Analysis of Greedy Heuristic Scheme

We analyze the effectiveness of greedy heuristic scheme based on studying properties of trellis diagrams. Started with convex segments across single trellis and along one dimension, we discovered some satisfying conditions to construct convex extraction paths or even optimal extraction paths.

4.3.1 Convex Segments and Global Condition

All R-D convex extraction paths can be constructed from two elementary types of R-D convex segments as shown in Figure 3.1 (c):

1. Intra-trellis (local) convex segments, which consist of two refinement steps, one of each in L and T dimensions:

Π⁰_L(L, T ) = πL(L, T ) k π^T(L⁰, T ) (4.2a) Π⁰_T(L, T ) = πT(L, T )k π^L(L, T⁰) (4.2b)

Each of these convex segments traverses a single four-node trellis.

2. Inter-trellis (global) convex segments, which also consist of two refinement steps, both of them in either L or T dimensions:

ΠL(L, T ; L⁰⁰, T ) : πL(L, T )k π^L(L⁰, T ) (4.3a) ΠT(L, T ; L, T⁰⁰) : πT(L, T ) k π^T(L, T⁰) (4.3b)

Each of these inter-trellis convex segments traverses two connected trellises in L or T dimensions.

The existence of intra-trellis segments Π⁰_L and Π⁰_T cannot be controlled directly by the setting of SVC encoding process. However, they can be verified by comparing the R-D improvement γL or γT of their first refinement steps {π^L, πT} against the R-D

Chapter 4. Searching for Optimal Extraction Paths

improvement Γ⁰ of the intra-trellis segments {Π⁰L, Π⁰_T}:

Π⁰_L(L, T ) exists iff γL(L, T )≥ Γ⁰(L, T ) (4.4a) Π⁰_T(L, T ) exists iff γT(L, T )≥ Γ⁰(L, T ) (4.4b)

The existence of inter-trellis segments ΠLand ΠT, nonetheless, can be manipulated indirectly by the setting of quantization parameter QP, inter-layer dependencies and temporal dependencies among the SVC coding layers. In fact, as mentioned in Chapter 5, R-D convex paths in L and T dimensions may exist at every L and T values if para-meter setting satisfy certain constraints for well-adapted SVC encoding. The discovery of this correlation between SVC encoder setting and decoder (extraction) operation is a major contribution of this thesis. Here, since the existence of convex R-D curves in every spatial/quality and temporal layer was essential for forming convex extraction paths, we referred it as the global condition.

4.3.2 Strong Local Conditions

The simplest composition of trellis diagram is single four-node trellis. We looked into four-node trellises to figure out the conditions for existence of intra-trellis (local) convex segments. We defined that it is strong local condition satisfied if one and only one intra-trellis convex segment exists in every intra-trellis. This situation arises when there is a clear domination of R-D improvements in either L or T dimension:

Only Π⁰_L(L, T ) exists iff min (γL(L, T ) , γL(L, T⁰)) > max (γT (L, T ) , γT (L⁰, T )) (4.5a) Only Π⁰_T(L, T ) exists iff min (γT (L, T ) , γT (L⁰, T )) > max (γL(L, T ) , γL(L, T⁰))

(4.5b) With this strong local condition and the global condition, the search for the optimal extraction path can be perfectly performed using greedy heuristic scheme (steepest descent method). This simple search strategy is feasible because there exists a unique convex extraction path between the base unit S(L, T ) and any target layer S( ˆL, ˆT ) if both strong local and global conditions of R-D performance are satisfied in an SVC bitstream. Figure 4.4 illustrates a typical example. Notice that the intra-trellis convex

Temporal (T) Layer (L)

0 1 2 3 0

1 2 3

) , (LT S

ˆ) ˆ, ( TL S

Figure 4.4: A trellis diagram with convex segments satisfying strong intra-trellis (local) and inter-trellis (global) R-D conditions

segments Π⁰_Land Π⁰_T tend to concentrate in two separate regions of the trellis diagram:

Π⁰_L (drawn as magenta arrows) gathers in the upper-left corner while Π⁰_T (drawn as green arrows) gathers in the lower-right corner. Both types of convex segments bend their paths towards the boundary that separates the two regions. This is owing to the contradiction between global and strong local conditions. The inequalities in Equations 4.5a and 4.5b eliminate the chance for Π⁰_L(a magenta arrow) to appear underneath or to the right of Π⁰_T (a green arrow). The boundary between the two regions defines a convex and optimal extraction path (with maximum convexity and minimum underlying area) of the SVC bitstream because any other extraction path between the same end points would inevitably traverse at least one intra-trellis non-convex segment and thus yield a worse R-D performance. Hence, the traversal from S(L, T ) to S( ˆL, ˆT )through any four-node trellis would follow the intra-trellis convex segments, which can be reduced as choosing steepest descent refinement steps at any steps.

4.3.3 Weak Local Conditions

Among all the R-D trellises of an SVC bitstream, some of them contain R-D convex segments but lack a clear domination of R-D performance in either L or T dimension.

We named it weak intra-trellis (local) condition if R-D performance of the four refine-ment steps in four-node trellis satisfies Equations 4.4a and 4.4b but not Equations 4.5a

Chapter 4. Searching for Optimal Extraction Paths

Temporal (T) Layer (L)

0 1 2 3 0

1 2 3

) , (LT S

ˆ) ˆ, ( TL S

Figure 4.5: A trellis diagram with convex segments satisfying weak intra-trellis (local) and inter-trellis (global) R-D conditions

and 4.5b. In these cases, both Π⁰_L and Π⁰_T exist in each of these trellises. The existence of multiple convex segments in one or more trellises revokes the unique existence of convex extraction path. Hence, the greedy heuristic scheme could not promise to fine the optimal extraction path. However, the difference in underlying area of two convex R-D curves in single four-node trellis is usually insignificant. Furthermore, the trellises that satisfied weak local condition almost appeared along the boundary between two regions of strong local conditional trellises empirically. Figure 4.5 shows an example of this situation. As a result, all the convex extraction paths may have similar underlying area of R-D curves and the greedy heuristic scheme can find one of them. Even though it may not be the optimal extraction path, it would have similar R-D performance.

4.3.4 Fractional Violation of Local Conditions

In some rare cases (when a subjective measures such as the mean opinion scores is used to quantify playback picture quality), the local R-D condition (i.e. the existence of intra-trellis R-D convex segments) may fail to be upheld. As a result, no convex ex-traction path exists between some S(L, T ) and S( ˆL, ˆT )pairs. A near-optimal extraction path with a slightly non-convex R-D curve [Section 3.3] may have to be accepted as a substitute instead. In the search for the near-optimal extraction path, extraction path segments with R-D curves that contain slight deviation from convexity [Criterion 3] are included into consideration. Figure 4.6 provides an example that contains a violation

(0,0)

Figure 4.6: R-D mesh and trellis diagram of an SVC bitstream with fractional viola-tion of intra-trellis (local) R-D condiviola-tions

of local R-D condition in the lower-left trellis. A slightly non-convex segment Π⁰_T(0, 0) shown as a dashed magenta arrow would not be pruned during searching. In these cases, the greedy heuristic scheme generally has no promise to find the optimal/near-optimal extraction paths. However, the violation of local conditions rarely occurred.