blend region range;
and
for signals in the overlapping region having a frequency lower than the
threshold frequency, blending the signals over a blend region range wider
than the first blend region range.
48. A computer program product for creating a panoramic image from a
plurality of rectilinear images obtained from a camera, the computer
program product comprising:
a computer-readable medium; and
computer program code, encoded on the medium, for:
estimating a projective transformation during a pairwise registration
process between at least two overlapping rectilinear images obtained from
a camera, said projective transformation providing at least one Hessian
matrix relating the at least two rectilinear images;
determining a vector of internal parameters and a vector of external
parameters from the at least one Hessian matrix to minimize image
discrepancy in an overlapping region between the at least two overlapping
rectilinear images;
blending the at least two images to provide a smooth transition between the
at least two images; and
mapping the blended images onto a three-dimensional geometry surface.
49. The computer program product according to claim 48, wherein the
computer program code for determining the vectors comprises computer
program code for determining the vectors to minimize image discrepancy in
the overlapping region during a process of global optimization.
50. The computer program product according to claim 48, wherein the
three-dimensional geometry surface is a sphere. |
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Claims |
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Description |
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BACKGROUND OF THE INVENTION
1. Field of Invention
The present invention relates generally to an improved system for creating
a full 360-degree virtual reality panorama from rectilinear images.
A panorama is a compact representation of the environment viewed from a 3D
position. While an ordinary image can capture only a small portion of the
environment, a panorama can capture it all, or any portion of it,
depending on the geometry in which the panoramas are represented. Recently
there has been an explosive popularity of panoramas on the world wide web
and in multimedia as an effective tool to present a photo-realistic
virtual reality. However, creating high-quality panoramas, especially
those that completely enclose space, has been difficult.
2. Description of Related Art
Various systems have been proposed for simulating a virtual reality
environment using photographic quality images. Many virtual reality
environments use 3D models or mathematical equations to create a simulated
world. The user explores this simulation in real time. Though 3D modeling
via equations has certain advantages, such as a depiction of a scene from
any arbitrary vantage point, creating images from equations generated by a
computer is seriously limited by the speed of the computer. To avoid this
problem, technology such as QuickTime.TM. VR from Apple Corporation uses
images that have already been produced, either photographically or
generated by a 3D modeling program, and stored in secondary memory.
Software only has to read the image files from a disk and display the
scene as needed, rather than calculating the scene from mathematical
models. However, a limitation of the QuickTime.TM. VR program is that it
requires that the view direction for photos reside in a single plane, such
as that obtained by rotating a camera on a tripod. It also requires that
the vertical field of view (or equivalently, the focal length) be known,
and that there be roughly equal angular increments between one photo and
the next.
Further, a panoramic movie or image can be created using specialized
hardware, such as with a panoramic camera or a fisheye lens camera.
However, such hardware is inconvenient for the average novice
photographer. In the alternative, software can be used to simulate a
panorama. This obviates the need for specialized hardware.
Though various software programs have been proposed to simulate panoramas
without the use of special hardware, these programs have certain serious
drawbacks that have not been successfully overcome to date. These include,
but are not limited to, unrealistic representations of images, lack of
proper registration and calibration of images, lack of proper blending of
images, and slow speed in registering, calibrating and blending images to
create a panorama.
SUMMARY OF THE INVENTION
Accordingly, one aspect of the present invention is to provide an improved
system and method for overcoming the drawbacks of prior techniques
discussed above.
Another aspect of the present invention is to provide for the registration,
calibration and global optimization of images, preferably captured from a
substantially single nodal position. The solution to creating a full
360-degree panorama quickly and seamlessly is divided into three steps.
The first step registers all overlapping images projectively. A
combination of a gradient-based optimization method and a
correlation-based linear search has proved to be robust in cases of
drastic exposure differences and small amount of parallax. The second step
takes the projective matrices and their associated Hessian matrices as
inputs, and calibrates the internal and external parameters of every image
through a global optimization. The objective is to minimize the overall
image discrepancies in all overlap regions while converting projective
matrices into camera parameters such as focal length, aspect ratio, image
center, 3D orientation and the like. Improved techniques for global
optimization are disclosed that give order of magnitude improvements over
prior systems of optimization. The third step re-projects all images onto
a panorama by a method employing Laplacian-pyramid based blending using a
Gaussian blend mask generated by the grassfire transform. The purpose of
the blending is to provide a smooth transition between images and
eliminate small residues of misalignments resulting from parallax or
imperfect pairwise registrations. The invention further provides for human
interaction, where necessary, for initialization, feedback and manual
options.
Further, the present invention, unlike some of the prior art, allows for
multiple views, from multiple planes and rows of images, and allows for
the arbitrary orientation of photographic images to be constructed into a
panorama, without specialized hardware such as a tripod or fisheye lens.
In addition, the present system and method can be several orders of
magnitude faster than the prior art.
The numerous aspects of the invention described herein result in a system
for registration, calibration and blending that creates high quality
panoramas from rectilinear images that is up to several orders of
magnitude faster than prior systems. In one calculation, the present
invention is up to 100,000 times faster than prior techniques. As a
consequence, the present invention could be used to construct panoramas
much quicker than previous methods. These panoramas can be used in
applications where real-time image rendering is important, such as in
real-time 3D virtual reality, the construction of background images,
computer animation, multimedia, and the like.
The above described and many other features and attendant advantages of the
present invention will become apparent from a consideration of the
following detailed description when considered in conjunction with the
accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
Detailed description of preferred embodiments of the invention will be made
with reference to the accompanying drawings.
FIGS. 1(a) and 1(b) is an artist's rendition of a composite photograph
before and after the application of the present invention.
FIG. 2 is a generalized flowchart of the various function modules
comprising one embodiment of the present invention.
FIG. 3 is a generalized flowchart of the method of operation of the
Pairwise Registration function module of one embodiment of the present
invention.
FIG. 4 is a generalized flowchart for the method of operation of the
Calibration and Global Optimization function module of one embodiment of
the present invention.
FIG. 5 is a generalized flowchart for the method of operation of the
blending function module of one embodiment of the present invention.
FIG. 6 is a screen shot of a user interface dialog window for the user
interface of one embodiment of the present invention.
FIG. 7 is a conceptual illustration on the problem of finding the proper
Laplacian pyramid level using the minor axis of an inertial ellipse.
FIG. 8 is a graphical illustration of the transition lengths for different
frequency image components (low, middle and high) used in one embodiment
of the blending function module of the present invention.
FIG. 9 conceptually illustrates the weighted average method for blending.
FIGS. 10(a) and (b) illustrate the blend mask used for blending two images
during the blending phase of one embodiment of the present invention.
FIGS. 11(a) and (b) illustrate a particular problem overcome during
blending in one embodiment of the present invention.
FIG. 12 illustrates a particular virtual reality orientation of images for
the user interface for one embodiment of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Disclosed herein is a detailed description of the best presently known mode
of carrying out the invention. This description is not to be taken in a
limiting sense, but is made merely for the purpose of illustrating the
general principles of the invention. The section titles and overall
organization of the present detailed description are for the purpose of
convenience only and are not intended to limit the present invention.
Turning to FIG. 1, there is shown a simulation of different overlapping
rectilinear images, or 2D photographs, framed by dashed lines during the
authoring portion of the present invention, as indicated by dashed lines
110. Where the images overlap there is potential for misalignment when
constructing a 3D panorarma, as indicated by blurry lines 112, for a
variety of reasons, including the arbitrary position of the camera, errors
in internal and external camera pararmeters, and, distortions that occur
when warping a 2D image to construct a 3D image space. The present
invention is designed to calibrate and align all such 2D rectilinear
images with respect to one another and globally, blend the images where
they overlap, and construct a reconstructed and relatively error free 3D
panorama image, shown conceptually in 2D form as FIG. 1(b), for any
arbitrary geometry.
FIG. 2 discloses a generalized flowchart for overall operation of the
invention. The invention is part of a system 210 comprising a computer,
having all necessary hardware, such as processing unit 212, e.g., a G3
microprocessor chip; I/O 214, such as a keyboard 216, video monitor 218,
and mouse 217; memory 220, which may be any sort of memory buffer,
preferably primary memory, e.g. RAM, that can be cached to secondary
memory, such as a hard drive. The system 210 is controlled by a program
residing in system memory 220, which also stores output data and other
data. The program is preferably written in the C or C++ language, using
classes, structures, functions, calls, translation units, headers,
subroutines, modules, and other features of structured programming, where
appropriate, of both data and source code, suitably compiled and in
executable form, in accordance with the teachings of the present
disclosure to practice the invention. The present invention is also
suitable for implementation with an interpreted language such as Java.
