What Does 'Zero Order Hold' and 'First Order Hold' Mean?

Question

While studying the Image Magnification in spatial domain, I have come across this definition of Image Magnification by Replication:

Replication is a zero order hold where each pixel along a scan line is repeated once and then each scan line is repeated.

And the definition for Image Magnification by Interpolation is given as:

Linear interpolation is a first order hold where a straight line is first fitted in between pixels along a row. Then the pixels along each column are interpolated along a straight line.

Can you please explain the terms 'zero order hold' and 'first order hold' in layman's terms, so that I could understand these two definitions? Thanks.

Answer 1

We need to assume the reader knows some basic stuff to answer that.
Let's give it a try.

Lets understand the sentence - Zero / First Order Hold.
We have the Zero / First Order and the Hold.

Zero / First Order hold means the order of the Taylor Series of the function we use to interpolate. In other words, the degree of the Polynomial we can write the function with. Zero Order means the function is constant, we interpolate the same value in the missing parts.
First order means we can use linear function to interpolate (Line with a slope).

Hold means we hold the parameters to be the same until the next sample.

Answer 2

The terms are clearly defined, in the excerpts and comments, so I assume you are looking for the origins of the terms. The word “hold” is my guess where the problem is.

My understanding is that the terms originally were used in mixed digital-analog control systems.

Feedback in real-time analog control systems has very little, often assumed negligible, time latency.

Reconstruction filters at the output of D/A converters exhibit a tradeoff between latency and reconstruction error for a given sample rate.

The zero order hold is reconstruction with steps and has a low latency, which is a favorable attribute in real-time control loops.

Lathi’s textbook, Signals Systems and Control, introduces a mixed Laplace-Z transform based on the zero order hold.

The first order hold is a refinement of the zero order but you have to wait for the next value to perform the linear interpolation.

I’m not sure why the terms zero and first order hold would be used in image processing because one typically has all the original pixels at once (do you repeat pixels from the left or right? up or down?), but the original context is in hybrid analog-digital electrical circuits. Perhaps in a raster imaging system like NTSC video, the idea of a hold makes sense.

Incidentally, zero and first order hold reconstruction sound terrible in audio applications without further filtering.

In summary, I believe that your confusion stems from the use of the word “hold” which really makes sense when you have a notion of “last”, ”current”, and “next” natural ordering.