Our distance and connection metrics are not enough to uniquely identify each of the twelve clip-sequences constructed from an original clip. For example, each of the following clip-sequences are distinct but share the same distance and connection values:

B’ = {21, 10, 32}        d(B’) = 3         c(B’) = 1

-B = {23, 01, 12}       d(-B) = 3         c(-B) = 1

C’ = {10, 32, 21}       d(C’) = 3         c(C’) = 1

-C = {12, 23, 01}       d(-C) = 3        c(-C) = 1

 

With this need for identification in mind, we introduce the notion of displacement of zero. We begin with some preliminary definitions.

 

Definition:

  • A clip-sequence can be described as an ordered set of six vertices {v0, v1, v2, v3, v4, v5}. We assign each of these vertices a position value: v0 = 0, v1 = 1, v2 = 2, v3 = 3, v4 = 4, v5 = 5.

 

Recall that {0”- 1”, 1”- 2”, 2”- 3”} is the “original clip” of each clip-sequence. The ordered set of vertices of the original clip is {0”, 1”, 1”, 2”, 2”, 3”} with the position values 0’’ = 0, 1’’ = 1, 1’’ = 2,  2’’ = 3, 2’’ = 4, 3’’ = 5.  

 

Definition: 

  • Displacement of zero in a clip-sequence is the difference between the position value of 0’’ in the clip-sequence and the position value of 0’’ in the original sequence. We use the letter ‘z’ to denote this value.

 

Example 1:  

B’ = {21, 10, 32}      z(B’) = z({21, 10, 32}) = z({2”, 1”, 1”, 0”, 3”, 2”}) = 3 - 0 = 3

-B = {23, 01, 12}      z(-B) = z({23, 01, 12}) = 2

 

Example 2: 

The ‘z’ value of a set of clip-sequences is the sum of the ‘z’ values of each clip-sequence in the set.

z({C’, -C}) = z({10, 32, 21}, {12, 23, 01}) = z({10, 32, 21}) + z({12, 23, 01}) = (1 - 0) + (4 - 0) =5

 

Definition: 

  • The three values  — distance (d), connection (c), and displacement of zero (z) — label the clip-sequences. The label, given in this order (d, c, z), is unique to each of the twelve clip-sequences.

 

Example 3: 

Below is a list of labels for the clip-sequences.

Clip-Sequence          Distance          Connection          Displacement          Label

A = {01, 12, 23}           d(A) = 0               c(A )= 2               z(A) = 0                    (0, 2, 0)

A’ = {10,21,32}             d(A’) = 4              c(A’) = 0               z(A’) = 1                    (4, 0, 1) 

-A = {23, 12, 01}          d(-A) = 4             c(-A) = 0               z(-A) = 4                  (4, 0, 4)

-A’ = {32, 21, 10}         d(-A’) = 0             c(-A’) = 2               z(-A’) = 5                 (0, 2, 5)

B = {12, 01, 23}           d(B) = 3               c(B) = 0                z(B) = 2                   (3, 0, 2)

B’ = {21, 10, 32}          d(B’) = 3               c(B’) = 1                z(B’) = 3                  (3, 1, 3)

-B = {23, 01, 12}          d(-B) = 3              c(-B) = 1                z(-B) = 2                  (3, 1, 2)  

-B’ = {32, 10, 21}         d(-B’) = 3              c(-B’) = 0              z(-B’) = 3                (3, 0, 3)

C = {01, 23, 12}           d(C) = 3                c(C) = 0               z(C) = 0                  (3, 0, 0)

C’ = {10, 32, 21}          d(C’) = 3               c(C’) = 1                z(C’) = 1                  (3, 1, 1) 

-C = {12, 23, 01}         d(-C) = 3               c(-C) = 1                z(-C) = 4                 (3, 1, 4)

-C’ = {21, 32, 10}         d(-C’) = 3             c(-C’) = 0              z(-C’) = 5                (3, 0, 5)

 

Discussion: 

  • “Displacement of zero” has been defined so that analogous displacement functions —  “displacement of one,” “displacement of two,” etc. — can be defined and used if necessary. In this paper, we have constructed and analyzed twelve clip-sequences based on an “original clip”. However, it is possible to construct many more; for instance, there are 120 possible permutations of a clip. If we introduce additional permutations of a clip then we can utilize the displacement functions to uniquely identify them. In other words, displacement functions complete our “labeling” system.
  • For example, if we are working with the 120 permutations of a clip-sequence, then utilizing all six displacement functions would be sufficient for labeling— clearly, because these functions are just another way of describing a sequence with six vertices. A more interesting question is:

 

Open Question:

  • How many displacement functions would we need to use — in addition to the distance and connection metrics — to uniquely label the 120 permutations of a clip-sequence? For the 12 clip-sequences that we are currently working with, we only need to add displacement of zero.
  • In terms of visual effect, the displacement functions identify points of “intersection” or synchronicity within a clip-sequence: If two clip-sequences have the same ‘z’ value, then the images that correspond to the vertex ‘0’ in each sequence will occur at the same time. However, this intersection will only be apparent if the clip-sequences are constructed from the same clip (see Examples 4 and 5 below).

 

Example 4:

 

Example 5: