A movie camera is a sampling system in time

Motion picture cameras acquire images sequentially in time, with each image representing a sample of the real world in time.

In both digital and film based motion picture cameras, the time varying signal is measured at a fixed frame rate, usually 24 frames per second (fps).  The sampling rate is 24 cycles per second (or 24 hertz), the same as the frame rate in frames per second.

This type of system can be considered a time-sampling system, and the performance of such a sampling system can be analyzed and predicted with the well-known Nyquist-Shannon sampling theorem.

To understand the time-domain sampling of a motion picture camera, consider a simple light source, such as a light bulb, which is being photographed with a motion picture camera.  If the intensity of the light bulb is modulated sinusoidally, the intensity recorded by the film or digital sensor should correspondingly represent samples of the time-varying brightness of the light bulb, and upon playback the light intensity varying over time should match the sine wave of the original light bulb.  The real world continuously varying intensity of the light bulb was recorded as a finite string of discrete values, one value for every frame of the movie.

Aliasing can occur in sampling systems

The Nyquist frequency is defined as half the sampling frequency.  For example, in a 24 frame per second (or 24 cycles per second, or 24 hertz) motion picture camera, the Nyquist frequency would be 12 hertz.

A well-understood property of sampling systems is aliasing, which the Nyquist-Shannon theorem predicts when real-world signals with frequencies above the Nyquist frequency are sampled.   Any real world signal frequency above the Nyquist rate will be aliased, or shifted into another—false—frequency that can be represented by the sampling system.

Since motion picture cameras are sampled systems, aliasing can and does occur when the real-world frequencies exceed the Nyquist rate.  Motion picture cameras measure in three dimensions: two spatial dimensions (the two-dimensional image produced for every frame) and also time.  For our purposes, we are considering only the time sampling, i.e. temporal sampling.  Aliasing in time dimension sampling is known as temporal aliasing.

In the sinusoidally varying light bulb example, if the frequency of the sine wave were 10 hertz, and the light was sampled with a normal 24 frame per second camera system, the 10 hertz signal would be accurately recorded and reproduced because it is less than the Nyquist frequency of 12 hertz.

However, if the light bulb were varied sinusoidally at 14 hertz, the recorded and reproduced frequency from a 24 frame per second camera would incorrectly be 10 hertz.  This is because 14 hertz is 2 hertz above the Nyquist frequency, so the resulting frequency is 2 hertz below the Nyquist frequency.  This is an example of signal aliasing when a frequency higher than the Nyquist frequency is sampled.

Examples of aliasing in motion picture cameras

Temporal aliasing in motion picture cameras is exhibited in many ways.  The most common and popularly understood manifestation of temporal aliasing is known as the “wagon wheel” effect.  This is a rapidly moving wagon wheel captured by a motion picture camera appears to stop, reverse direction, or move slowly.  The higher frequencies of the motion are aliased, or falsely shifted, to appear as different frequencies.

Temporal aliasing in motion picture cameras is exhibited in nearly all situations where fast movement is being captured.

Prefiltering as a method for eliminating aliasing

In sampling systems, the method employed to eliminate aliasing is to band limit the real-world signal before the sampling takes place to ensure that no frequencies above the Nyquist frequency are allowed to enter the sampling system.  This is known as prefiltering.

Prefiltering is a procedure performed on the unsampled signal before sampling.  It is usually a low-pass frequency filter.

The ideal low-pass frequency filter for prefiltering would be unity (signal unaffected) below the Nyquist frequency, and 0 (no signal allowed) above the Nyquist frequency.

Normal motion picture cameras have some frequency prefiltering, but it doesn't work that well

Motion picture cameras do have some inherent prefilter, as the amount of time the shutter is open causes some motion blurring on a single frame (sample).

Exposure time for a frame is typically indicated as a shutter angle.  A 360 degree shutter angle indicates the frame is exposed for the entire time of the sample, while a 180 degree shutter angle indicates the frame is exposed for half of the time between samples.  For example, in a 24 frame per second motion picture system, a 180 degree shutter would expose each frame for 1/48 of a second, while a 360 degree shutter would expose each frame for 1/24 of a second.

If the amount of light allowed to pass to the sensor (film or digital sensor) during the time of the frame is plotted as a function of time, the resulting plot describes how the incoming image intensity changes over time.  This change in intensity over time is called the exposure window function, or simply the window function.

It can be seen that the exposure window functions for motion picture shutters have a sharp transition between 0 (no light) and 1 (full exposure).  Existing motion picture cameras do not implement any other values other than 0 and 1, as the shutter is either open or closed.

Filters can be represented by their response to a given frequency, and one such representation is called the modulation transfer function, or MTF.  The modulation transfer function when expressed linearly is normalized between 0 and 1, where 1 is full response to a given frequency and 0 is no response.

There is a direct mathematical relationship between the exposure window function and the prefiltering.  If an exposure window function is known, the resulting modulation transfer function of the prefilter can be calculated.

The following figure shows the MTFs of the effective prefiltering of a 180 degree and a 360 degree shutter angle compared with an ideal prefilter for a 24 frame per second system (Nyquist frequency is therefore 12 hertz).

 
Transient
 

The Tessive Time Filter

If the exposure window function were shaped differently, and had transitional values other than 0 and 1 (fully closed and fully open), the resulting modulation transfer function of the prefilter could be shaped in new ways.

The patent pending Tessive Time Filter has a continuously varied exposure windows which creates a prefilter with a MTF that better reduces aliasing frequencies.  The following figure shows the exposure window and the resulting MTF, compared to a normal 180 degree shutter MTF.  The resulting MTF has substantially less response above the Nyquist frequency, but a slightly reduced response below Nyquist.

Transient

The Tessive Time Compensator

The slight reduction in response in the region below Nyquist frequency can be adjusted with a postfilter.  A postfilter is a digital finite impulse response (FIR) convolutional filter.  The following figure shows the combined MTF of a prefilter and a postfilter employed to provide good response below Nyquist frequency while reducing response above Nyquist, compared with the MTF of a typical 180-degree shutter.

Transient