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Image formation

Camera and lens

  • Pinhole camera: use a barrier to block off most of the ways. The opening on the barrier is aperture Pinhole Camera

  • Lens: same projection as pinhole, but gather more light Lens1

If \(o=\infty, f=i\)

Lens1

  • Field of View:

Field of View

  • Aperture: light receiving area of lens, indicated by lens diameter。 Aperture can be reduced/increased to control image brightness.

Aperture

  • F-number: It is more convenient to represent aperture as a fraction of focal length: \(D=\frac{f}{N}\). So F-number \(N=\frac{f}{D}\)

  • Lens Defocus

Lens Defocus

From similar triangles:

\(\frac{b}{D}=\frac{|i'-i|}{i'}\)

Blur circle diameter \(b=D\frac{i'-i}{i'} \propto D \propto \frac{1}{N}\)

Depth of Field

  • range of object distances over which the image is sufficiently well focused,
  • i.e. range over which blur b is less than pixel size.

Depth of Field

From the formular above we can get the way to increase DoF⭐ :

  • Increase \(o\), the object distance
  • Increase \(N\), i.e. decrease \(D\) the size of the aperture for \(D=f/N\)
  • Decrease \(f\) the focal length
  • Apart from above: Increase the sensor size

To blur the background, decrease DoF:

  • Near foreground, far background
  • Large aperture
  • Long focal length

Geometry image formation

The following are based on the pinhole camera model.

Perspective projection

Goal: 3D world coordinates -> 2D image coordinates

Perspective projection

The coordinate \(p\) can be abtained by similar triangles. The origin is the central point of the lens.
The above \(p\) can't be written as matrix mutiply vector, so it is not linear transformation.
Use homogenous coordinates to get linear transformation:

Homogeneous coordinates

homogeneous coordinates are invariant to scaling

  • Each point has an infinite set of homogeneous coordinate
  • Projection is a matrix multiplication using homogeneous coordinates.

Visualize the perspective projection: put the image plane in front of the camera

Visualize perspective projetion

Vanishing point

Properties:

  • Any two parallel lines have the same vanishing point v

  • The ray from C through v is parallel to the lines

    • v tells us the direction of the lines
    • v may be outside the image frame or at infinity

  • vanishing lines:

Multiple Vanishing Points - Any set of parallel lines on the plane define a vanishing point - The union of all of these vanishing points is the vanishing line - Note that different planes define different vanishing lines - The direction of the vanishing line tells us the orientation of the plane

Perspective distortion

  • The exterior columns appear bigger
  • The distortion is not due to lens flaws

Radio distortion

  • Caused by imperfect lenses
  • More noticeable for rays that pass through the edge of the lens
  • long focal length->pin cushion
  • short focal length/wide angle->barrel

Photometric image formation

Rolling shutter effect

  • small shutter speed-> short exposure time -> small light intensity
  • 在曝光的时间窗口内，如果相机或物体发生了移动，那么先曝光的行记录的是物体在“时刻A”的位置，后曝光的行记录的是物体在“时刻B”的位置。将不同时刻的“切片”拼接成一幅完整的图像时，就会产生扭曲。

Color space

RGB

HSV: A More Intuitive Way to Think About Color

  • Hue:
  • Saturation:
  • Value(Brightness):

Shading

  • Compute light reflected toward camera at a specific point.
  • Inputs:
    • Viewer direction, v
    • Surface normal, n
    • Light direction, I (for each of many lights.)
    • Surface parameter

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