Fourier Analysis of Video Signals. Yao Wang
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1 Fourier Analysis of Video Signals & Frequency Response of the HVS Yao Wang Polytechnic Institute of NYU, Brooklyn, NY11201
2 Outline Fourier transform over multidimensional space (review on your own) Frequency domain characterization of video signals Frequency response of the HVS Video sampling a brief discussion Frequency Domain Analysis
3 Frequency domain characterization of video signals Spatial frequency Temporal frequency Temporal frequency caused by motion Yao Wang, 2003 Frequency Domain Analysis
4 Spatial Frequency Spatial frequency measures how fast the image intensity changes in the image plane Spatial frequency can be completely characterized by the variation frequencies in two orthogonal directions (e.g horizontal and vertical) f x : cycles/horizontal unit distance f y : cycles/vertical unit distance It can also be specified by magnitude and angle of change f s f 2 x f 2 y, arctan( f y / f x ) Yao Wang, 2003 Frequency Domain Analysis 4
5 Illustration of Spatial Frequency f s 125, arctan(2) Yao Wang, 2003 Frequency Domain Analysis 5
6 Angular Frequency Previously defined spatial frequency (cycles per pixel) depends on viewing distance. Angular frequency is what matters to the eye! f 2arctan( h / 2d)(radian) f s 180 d h f s (cycle/degree) 2h/2d(radian) 180 h d (degree) Yao Wang, 2003 Frequency Domain Analysis 6
7 Temporal Frequency Temporal frequency measures temporal variation (cycles/s) In a video, the temporal frequency is actually 2-dimensional; each point in space has its own temporal frequency Non-zero temporal frequency can be caused by camera or object motion Start simple: single object with constant velocity From Yao Wang, 2003 Frequency Domain Analysis 7
8 Temporal Frequency caused by Linear Motion Yao Wang, 2003 Frequency Domain Analysis 8
9 Relation between Motion, Spatial and Temporal Frequency Consider an object moving with speed ( v is 0( x, y), the image pattern at time t is x, v y ). Assume the image pattern at t 0 ( x, y, t) 0( x vxt, y v yt) ( fx, f y, ft ) 0 ( fx, f y ) ( ft vx fx vy f y ) Relation between motion, spatial, and temporal frequency: f t ( v x f x v y f y ) The temporal frequency of the image of a moving object depends on motion as well as the spatial frequency of the object. Example: A plane with vertical bar pattern, moving vertically, causes no temporal change; But moving horizontally, it causes fastest temporal change Yao Wang, 2003 Frequency Domain Analysis 9
10 Illustration of the Relation Yao Wang, 2003 Frequency Domain Analysis 10
11 Video in Frequency Domain Spatio-temporal temporal frequency domain is pretty empty Individual objects are either still (ie, defined only for zero temporal frequency) or moving at (nearly) constant velocity (ie, defined only on a plane in the 3d frequency-domain) Yao Wang, 2003 Frequency Domain Analysis 11
12 Frequency response of the HVS Temporal frequency response and flicker t B 1 mcos ft 2 Spatial frequency response x, y, t B 1 mcos f x, 2 Spatio-temporal response x, y, t B 1 mcos 2 f x cos 2 f t x t Smooth pursuit eye movement Yao Wang, 2003 Frequency Domain Analysis
13 Contrast Sensitivity Function t B 1 m cos fx 2 B brightness, f frequency, m modulation level What is minimum modulation level at which sinusoidal grating is visible? 1/mmin at a given frequency is the sensitivity Contrast sensitivity function also known as the Modulation Transfer Function of the human eye Humans less sensitive to variations in chrominance Yao Wang, 2003 Frequency Domain Analysis 13
