IMAGE FORMATION. Light source properties. Sensor characteristics Surface. Surface reflectance properties. Optics

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1 IMAGE FORMATION Light source properties Sensor characteristics Surface Exposure shape Optics Surface reflectance properties

2 ANALOG IMAGES An image can be understood as a 2D light intensity function f(x,y) where: x and y are spatial coordinates The value of f at any point (x, y) is proportional to the brightness or gray value of the image at that point.! Cannot be stored as such on a digital computer.

3 DIGITAL IMAGES A digitized image is one in which: Spatial and grayscale values have been made discrete. Intensities measured across a regularly spaced grid in x and y directions are sampled to 8 bits (256 values) per point for black and white, 3x8 bits per point for color images. They are stored as a two dimensional arrays of gray-level values. The array elements are called pixels and identified by their x, y coordinates.

4 PIXELS

5 IMAGE FORMATION Projection from surfaces to 2-D sensor. Where: Geometry How bright: Radiometry Stored how: Sensing

6 PINHOLE CAMERA MODEL Idealized model of the perspective projection: All rays go through a hole and form a pencil of lines. The hole acts as a ray selector that allows an i nverted i m a g e t o form.

7 MAGNET LIKE SLOPES

8 VIRTUAL IMAGE

9 CAMERA GEOMETRY Pinhole geometry without image reversal

10 COORDINATE SYSTEMS World, Camera, Image Coordinate Systems

11 CAMERA COORDINATE SYSTEM The center of the projection coincides with the origin of the world. The camera axis (optical axis) is aligned with the world s z-axis. To avoid image inversion, the image plane is in front of the center of projection.

12 1D IMAGE

13 2D IMAGE

14 DISTANT OBJECTS APPEAR SMALLER

15 PARALLEL LINES MEET

16 VANISHING POINTS The projections of parallel lines all meet at one point, called the vanishing point. As focal length and distance to camera increase, the image remains the same size but perspective effects diminish.

17 ROAD FOLLOWING Rasmussen et al., BMVC 04

18 PROJECTION IS NON LINEAR!Reformulate it as a linear operation.

19 HOMOGENEOUS COORDINATES Homogeneous representation of 2D point: Homogeneous representation of 3D point:! Projections become linear transformations.

20 SIMPLE PROJECTION MATRIX

21 INTRINSIC AND EXTRINSIC PARAMETERS Camera may not be at the origin, looking down the z-axis! Extrinsic parameters One unit in camera coordinates may not be the same as one unit in world coordinates! Intrinsic parameters

22 LINEAR CAMERA MODEL

23 <latexit sha1_base64="vcki0xckjcpbd39difrszkiyccm=">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 PRINCIPAL POINT u = X i + p u = fx/z + p u v = Y i + p v = fy/z + p v 2 3 f 0 p u K = 4 0 f p v

24 <latexit sha1_base64="7s3+g0bjnvs8dleehmwhxicgowq=">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</latexit> INHOMOGENEOUS SCALING u = u X i + p u = u X/Z + p u v = v Y i + p v = v Y/Z + p v K = 2 4 u 0 p u 0 v p v The pixels are not necessarily square.

25 <latexit sha1_base64="l0b1q5kwgxjhoffsqjpju2jrxiw=">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</latexit> s encodes the non-orthogonality of the u and v directions. Very close to zero in modern cameras. AXIS SKEW K = 2 4 u s p u 0 v p v

26 ROTATION / TRANSLATION!Rotations and translations also expressed in terms of matrix multiplications in projective space.

27 FULL PROJECTION MATRIX Hartley, Chap 6.

28 CAMERA CALIBRATION Internal Parameters: Horizontal and vertical scaling (2) Principal points (2) Skew of the axis (1) External Parameters: Rotations (3) Translations (3)! 11 free parameters.

29 LIMITATIONS Idealization because the hole cannot be infinitely small Image would be infinitely dim Diffraction effects! Use of Lenses.

30 IMAGING WITH A LENS P: point emitting light in all directions. Image plane An ideal lens realizes the same projection as a pinhole but gathers much more light!

31 THIN LENS PROPERTIES Any incident ray traveling parallel to the optical axis, will refract and travel through the focal point on the opposite side of the lens. Any incident ray traveling through the focal point on the way to the lens will be refracted and travel parallel to the principal axis. An incident ray which passes through the center of the lens will in effect continue in the same direction that it had when it entered the lens. All rays emanating from P and entering the lens will converge at P optical axis Image plane

32 CAMERA OBSCURA Used by painters since the Renaissance to produce perspective projections. Direct ancestors to the first film cameras.

