Bài giảng Chapter 2: Projective geometry and camera models

Lecture with the content mapping between image and world coordinates, projective geometry vanishing points and lines, pinhole camera model, cameras & lenses, projection matrix.

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Department of Mechatronics

Contents

Mapping between image and world coordinates

• Projective geometry

• Pinhole camera model

• Cameras & lenses

• Projection matrix

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Camera and World Geometry

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Projection can be tricky…

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Projection can be tricky…

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Projective Geometry

• What is lost?

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Length is not preserved

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Vanishing points and lines

• Parallel lines in the world intersect in the image at a

“vanishing point”.

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Note on estimating vanishing points

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Camera obscura: the pre-camera

• Known during classical period in China and Greece

(e.g Mo-Ti, China, 470BC to 390BC)

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Parallel light rays which pass through a small aperture

begin to diverge and interfere with one another This

becomes more significant as the size of the aperture

decreases relative to the wavelength of light passing

through, but occurs to some extent for any size of

aperture or concentrated light source.

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Cameras and Lenses

• A lens focuses light onto the film

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 Rays passing through the center are not deviated.

 All parallel rays converge to one point on a plane

located at the focal length f.

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 There is a specific distance at which objects are “in

focus” [other points project to a “circle of confusion”

in the image].

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and reflection are co-planar.

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Large (top) and small (bottom) apertures

Khẩu độ không hoạt động độc lập, mà nó phải có kết hợp chặc chẽ với độ dài tiêu cự để có được hiệu ứng ánh sáng tốt nhất.

Máy ảnh số đã có chế độ đo sáng tự động, các chế độ chụp cài sẵn Những chương trình này đã được lập trình sẵn và dựa vào hệ thống đo sáng, máy sẽ lựa chọn khẩu độ phù hợp với khung

cảnh tại thời điểm chụp.

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Field of View (Zoom, focal length)

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view of a camera, while varying the camera distance, keeping the object in frame, results in vastly differing images At narrow angles, large distances, light rays are nearly parallel, resulting in a "flattened" image At wide angles, short distances, the object appears

distorted.

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Mathematically, for a linear system, F, defined by F(x) = y, where x is some sort of stimulus (input) and

y is some sort of response (output), the superposition (i.e., sum) of stimuli yields a superposition of the respective responses:

In the field of electrical engineering, where the x and y signals are allowed to be complex-valued (as

is common in signal processing), a linear system must satisfy the superposition property, which

requires the system to be additive and homogeneous

F(x1 + x2) = F(x1) + F(x2) F(ax) = aF(x)

F x  x   F x  F x 

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Projection

Camera Center (tx, ty,

v

u

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Finally, the camera coordinate system may be skewed due to manufacturing error, so that angle 

b etween two image axes is not equal to 90º.

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