14 Temporal Response Critical flicker frequency: The lowest frame rate at which the eye does not perceive flicker. Contrast sensitivity Provides guideline for determining the frame rate when designing a video system trolands 850 trolands 77 trolands 7.1 trolands 0.65 trolands 0.06 trolands Frequency (Hz) Figure 2.5 The temporal frequency response of the HVS obtained by a visual experiment. Different curves represent the responses obtained with different mean brightness levels, B, measured in trolands. The horizontal axis represents the flicker frequency f, measured in Hz. Reprinted from D. H. Kelly, Visual responses to time-dependent stimuli. I. Amplitude sensitivity measurements, J. Opt. Soc. Am. (1961) 51:422 29, by permission of the Optical Society of America. Critical flicker frequency depends on the mean brightness of the display: 60 Hz is typically sufficient for watching TV. Watching a movie needs lower frame rate than TV Yao Wang, 2003 Frequency Domain Analysis 14
15 Spatial Response trast sensitivity Cont Spatial frequency (cpd) Figure 2.6 The spatial frequency response of the HVS, obtained by a visual experiment. The three curves result from different stabilization settings used to remove the effect of saccadic eye movements. Filled circles were obtained under normal, unstablized conditions; open squares, with optimal gain setting for stabilization; open circles, with the gain changed about 5 percent. Reprinted from D. H. Kelly, Motion and vision. I. Stabilized images of stationary gratings, J. Opt. Soc. Am. (1979), 69: , by permission of the Optical Society of America. Yao Wang, 2003 Frequency Domain Analysis 15
16 Spatial sensitivity x, y, t B 1 mcos f x 2 Image provided by Amy Reibman Yao Wang, 2003 Frequency Domain Analysis 16
17 Spatiotemporal p Response itivity Contrast sensi The reciprocal relation between spatial and temporal sensitivity was used in TV system design: itivity Contrast sensi Interlaced scan provides tradeoff between spatial and temporal resolution Spatial frequency (cpd) Temporal frequency (Hz) 30 (a) (b) Figure 2.7 Spatiotemporal frequency response of the HVS. (a) Spatial frequency responses for different temporal frequencies of 1 Hz (open circles), 6 Hz (filled circles), 16 Hz (open triangles), and 22 Hz (filled triangles). (b) Temporal frequency responses for different spatial frequencies of0.5 cpd (open circles), 4 cpd (filled circles), 16 cpd(open triangles), and22cpd d (filled triangles). Reprinted from J. G. Robson, Spatial and temporal contrast sensitivity functions of the visual systems, J. Opt. Soc. Am. (1966), 56: , by permission of the Optical Society of America. Yao Wang, 2003 Frequency Domain Analysis 17
18 Smooth Pursuit Eye Movement Smooth Pursuit: the eye tracks moving objects Net effect: reduce the velocity of moving objects on the retinal plane, so that the eye can perceive much higher raw temporal frequencies than indicated by the temporal frequency response. Temporal frequency caused by object motion when the object is moving at ( v f t ( v x f x v y f y ) Observed temporal frequency at the retina ~ ft ft ( v ~ x fx v~ y f y ) ~ f 0 if v~ v, v ~ v t x x y y when the eye is moving at ( v ~ x, v~ y x ) :, v y ) : Yao Wang, 2003 Frequency Domain Analysis 18
19 ivity Contrast sensiti 0.15 No SPEM Spatial frequency (cpd) Temporal frequency (Hz) (a) SPEM 2 deg/s Spatial frequency (cpd) Temporal frequency (Hz) Spatial frequency (cpd) Temporal frequency (Hz) SPEM 10 deg/s (b) (c) Figure 2.8 Spatiotemporal response of the HVS under smooth pursuit eye movements: (a) without smooth pursuit eye movement; (b) with eye velocity of 2 deg/s; (c) with eye velocity of 10 deg/s. Reprinted from Girod, B. Motion compensation: visual aspects, accuracy, and fundamental limits. In Sezan, M. I., and R. L. Lagendijk, eds., Motion Analysis and Image Sequence Processing, Boston: Kluwer Academic Publishers, 1993, , by permission of Kluwer Academic Publishers.
