# imaging geometry for the pinhole camera outline: motivation |the pinhole camera

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• Imaging Geometry for the Pinhole CameraOutline: Motivation

|The pinhole camera

• Example 1: Self-Localisation

View 3View 2View 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 detectionfrom the atlastest slicedeform. fieldbefore registrationafter

• Example 6: Recognition and Localisation of ObjectsObject Models:What objects are in the image? Where are they?

• 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. Zissermans EPSRC* tutorialImage formation by common cameras is well modeled by a perspective projection:If expressed as a linear mapping between homogeneous coordinates:*

• Imaging Geometry: Internal camera parametersC is the camera calibration matrix.(u0, v0) is the principal point, the intersection of the optical axis and the image plane au=f ku, av = 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 CoordinatesThe 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 transformationsChoose 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 1The plane projective transform is called a homographyFour point-to-point correspondences define a homographyFrom the model of pinhole camera, we know the form ( denotes similarity up to scale):or, equivalently:

• Image Geometry: Computing Plane Projective Transform 2Multiplying 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 3The constrains from four points can be expressed as a linear (in unknowns hij) into an 8x8 matrix:

• Removing Perspective DistortionHave coordinates of four points on the object plane

Solve for H in x=Hx from the and corresponding image coordinates.

Then x=H-1 x

(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 tableR = [rij] is a rotation matrix, i.e. R R>=1, also

• Taxonomy of planar projective transforms IIn 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: