Time series and images : difference and terminology

Question

A time series is an ordered collection of random variables. Considering a one-dimensional time series $A_i = {a_{i1},a_{i2},\ldots,a_{it}}$ where $t$ denotes the time index. So, the time series is a sequence $\{a\}_{j=1}^t$. Let, $B = \{A_1,A_2,\ldots,A_N\}$ be N time series.

An image consists of pixels and characterized by features. Let the feature vector $f_{object} = {f_1,f_2,\ldots,f_d}$ for a $d$ dimensional image. If there are multiple images, say N different images then for each image object there will be a feature set $F = \{f_{object1}, f_{object1}, \ldots, f_{objectN}\} $. Let the feature vector set be a N by d matrix.

My questions are,

(1) Whether the feature vector for each object $f_{object}$ is a time-series, OR

is the set $F$, that is consisting of a matrix N by d, a d dimensional time series?

(2) Is each feature vector for an object considered as a d dimensional time-series or are there d time series and then the dimension of the time series is N ?

Answer 1

If I understood correctly, $f_{1}$ is a value which describes pixel 1 in an image which has a total number of $d$ pixels.

Now you have several such images, which you call $f_{object1}$ and $f_{object2}$. In your description you do neither use the word time nor the word video regarding these objects. Therefore I would interpret that you have independent images - thus no time series at all.

Yet I assume that you are actually meaning that $F$ is video which contains subsequent images $f_{object1}$ and $f_{object2}$, ordered in time. In this case:

$f_{object}$ is not a time series, but an image.

$F$ is a time series, a video, where N is the index of time.

To be very concrete, you have a d dimensional array which changes over time. The time index is N.

Answer 2

Trivially speaking, an image is also an ordered sequence of random variables, if you imagine unrolling the image along one axis. The idea of 'time' is really just there to give an intuitive understanding to the order; that could just as easily be replaced by 'location' and then you have an image. In the context of your questions:

(1) Each feature vector for $f_{object}$ is an ordered set of random variables of size $d$

(2) Both are acceptable ways of considering this. If your $N$ different images are ordered in time, as in a video, then you might prefer to work with it as a $N$ member time series, where each member is of dimension $d$, but in terms of the object itself it has no preference.