The fundamental idea of efficient allocation process as an intellectual foundation for bit allocation analysis was explored in this paper. It should serve only to eliminate a set of clearly wasteful modes of quantizer-based allocation. To this aim we replace the goal of finding efficient allocations defined by the rate distortion function (that is, to achieve some distortion constraint at the minimum bit rate) with the goal of finding an efficient combination of allocations such that an increase in one quantizer allocation can be achieved only at the cost of a decrease in some other quantizer allocation or increasing bit consumption.
The characterization theorem in this case is based on the existence of a hyperplane that separates two disjoint convex sets, which is probably the most fundamental theorem in the mathematical theory of optimization. It allows to characterize the concept of efficient allocation process by profit maximization. The fundamental theorems of mathematical optimization play an important role in bit-allocation analysis just as derivatives and marginal analysis played an important role in the rate distortion theory. Since the bit-allocation analysis is a set-theoretic approach, it is more fundamental and powerful than a differentiable function approach.
An important result of this paper is that the technological possibilities in bit allocation (i.e., Axiom 1 through Axiom 4) can be represented in a linear model by the characterization of the set of allocation processes with resource limitations using linear inequalities. Hence the bit allocation analysis has practical and computational relevance for quantizer-based allocation. It is a typical linear programming problem, of which the computational method is well known and widely used in practice.
Here we have proposed a new technique for progressive transmission following (i) attention-based quantizer formation, (ii) intra-quantizer prioritization using embedded zerotree coding, and (iii) inter-quantizer prioritization by bit allocation analysis. One possible use of our results would be for a video codec to possess several alternative definitions of a profit vector 
. By choosing the appropriate strategy for computing vector at different bit rates, the system may attend to different parameters of interest at different bit rates within the same spatial locations.
The comparative performance of the resulting technique and the MC 3D-SPIHT was explored using a perception experiment of target detection where observers were presented with the progressive transmission of moving target sequences at extremely low bit rates. The video coder using bit allocation analysis produced the lower average bit rate for achieving target detection. A key feature of the video coder developed in this paper was to avoid the use of motion compensated temporal filtering, and consequently, motion vector components did not need to be transmitted.
Acknowledgments. We wish to thank Prof. William A. Pearlman (Rensselaer Polytechnic Inst., USA) for providing us with the N-dimensional SPIHT coding software (under patent protection). We would like to thank Dr. Alexander Toet (TNO Research Institute, The Netherlands) for providing us with image data, search times, and cumulative detection probabilities from search experiments made during the DISSTAF field test. This research was sponsored by the Spanish Board for Science and Technology (CICYT) under grant TIC2003-00473.