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Computer Vision

Explore the fundamentals of cameras, visual sensors, and imaging devices in computer vision. From pinhole cameras to modern lens-equipped cameras, delve into imaging surfaces, signals, and projection equations. Learn about the different models and geometries used in capturing and processing visual data.

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Computer Vision

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  1. Computer Vision 一目 了然 一看 便知 眼睛 頭腦 影像 智慧 Computer = Image + Artificial Vision Processing Intelligence

  2. Chapter 1: Cameras ○ Human Eye Field Of View (FOV) Width × Height = 160 deg × 135 deg

  3. Eyeball • Camera

  4. CCD camera CCD (Charge-Coupled Device): rectangular grid of electron collection site laid over a silicon wafer CCD Image plane:

  5. Color camera Color image plane Color image plane: successive rows or columns are made sensitive to R, G or B light using a filter that blocks the complementary light. Bayer pattern: a filter pattern of 2 by 2 blocks, each of which is formed by 2 G, 1 R, and 1 B receptors.

  6. ○ Visual Sensors Animal eyes: birds, bats, snakes, fishes, insects (fly, bee, locust, grasshopper, cricket, cicada) Cameras: fish-eye, panoramic, omni, PTZ Imaging Devices : telescopes, microscopes

  7. ○ Imaging Surfaces Planar, Spherical, Cylindrical ○ Signals Single value (gray images) A few values (color images) Many values (multi-spectral images)

  8. Pinhole Cameras 1.1. Pinhole Cameras and Model Pinhole imaging model

  9. 2 mm ○ Big pinhole - Averaging rays blurs image Small pinhole - Diffraction effect blurs image 0.6 mm 1 mm 0.35 mm 0.15 mm 0.07 mm

  10. Diffraction (light wavelength > hole size) ＊In general, images acquired by pinhole cameras are relatively dark because a very small set of rays from a particular point hits the screen ＊ Pinholes Lenses Lenses: gather light, sharp focus

  11. 1.1.1. Perspective Projection Perspective Projection Equations

  12. Property: • theapparentsize of objects depends on • their distances from the pinhole

  13. (2) The projections of two parallel lines lying in a plane converge on a horizontal line formed by the intersection of the image plane with the plane passing through the pinhole.

  14. Assume a coordinate system x-y-z and

  15. (B) Algebraic method (1) Define a) Camera coordinate system b) Image coordinate system (2) Prove by the projective projection equations and the limit theory How about if the image plane and the floor plane are not perpendicular to each other?

  16. Assignment 1 The projections of two parallel lines, l1 and l2, which lie on the ground plane G, onto the image plane I converge at a point p. Give the image coordinates of p and its height. 19

  17. 1.1.2. Affine Projection ○ Three Models: (a) Weak-perspective projection – when the scene relief is small relative to the average distance (z0) from the camera

  18. (b) Orthographic Projection– when the scene is far away from the camera, i.e., remote scene (c) Para-Perspective Projection (see Ch. 2)

  19. Omni-Directional Imaging Geometry

  20. Mapping between image coordinates (x,y) and ground coordinates (Qx,Qy,k)

  21. Image de-warping -- Transforms an omni-directional (OD) image into a perspective (PP) image OD image PP image

  22. Hyperbola function for the mirror in the (X,Y,Z) space: where a: the semiminor axis of the hyperbolab: the semimajor axis of the hyperbola

  23. 1.2. Camera with Lenses ○ Reasons for equipping lenses: a) gather light, b) sharp focus Pinhole Cameras Modern Cameras

  24. ○ Snell’s law • The laws of geometric optics (i) Light travels in straight lines in homogeneous media. (ii) Reflection: (iii) Refraction:

  25. Proof: Find a path of light traveling from A to B with the minimal time (Fermat’s Principle) ? Show

  26. 1.2.1. Paraxial Geometric Optics -- Consider light rays close to the optical axis are all small.

  27. Approximation: Substituting into Snell’s law, Paraxial refraction equation

  28. Taylor expansions Approximation: Substituting into

  29. Paraxial refraction equation :

  30. 1.2.2. Thin Lenses (a) Rays passing through O are not refracted; (b) Rays parallel to the optical axis are focused on the focal point F’ (c) Rays passing through the focal point F are refracted to parallel the optical axis

  31. 1.2.3. Thick Lenses (real lenses) -- There is a thickness between the two spherical interface surfaces. .

  32. ○ Terminologies (a)Field of view (FOV): the scene space that projects onto the image plane of the camera FOV = , where (b) Depth of fieldorDepth of focus (DOF): the range of distances within which objects are in acceptable focus.

  33. ○Types of aberrations (A) Spatial aberration -- The rays from P striking the lens farther from the optical axis are focused closer to the lens -- The image of P in the image plane forms a circle of confusion (COC)

  34. -- Longitudinal spherical aberration (LSA): The distance between P’ and the intersection of the optical axis with a ray issued from P and refracted by the lens -- Transverse spherical aberration (TSA): The distance between P’ and the intersection of the ray with the image plane Result in shape distortion

  35. Assignment 2 Consider the camera equipped with a lens, with its image plane at position z' and the plane of scene points in focus at position z. Now suppose that the image plane is moved to . Show that the diameter of corresponding blur circle is , where d is the lens diameter. Use this result to show that the depth of field is given by and conclude that, for a fixed focal length, the depth of filed increases as the lens diameter decreases, and thus the f number increases.

  36. Fish-eye Imaging Geometry Image Scene Fisheye camera Undistorted image point Distorted image point Distorted function

  37. FOV distortion model 41

  38. Input images Recovered images 42

  39. (B) Chromatic aberration -- Due to both (i) the index of refraction of a medium and (ii) the focal length of the lens depend on the wavelength of the incident light rays

  40. ○Compound lenses: for minimizing aberrations ○ Vignetting effect: Light beams emanating from object points located off-axis are partially blocked by lenses behind the aperture

  41. Vignetting Compensation law

  42. , . 48

  43. 1.4.2. Sensor Models ○ The number of electrons recorded at the site (r, c) of a CCD array where : irradiance : reflectance T : time S(r,c) : spatial domain of the cell : quantum efficiency (the number of electrons generated per unit of incident light energy)

  44. ○ Imaging process: Current (I) Voltage Signal Digit CCD camera frame amplifier electronics grabber ○ The model for digital signal

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