INT345: COMPUTER VISION

UNIT-3
 Lecture-1
Stereo Geometry
Stereo Geometry

• It refers mathematical relationship between two

cameras
• That is left and right camera
• That capture same scene from different viewpoints.
• It is the foundation of stereo vision,
• which helps in depth estimation,
• 3D reconstruction, and robotic vision.
Stereo Geometry

• (A) Two-View System

• A stereo system consists of two cameras
• capturing the same scene from different
positions.
• Left Camera (CL) and Right Camera (CR).
• Each 3D point P in space appears as:
• pL in the left image
• and pR in the right image.
Stereo Geometry

• B. Camera Projection Model

• A 3D point P(X,Y,Z) in world coordinates projects onto
the left and right image planes:

• K is the intrinsic matrix (camera parameters).

• R and t are extrinsic parameters (rotation and
translation).
Fundamental Elements Stereo Geometry

• (A) Disparity
• Disparity measures the difference in pixel
coordinates
• of corresponding points in two images.
• It is used to estimate depth.
• where: d is disparity.
• xL, xR are the x-coordinates of the point
• in the left and right images.
Fundamental Elements Stereo Geometry

• Depth Estimation
• From disparity, we compute depth Z
using:

• f is the focal length.

• B is the baseline distance
• (distance between the two camera
centers).
• d is the disparity.
Fundamental Elements Stereo Geometry

• Rectification
• Rectification aligns stereo images
• corresponding points appear on the same
horizontal line,
• simplifying disparity computation.
• In rectified images, epipolar lines are parallel.
Example: Depth Computation
• Given:
• Baseline: B=0.2m
• Focal length: f=700pixels
• Left image coordinate: xL=120
• Right image coordinate:
xR=110
Example: Depth Computation
Epipolar Geometry
• Epipolar geometry describes the relationship
• between corresponding points
• in two images taken from different viewpoints.
• Instead of searching the entire second image for a
match,
• It restricts the search to a line called the epipolar
line.
Epipolar
Geometry
Real world scene captured from two different
view points
 Elements of Epipolar Geometry
• gray region is the epipolar plane.
• The orange line is the baseline,
• while the two blue lines are the epipolar
lines.
Elements of Epipolar Geometry

• Two cameras observing the same 3D point P,

• whose projection in both planes is located at p and
p’
• The camera centers are located at O1 and O2,
• and the line between them is referred to as the
baseline.
• plane defined by the two camera centers and P the
epipolar plane.
Elements of Epipolar Geometry
• Epipoles
• The epipole is the projection of one camera center
• into the other camera's image plane.
• All epipolar lines pass through the epipole.
• Epipolar Plane
• A plane formed by the two camera centers and a 3D point.
• It intersects both image planes, defining the epipolar lines.
• (C) Epipolar Lines
• The line along which a point in one image must lie in the other
image.
Elements of Epipolar Geometry

• The locations of where the baseline intersects

• the two image planes are known as the epipoles e and
e’.
• the lines defined by the intersection of the epipolar
plane
• and the two image planes are known as the
epipolar lines.
• relationship between two image points is given by
the
• fundamental matrix F
 Fundamental Matrix (F)

• The fundamental matrix (F) is a 3×3 matrix that

• relates corresponding points between two images.
• If x and x′ are corresponding points in the two
images, then:
Fundamental Matrix (F)
• The fundamental matrix encapsulates the epipolar
constraint

• If F is known, corresponding points can be

found :
• by computing epipolar lines.
• F can be computed using at least :
• 8 point correspondences (8-Point Algorithm).
Example:
Thank
You
INT345: COMPUTER VISION
UNIT-3
Lecture-2
Normalized 8-Point Algorithm
• It is a robust method
• for computing the fundamental matrix
• from at least 8 corresponding points in two
images.
Normalized 8-Point Algorithm
• Step 1. Normalize the points

• Step 2. Solve for step using linear system

SVD
Normalized 8-Point Algorithm
Ranks of
matrix
• Rank 0 : All elements are zero.
• Rank 1: Row 2 = 2 × Row 1,
• Only one independent row exists.
• Rank 2: Row 3 = Row 1 + Row 2,
• meaning it is a linear combination of the first two
rows.
• Only two independent rows exist.
• Rank 3: All three rows are independent
Geometric distance computaion
Geometric distance computaion
Geometric distance computaion
Camera motion
• It refers to the movement of a camera in a 3D
scene.
• camera motion is essential for applications
like:
• 3D reconstruction,
• AR
Types of Camera Motion
• A camera can move in six degrees of freedom (6-
DoF):
• Translation: Moving along the X, Y, or Z axis.
• Rotation: Rotating around the X, Y, or Z axis.
• This can be mathematically represented using:
• Rotation Matrix R (3×3)
• Translation Vector t (3×1)
Types of Camera Motion
3-DOF (IN STRAIGHT LINE)
• Forward / Backward (Z-translation)
• Moves along the Z-axis (closer to or farther from a
subject).
• Example: Moving toward an object (zooming in physically)
or away
from it.
• Left / Right (X-translation)
• Moves along the X-axis (side-to-side motion).
• Example: A camera moving left or right in a film scene.
• Up / Down (Y-translation)
• Moves along the Y-axis (vertical motion).
• Example: A drone rising or lowering without tilting.
Rotational Movements (3-DoF) – Rotating
Around an Axis
• These describe how the camera rotates in place without
changing its position:
• Pitch (X-rotation, Tilting Up/Down)
• Rotates around the X-axis (tilting the camera up or down).
• Example: Looking up at a tall building or down at the ground.
• Yaw (Y-rotation, Panning Left/Right)
• Rotates around the Y-axis (turning left or right).
• Example: A security camera scanning an area from left to right.
• Roll (Z-rotation, Rotating Clockwise/Counterclockwise)
• Rotates around the Z-axis (tilting sideways).
• Example: A GoPro mounted on a car tilting sideways during a sharp
turn.
Motion Models
• Rigid Body Motion (Most Common)
• The camera moves with rotation R and translation t
• but does not deform.

• Affine Motion
• Includes scaling, shearing, rotation, and translation.
• Used in cases where the camera model is not strictly
pinhole.
• Homography (Planar Motion)
• Assumes the scene is planar.
• Useful in image stitching and panorama generation.
 Motion Type Description Example
Dolly Moves forward/backward Pushing in on an actor’s
face
Side-tracking a
Tracking Moves left/right
running character

Crane (Jib) Moves up/down Aerial views or rising shots

Pan Rotates left/right Scanning a city skyline
Tilt Rotates up/down Revealing a tall building
Roll (Dutch Angle) Tilts diagonally Creating disorientation
Dramatic zoom-in
Zoom Changes focal length
on a character
Following a character
Steadicam Smooth motion
through a
house
Handheld Unsteady movement Documentary-style realism
Dolly Zoom Zoom + Dolly The Vertigo effect
Optical Flow for Motion Estimation
• Optical flow estimates the pixel displacement
• between two consecutive frames,
• which helps determine the motion of the camera.
• Lucas-Kanade Optical Flow: Works well for small
motion.
Linear Triangulation Method in
Stereo Vision
• It is a mathematical technique used to find
• 3D position of point given its
• 2D projections in two camera images.
• It helps reconstruct 3D scenes from stereo
images.
Thank
You