Imaging Geometry for the Pinhole Camera
Outline:
• Motivation
• |The pinhole camera
Example 1: Self-Localisation
View 3
View 2
View 1
Example 2: Build a Panorama(register many images into a common frame)
M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003
Example 3: 3D Reconstruction: Detect Correspondences and triangulate
Example 4: Camera motion tracking ⇒ image stabilizationbackground part of the image registered
original stabilized
original stabilized
Example 5: Medical imaging – non-rigid image registration for change detection
from the atlas
test slice
deform. field
before registration
after
Unis, 3 Oct 2005 7.
Example 6: Recognition and Localisation of Objects
• Object Models: • What objects are in the image? • Where are they?
Unis, 3 Oct 2005 8.
Example 7: Inspection and visual measurement(in the registered view angles and lengths can be checked)
Imaging Geometry: Pinhole Camera ModelThis part of the talk follows A. Zisserman’s EPSRC9 tutorial
• Image formation by common cameras is well modeled by a perspective projection:
• If expressed as a linear mapping between homogeneous coordinates:
9
Imaging Geometry: Internal camera parameters
C is the camera calibration matrix.
• (u0, v0) is the principal point, the intersection of the optical axis and the image plane
• u=f ku, v = f kv define scaling in x and y directions
Moving from image plane (x,y) to (u,v) pixel coordinates:
Imaging Geometry: From World to Camera Coordinates
The Euclidean transformation (rigid motion of the camera) is described by
Xc = R Xw + T.
Chaining all the transformations:
This defines a 3x4 projection matrix P from Euclidean 3-space to an image:
Imaging Geometry: Plane projective transformations
Choose the world coordinates so that the plane of the points has zero Z coordinate. The 3x4 projection matrix P reduces to:
Image Geometry: Computing Plane Projective Transform 1
• The plane projective transform is called a homography
• Four point-to-point correspondences define a homography
• From the model of pinhole camera, we know the form (» denotes similarity up to scale):
or, equivalently:
Image Geometry: Computing Plane Projective Transform 2
• Multiplying out:
• Each point correspondence defines two constraints:
• Two approaches can be used to address the scale ambiguity. We will use the simpler one that sets h33=1. This is OK unless points at infinity are involved
Image Geometry: Computing Plane Projective Transform 3
• The constrains from four points can be expressed as a linear (in unknowns hij) into an 8x8 matrix:
Removing Perspective Distortion
1. Have coordinates of four points on the object plane
2. Solve for H in x’=Hx from the and corresponding image coordinates.
3. Then x=H-1 x’
4. (E.g.) inspect the part, checking distances or angle
Taxonomy of planar projective transforms II
Notes:
•Properties of the more general transforms are inherited by transformations lower in the table
•R = [rij] is a rotation matrix, i.e. R R>=1, also
Taxonomy of planar projective transforms I
• In many circumstances, we know from the imaging set-up, that the image-to-image transformation is simpler than homography or can be well approximated by a transformation with a lower number of degrees of freedom.
• Three types of transforms are commonly encountered:– Euclidean (shifted and rotated, e.g. two flatbed scans of the
same image )
– Similarity (shift, rotation, isotropic scaling, e.g. two photos from the same spot with different zoom)
– Affine transformation
Image Geometry: Computing Affine Transform
• An affine transform is defined as:
• Each point-to-point correspondence provides to constraints, 3 correspondences are needed to uniquely define the transformation.
• Solving the problem requires inversion of a single 3x3 matrix:
Unis, 3 Oct 2005 20.