20 Video Sampling A Brief Discussion Review of Nyquist sampling theorem in 1-D Extension to multi-dimensions Prefilter in video cameras Interpolation filter in video displays Yao Wang, 2003 Frequency Domain Analysis
21 Nyquist Sampling Theorem in 1-D Given a band-limited signal with maximum frequency f max, it can be sampled with a sampling rate f s >=2 f max. The original continuous signal can be reconstructed (interpolated) from the samples exactly, by using an ideal low pass filter with cut-off frequency at f s /2. Practical interpolation filters: replication (sample-and-hold, 0 th order), linear interpolation (1 st order), cubic-spline (2 nd order) Given the maximally feasible sampling rate f s, the original signal should be bandlimited to f s /2, to avoid aliasing. The desired prefilter is an ideal low-pass filter with cut-off frequency at f s /2. Prefilter design: Trade-off between aliasing and loss of high frequency Yao Wang, 2003 Frequency Domain Analysis 21
22 Extension to Multi-dimensions If the sampling grid is aligned in each dimension (rectangular in 2-D) and one performs sampling in each dimension separately, the extension is straightforward: Requirement: f s,i >= 2f max,i Interpolation/pre-filter: ideal low-pass in each dimension If the sampling grid is an arbitrary lattice, the support region of the signal spectrum must be limited within the Voronoi region of the reciprocal of the sampling lattice See Chapters 3 and 4 for sampling and sampling-rate conversion for K-D signals and for video in particular. Interlaced scan uses a non-rectangular lattice in the verticaltemporal plane. Yao Wang, 2003 Frequency Domain Analysis 22
23 Aliasing in 2D Example provided by Amy Reibman
24 Pictures off the web Example provided by Amy Reibman
25 Fourier examples of aliasing: jaggies ARReibman 2009 Example provided by Amy Reibman
26 Fourier examples of aliasing: jaggies ARReibman 2009 Example provided by Amy Reibman
27 Fourier examples of aliasing: jaggies ARReibman 2009 Example provided by Amy Reibman
28 Fourier examples of aliasing: jaggies ARReibman 2009 Example provided by Amy Reibman
29 Fourier examples of aliasing: jaggies ARReibman 2009 Example provided by Amy Reibman
30
31 Video Cameras Sampling mechanism All perform sampling in time Film cameras capture continuous frames on film Analog video cameras sample in vertical but not horizonal direction, arrange the resulting horizontal scan lines in a 1-D continuous signal Digital cameras sample in both horizontal and vertical direction, yielding pixels with discrete 3-D coordinates Sampling frequency (frame rate and line rate) Depending on the maximum frequency in the underlying signal, the human visual thresholds, as well as technical feasibility and cost Prefilter Controlled by temporal exposure, scanning beam, etc. Digital cameras may capture at higher sampling rates and then implement explicit i filtering i before converting to lower resolution Yao Wang, 2003 Frequency Domain Analysis 31
32 Typical Camera Response Temporal prefilter: the value read out at any frame is the average of the sensed signal over the exposure time Spatial prefilter: the value read out at any pixel is a weighted integration of the signal in a small window surrounding it, called the aperture, can be approximated by a box average or a 2-D Gaussian function Yao Wang, 2003 Frequency Domain Analysis 32
33 Video Display The display device presents the analog or digital video on the screen to create the sensation of continuously varying signal in both time and space. With CRT, three electronic beams strike red, green, and blue phosphors with the desired intensity at each pixel location. No explicit interpolation filters are used. Spatial filtering determined by the size of the scanning beam, temporal filtering determined by the decaying time of the phosphors. The eye performs the interpolation task: fuses discrete frames and pixels as continuously varying, if the temporal and spatial sampling rates are sufficiently high. Yao Wang, 2003 Frequency Domain Analysis 33
34 Homework 2 Reading assignment: Chapter 2 Section 3.3, Section 3.4 Written assignment Prob. 2.1,2.3,2.5,2.6,2.7 Yao Wang, 2003 Frequency Domain Analysis 34
To determine vertical angular frequency, we need to express vertical viewing angle in terms of and. 2tan. (degree). (1 pt)
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