The results of running our intensity and average change calculations on CS data on objects moving at different speeds shows that a fully deterministic relationship exists between our calculations and the speed of the object along the direction of motion.
Recall our three calculations: Total intensity is a measurement of non-zero area in a single frame. Mean absolute change is the absolute value of the average change between basis projections in subsequent frames. Mean change squared is the mean of the squared difference between basis projections in subsequent frames.
The three calculations were computed for every frame of a movie clip. Sample results of the metrics over time are shown below.
| Mean Magnitude of Coefficient Differences (2pixels/frame) |
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| Mean Magnitude of Coefficient Differences (18pixels/frame) |
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Since the video clips analyzed contained an object moving at constant velocity, it is expected that the average absolute and squared changes would be constant across a frame. The noise in the data comes from the random nature of the basis used.
To plot average mean and absolute change with respect to velocity, the average value of each calculation is taken from the above data. The results are graphed below.
The first three graphs show variations in absolute mean differences with respect to velocity for different shapes. Each shape has a unique curve because of the shape of image overlap for different speeds. As future work, these minor differences could allow us to distinguish shapes from such data. The trends observed here are not linear, but a fit curve can easily be generated from a few data points and used to classify new data. Since the calculation of this feature is very simple, it could be implemented in low power applications.
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With help from Ilan Goodman, we saw that Parseval's theorem dictates that the two-norm of the change in area is linearly proportional to the two-norm of the change in the compressed sensing coefficients. Since the change in area is linear in velocity for rectangles, this predicts that the average squared change will be linear for a rectangle. This relationship is supported by our data.
| Mean Change Square for Vertical Rectangle |
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The linear relationship does not hold for other shapes because the overlap between frames is not linear in velocity.
| Mean Change Squared for Triangle |
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| Mean Change Squared for Circle |
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The velocity of a known object can be determined using a fit of this data for any shape object. However, there exists an upper bound to the velocities that we can detect. Consider an object of size x pixels along the direction of motion. Once the velocity exceeds x pixels per frame, the amount of overlap has already gone to zero and the exact velocity cannot be determined. This can be described as analogous to aliasing.
| Velocity Detection Upper Limit |
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This work is licensed by Siddharth Gupta, Veena Padmanabhan, Grant Lee, and Heather Johnston under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.
Last edited by Heather Johnston on Dec 23, 2005 12:26 pm -0600.
