6 – 02 Brightness Constancy Assumption V3

Now, optical flow assumes that points in one image frame have the same intensity pixel value as those same points in the next image frame. That is optical flow assumes that the color of a surface will stay the same over time. In practice, this is not a perfect assumption but it’s close most of the time. So in these images, which I’ll call I_1 and I_2, the intensity at the point x, y in this first image is the same as the intensity in image two at the point x plus u, y plus v. So far, we have treated these as two separate images in x and y space, but we also know that they’re related in time. How do you think we can mathematically represent that image one comes right before image two? To relate image frames in space and time, we can think about these image frames in another way. This first image is just the 2D pattern of intensity that happens at time t, and the second image is the intensity pattern that happens at time t plus one, one time step later. In this way, we can think of a series of image frames I as a 3D volume of images with x and y coordinates, pixel values at each point, and a depth dimension of time. We can write this intensity equation as a function of x and y and t. This equation is known as the brightness constancy assumption. This function can be broken down into something called a Taylor series expansion, which represents this intensity function as a summation of terms. In this case, the terms I calculated as the derivatives of the intensity with respect to x, y and t. We can simplify this expansion, and the result is an equation that relates the motion vector quantities u and v to the change in image intensity in space and in time, which are measurable changes. This is the foundation of how optical flow estimates the motion vectors for a set of feature points in a video.

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