In constructing a panorama from rectilinear images, the system finds
solutions to three sub-problems: (1) the projective registrations of
overlapping images (shown as the "local pairwise registration" box 222 in
FIG. 2), (2) calibration and global optimization of these images, a
self-calibration in which 2D image planes are positioned as 3D planes in
space (shown as the "calibration and global optimization" box 224 in FIG.
2), and (3) the composing or blending problem in which images are ready to
be reprojected to a 3D environment map with pixels in overlap regions
being composed from multiple images, to smooth any transitional
discontinuities (shown as the "blending" box 226 in FIG. 2). Finally,
there is the projection or construction of the assembled panorama onto a
3D geometry surface, such as a cylinder, cube or sphere (defined as the
"projection" box 228 in FIG. 2).
The solutions to these sub-problems are performed by software function
modules 222, 224, 226, 228 residing in memory 220 and operating the
processor 212. The modules are designated, as explained further herein,
the pairwise registration function module 222, the calibration and global
optimization function module 224, the blending function module 226, and
the projection function module 228. A user interface module 230, also
residing in memory 220, may interact with the other modules to pass data
to and from the modules, and accept input from a human user of the system.
The modules may receive data from memory, manipulate that data as
described herein, and output the data to other modules. The three modules
222, 224 and 226 may perform feedback to pass data back to previous
modules, as indicated by arrows 233, and as described below. Although in a
preferred embodiment the modules are programmed as separate software
routines or classes, the modules may be combined into one module
performing all the designated tasks performed by separate modules.
As a final step, the fourth module, the projection function module 228,
constructs a panoramic scene by projecting the blended image onto any
designated geometry view surface, typically a cubic, polyhedral,
cylindrical or spherical surface. The projection module may be controlled
through the user interface 230 as well, to allow a user to select what
geometry will be projected onto and to control and modify other factors,
including the use of photo re-touching software such as PhotoShop.TM. for
modifying the final panorama.
Generally, the local registration, self-calibration and global
optimization, and blending involve a multi-step procedure.
First, regarding the initial local registration, and referring generally to
the generalized flowchart of FIG. 3, the system 210 reads in each
overlapping rectilinear image into main memory 220, as indicated by step
312. The images are assumed to roughly share a common nodal point (i.e.,
that point in the three-space where all rays of light converge through a
lens) with other overlapping rectilinear images. The object of the program
during local registration is to register the locally overlapping images,
by comparing common overlapping areas between overlapping images at
certain predetermined resolution levels on a Gaussian pyramid representing
the overlapped images. Different combinations of overlapping areas are
tried to achieve the optimal overlap between images (or, equivalently, the
smallest error in the error function or pairwise objective function
described herein) using the steps described herein, which generally
minimizes the average squared pixel intensity (e.g., brightness and
contrast) difference with respect to certain transformation parameters.
Initial values for parameters used in optimizing the pairwise objective
function are assumed by the computer, as indicated in step 314. The
initial values may optionally be input by a user, e.g., with a user
interface 230 as in FIG. 2, and in response to a user dialog window such
as of the kind shown in FIG. 6. Besides the global orientation (pan, tilt
and roll) the other parameters that are most likely to give instability in
convergence of the error function are bad initial estimates of the
brightness and contrast, as well as of the geometric image center of
projection of the overlapping rectilinear images. Certain parameters most
likely to create instability in the convergence of the local error
functions can be controlled (e.g., progressively dampened at different
levels of the Gaussian pyramid) to ensure convergence, as indicated by
step 318. The overlapping images are then perturbed and the local error
function with respect to these and other variables is calculated until a
minimal local error function is found, as indicated by step 320. The
minimal local error function is then stored for a particular level of the
Gaussian pyramid, as indicated in step 322, for each pairwise
registration, and is saved and later used to compute a global error
function for all the overlapping images. The local pairwise registration
module 222 iterates until the entire Gaussian pyramid is traversed,
starting from the coarsest level of the pyramid (sometimes called the
bottom, where the pyramid can be standing on its inverted top) and working
to the finest level resolution, as indicated in decision box 324. It
should be noted that at any stage throughout the registration, and
throughout the invention in general, the system may check for a user
interruption, through the user interface, that would require immediate
attention from the processor, such as to allow the user to interactively
adjust the parameters to avoid divergence or convergence to an undesired
local minimum.
As indicated in box 316 of FIG. 3, it must be determined at what level in
the Gaussian pyramid to start the local pairwise registration. One way to
find the lowest level is to select the resolution level at which it is
found that the images share at least some arbitrary number of overlapping
pixels, e.g., preferably about 30 pixels from each side, e.g., preferably
no less than 30 pixels across the overlapping area. If greater than 60
pixels of overlap is found in these areas, the size (resolution) of the
overlap region is decreased by half (going deeper into the pyramid) and
the procedure of the present invention is reiterated again. If, on the
other hand, the overlap is less than 30 pixels, then the size (resolution)
of the overlap region is increased by doubling. By utilizing
multi-resolution registration of overlapping images by way of the Gaussian
pyramid, convergence to the desired optimum is accelerated, and false
local minima are avoided.
On occasion, it may be visually apparent to a user that during registration
the images are not converging optimally. In this case user input may
manually abort the pairwise registration procedure, and the user may
manually help align the images closer before resuming automatic
registration, as before. This manual intervention is true for all aspects
of the invention. Nevertheless, the present invention is surprisingly
robust, and manual intervention is not a prerequisite for the invention to
work.
Non-optimal convergence or divergence has sometimes been found to be the
case whenever images for a spherical projection are used, especially those
in the "pole" regions of the sphere (though in general the invention can
adjust quite nicely for images that wrap around the poles). Divergence
sometimes results when the initial default parameters chosen are wildly
off or not suitable for convergence. During such instability, the images
will appear to a user to "run away" from each other. In this case, and
throughout the invention, provision may be provided in the user interface
230 of the embodiment of FIG. 2 of the present invention for manual
intervention, such as to abort the program, for the manual selection and
relative positioning of the images to be pairwise registered, and for the
selection and relative positioning of overlapping images for blending.
The iterative method of moving down a pyramid when an overlap region is
greater than, say, 30 pixels, is an attempt to prevent instability in the
error function due to problematic parameters, such as initial value errors
in the image center of projection of the images being registered, and
errors in setting initial brightness and contrast values. Techniques of
damping and annealing of problematic parameters (with damping
progressively diminished and finally set to zero as one moves up the
pyramid to finer levels) can be used to stabilize the local error function
for these problematic terms, as explained further herein.
One improvement over prior techniques has been to save the local error
function values and use them to compute and optimize the global error
function needed for optimization. This improvement also avoids having to
evaluate the entire global error function (global objective function) from
scratch. The pairwise objective functions (local error functions) are
approximated by a quadratic Taylor series, and, together with the chain
rule, the global objective function (global error function) is minimized.
Calculation of the global error function is greatly speeded up by this
procedure.
Further regarding rectilinear images taken in a non-arbitrary manner (such
as from a tripod that is rotated, or a photographer who manually "pans" a
field of view), the number of pyramid levels and optimal direction for the
blending of images overlapping in a region can be computed by the present
invention by computing the minimum eigenvalue of the 2.times.2 inertial
tensor of the overlapping region between two images. It has been found in
practice that for an arbitrary polygon representing an overlapping region,
the optimal direction for blending, as well as the width of the blending
region (which determines the level in the pyramid at which to start the
method of registration and optimization) is found along the minor axis of
the inertial ellipse found from solving for the inertial tensor of the
overlapping images. A similar method of finding the proper pyramid level
is by solving for the smallest eigenvalue of an inertial tensor of the
overlap region between images. Conceptually, such an ellipse is shown in
FIG. 7. The blending region in an arbitrarily shaped polygon region 710,
which represents the area of overlap between overlapping images, lies
along the width and direction of the minor axis 712 of the ellipse 714,
which is-calculated from the inertial tensor of the overlapping images
forming the polygon.
Thus, the results from computing the inertial tensor are used to determine
the pyramid level, blending width, and blending direction. The smallest
inertial eigenvalue is used to determine the number of pyramid levels. One
could also use the eigenvalue vector (eigenvector) to determine the
direction, or, preferably, use a blending mask, as explained herein, that
yields a grayscale ramp, which defines direction in a direction field from
taking the grayscale ramp gradient.
Next, after pairwise local registrat | | |
