1c. Projection-Computer Vision Introduction.pptx

PROJECTIONS
Prof. M Khairul Islam, PhD
Dept. of Computer Science & Engineering
University of Chittagong
Email: mkislam@cu.ac.bd

Projection
 Projection is the process of transforming a three-
dimensional (3D) scene into a two-dimensional (2D)
image.
 This transformation is essential for displaying 3D
objects on 2D surfaces like screens or printed images,
where depth and perspective are conveyed in a limited
visual space.

P
P’
P (3D) P’ (2D)
Plane of Projection
Projector
Center of Projection
Object Point

• Center of projection: The point from where
projection is taken. It can be either light source or
eye.
• Projection plane: The plane on which projection of
object is formed.
• Projectors: Line emerging from COP and hitting the
projection plane. When projectors hit object and
then hit projection plane the shadow of the object
will be formed on projection plane.

Parallel Projection
 Parallel projection is a kind of projection where the
projecting lines emerge parallelly from the polygon
surface and then incident parallelly on the plane.
 In parallel projection, the centre of the projection lies at
infinity.
 In parallel projection, the view of the object obtained at
the plane is less-realistic as there is no for-shortcoming
and the relative dimension of the object remains
preserves.
Chapter 14

Orthographic Projections
 The direction of projection is normal to the projection
of the plane.
 Projectors are parallel to each other making an angle
90 with view plane.
 Orthographic projections are most often used to
procedure the front, side, and top views.
 Engineering and architectural drawings commonly
employ these orthographic projections.
 Some special orthographic parallel projections involve
plan view, side elevations.
Chapter 14

Orthographic Projection
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 Orthographic Projection Matrix:
 Projection along z axis:
Transformation: x’ = x
y’ = y
z-coordinate information is lost!

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 Projection plane parallel to principal face
 Usually form front, top, side views
 Front, side and rear orthographic projections are called
elevations.
 Used in engineering drawing
Multiview Orthographic Projection

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 Allow projection plane to move relative to object
 classify by how many angles of
a corner of a projected cube are
the same
 none: trimetric
 two: dimetric
 three: isometric
q 1
q 3
q 2
Axonometric Projections

Axonometric Projections
 Isometric Projection: The direction of projection makes
equal angles with all the three principal axes.
 Di-metric Projection: The direction of projection makes
equal angles with exactly two of the principal axes.
 Tri-metric: The direction of projection makes unequal
angels with the three principal axes.

Oblique Projections
 Oblique projections are obtained by projectors along
parallel lines that are not perpendicular to the
projection plane.
 An oblique projection shows the front and top surfaces
that include the three dimensions of height, width and
depth.
 The front or principal surface of an object is parallel to
the plane of projection.
 Effective in pictorial representation.

Oblique Projection
 Oblique Projection Matrix:
 Transformation: x’ = x+k1z
y’ = y+k2z
 The z-coordinate value of the object point, leads
to a shift of x, y coordinates of the projected
point, proportional to z.
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 Arbitrary relationship between projectors and
projection plane
Oblique Projection

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 If we look at the example of the cube it
appears that the cube has been sheared
 Oblique Projection = Shear + Orthogonal
Projection
Oblique Projections

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Advantages and Disadvantages
 Can pick the angles to emphasize a particular face
 Architecture: plan oblique, elevation oblique
 Angles in faces parallel to projection plane are
preserved while we can still see “around” side

Types of Oblique Projections
 Cavalier: The direction of projections is chosen so that
there is no fore-shortening of lines perpendicular to xy-
plane.
 caviar: 45 degree
 Cabinet: The direction of projections is chosen so that lines
perpendicular to xy-plane are foreshortening by half their
length
 cavinet: 63.4 degree

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Advantages and Disadvantages
 Preserves both distances and angles
 Shapes preserved
 Can be used for measurements
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Building plans
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Manuals
 Cannot see what object really looks like
because many surfaces hidden from view
 Often we add the isometric

Perspective Projection
 The projectors intersect at a Center of Projection O.

 For convenience of deriving projection equations, we assume that
the center of projection is at the origin.
 The object and the plane of projection are now on the negative z
side.
 The distance D then refers to the z-coordinate of the object point
(towards –z).
 The plane of projection has the equation z = f.

 The z-coordinate value of the object point leads to
proportional scaling along x, y directions.
 Projections of objects located closer to the center of
projection O, appear to be larger in size compared to
objects that are farther away from O.

Perspective projections: Properties
 Projections map points from one space to
another coordinate space of lower dimension,
and hence involves loss of information.
 Projections are not invertible. All projection
matrices are singular.
 All points on a projector map to the same point
on the plane of projection.

 Parallel lines (not parallel to the projection plan)
on the object converge at a single point in the
projection (the vanishing point)
 Drawing simple perspectives by hand uses these
vanishing point(s)
vanishing point
Vanishing Points

 Vanishing Points:
These points are formed by the intersection of lines parallel
to one of the three principal axis. The number of principal
vanishing points is determined by the view plane.
 One point perspective
 Two point perspective
 Three point perspective

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One-Point Perspective
 One principal face parallel to projection plane
 One vanishing point for cube
Image plane is perpendicular to x
axis

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 On principal direction parallel to projection plane
 Two vanishing points for cube
Two-Point Perspective

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Three-Point Perspective
 No principal face parallel to projection plane
 Three vanishing points for cube

Perspective Characteristics/Anomalies
 Perspective Foreshortening: The farther an object is
from the center of projection, the smaller it appears.
 Example: Square A is larger in size than square B but at
vanishing point in viewing plane they appears to be of
same size.

Scaled Orthographic Projection
 Special case of perspective projection
 Object dimensions are small compared to
distance to camera
 Also called “weak perspective”
 What’s the projection matrix?
Image World
Slide by Steve Seitz
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Parallel vs Perspective Projections
Parallel Projection Perspective Projection
Size of 3D figure and 2D figure
remains same
Size of 3D figure is larger than 2D
figure
Shape of 3D figure and 2D figure
remains same
Shape of 3D figure and 2D figure
differs based on the viewpoint
location as well as angle
Lines of projection are straight
Lines of projection converges to a
viewpoint
The distance between view plane
and viewpoint is infinite
Distance between view plane and
viewpoint is finite since it
converges into the view point
Actual size of the 3D object is
known
Actual size of the 3D object is
unknown
An example is X-ray of teeth
An example is a picture of
mountain taken using camer

1c. Projection-Computer Vision Introduction.pptx

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