33 DURER He clearly knew all about the perspective transform!

34 SHIFTING PERSPECTIVE China, 8th century: The focal point moves from one part of the image to the other. The characters are always seen at eye-level as the picture is unrolled.

35 THIN LENS EQUATION! Lens with focal distance f equivalent to pinhole camera with similar focal distance but larger aperture.

36 DEPTH OF FIELD vs APERTURE Large Aperture: Large blur circles Shallow depth of field Small Aperture: Low intensity Long exposure time

37 DEPTH OF FIELD Range of object distances (d-d ) over which the image is s u f f i c i e n t l y w e l l focused. Range for which blur circle is less than the resolution of the sensor. Small focal length > Large depth of field.

38 APERTURE aperture iris diaphragm Diameter d of the lens that is exposed to light.

39 BLUR CIRCLE blur circle (circle of confusion) Sensor Plane Out -of- Focus Point In Focus Point aperture Simple geometry: Thin lens equation:

40 <latexit sha1_base64="niudt8/qosbzdhlsmdbp5ss8ut0=">aaacwxicbvbnb9naffybfgrtinajl6fgvcobymkfocbvcofybkgvyjd6xj8nq67xzncnilz/ss4iib+cxcbxgx48avejmxma3ukrkyynot+e/2bnd2+/9zb4dpd4ydp+s+dftvlrtlneyljfpghickvtk6yki0otfqmk8/tqw1o//07aifj9saukkgixsusco3xuvp8n4lkaep7co4hzjbzbtjenlcsscnvdcfnlpg2g2cmr/cumm3e3sralpb0oie6cmjoypivabj8llbjcdchcoldr8kcihe9ywnl/ei2izcbdmo7aghvznu//jlos14ul4 CHANGING APERTURE f/11 1/30sec f/2.8 1/500sec Small aperture, long exposure. Small aperture, long exposure. r b = a s 0 f 2 (d 0 f)(d f) Small a! Small r b

41 <latexit sha1_base64="/fhj0k6p0jt4k8+bcifvgghuhk0=">aaacwxicbvbnj9mwfhqc7jbwvdgjlycatovalfychjbwcog4cmqu1itqxxlurxwcydugkps/yquh8veqcnscyjcn2rrnznpyk9dkwpckp4pwxs1bb4ed29gdu/fupxg+fptjvo3hnoevqsx5jpau1dr30ik6rw1hmss6yy/ebvwzr2ssrprht6kpk3glpzacnaewwy+qliiwyxxeqyom8ha71h53kcos7rlnxodz146l4+figywlf3epkau1u4wgzakuk3zkpf5fh0pucmirq7x3odhvtxh63ufey+eomss7getg2omr6+d0ofyefhvvsh CHANGING FOCAL LENGTH Wide field of view (small f) r b = a s 0 f 2 (d 0 f)(d f) Small f! Small r b Narrow field of view (large f)

42 DISTORTIONS The lens is not exactly a thin lens: Different wave lengths refracted differently Barrel Distortion

43 CHROMATIC ABERRATION Different wavelengths are refracted differently.

44 RADIAL LENS DISTORTIONS No Distortion Barrel Distortion Pincushion Distortion Radial distance from Image Center: r u = r d + k 1 r d 3

45 LENS SYSTEMS Aberrations can be minimized by aligning several lenses with well chosen Shapes, Refraction indices.

46 UNDISTORTING

47 UNDISTORTING Once the image is undistorted, the camera projection can be formulated as a projective transform.! The pinhole camera model applies.

48 RADIOMETRY Scene Radiance: Amount of light radiation from a surface point (Watt / m2 / Steradian) Image Irradiance: Amount of light incident at the image of the surface point. (Watt / m2) Fundamental Radiometric Equation:

49 VIGNETTING Images can get darker towards their edges because some of the light does not go through all the lenses.

50 DE VIGNETTING Y. Zheng, S. Lin, and S.B. Kang, CVPR 08

51 SENSOR ARRAY Photons free up electrons that are then captured by a potential well. Charges are transferred row by row wise to a register. Pixel values are read from the register.

52 SENSING Conversion of the optical image into an electrical image :! Quantization in Time Space

53 IN SHORT Camera geometry can be modeled in terms of the pinhole camera model, which is linear in projective space. Image radiance is roughly proportional to surface radiance and the two can be used interchangeably for our purposes.

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