Jonathan D. Victor and Mary M. Conte, "Local image statistics: maximum-entropy constructions and perceptual salience," J. Opt. Soc. Am. A 29, 1313-1345 (2012)
The space of visual signals is high-dimensional and natural visual images have a
highly complex statistical structure. While many studies suggest that only a
limited number of image statistics are used for perceptual judgments, a full
understanding of visual function requires analysis not only of the impact of
individual image statistics, but also, how they interact. In natural images,
these statistical elements (luminance distributions, correlations of low and
high order, edges, occlusions, etc.) are intermixed, and their effects are
difficult to disentangle. Thus, there is a need for construction of stimuli in
which one or more statistical elements are introduced in a controlled fashion,
so that their individual and joint contributions can be analyzed. With this as
motivation, we present algorithms to construct synthetic images in which local
image statistics—including luminance distributions, pair-wise correlations, and
higher-order correlations—are explicitly specified and all other statistics are
determined implicitly by
maximum-entropy. We then apply this
approach to measure the sensitivity of the human visual system to local image
statistics and to sample their interactions.
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Conversion between Block Probabilities and Coordinatesa
blocks
blocks
blocks
Tabulated coefficients are determined from the inverse transformation
[Eq. (8)] and the
definitions [Eqs. (13) through (16)].
Table 2.
Specification of Maximum-Entropy Ensembles from Pairs of
Coordinatesa,b
Value
Coordinate Method Pair and
{Multiplicity}
{2} 1DM
(P2)
{2} 1DM
(P1)
{4} 2DM
(P1)
{1} 2DM
(P2)
{1} 2DM
(P2)
0
0
0
0
0
{4} 2DM (P1)
0
0
0
0
0
0
0
{1} 2DMO
0
0
0
0
0
0
0
{8} 2DM (P1)
0
0
0
0
0
0
0
{4} 2DM (P1)
0
0
0
0
0
0
0
0
{4} 2DMT-DA
0
0
0
0
0
0
0
{2} 2DM
(P2)
0
0
0
0
0
0
0
0
{2} 2DM
(P1)
0
0
0
0
0
0
0
0
{2} 2DM (P1)
0
0
0
0
0
0
0
0
{4} 2DM-DA
0
0
0
0
0
0
0
0
{4} 2DM
(P1)
0
0
0
0
0
0
0
0
The row headed by each pair of coordinates indicates the
subspaces in which simple maximum-entropy ensembles may be
sampled by specific algorithms. Algorithms designated as
follows: 1DM: one-dimensional Markov process; 2DM:
two-dimensional Markov process; P1: One set of Pickard
conditions [Eq. (58) or Eq. (59)] hold; P2: Both sets of Pickard conditions hold;
2DMO: two-dimensional Markov process on oblique axes; 2DMT:
two-dimensional Markov process on a tee-shaped glider. For this
algorithm, * denotes that the parameter values obtained are
highly accurate approximations, but not exact—see Appendix C, Section C2. DA: donut algorithm.
, , and denote the roots of specific
cubic polynomials (see Appendix C, Section C3).
Because of symmetry, the 45 pairs that can be drawn
from the 10 coordinates constitute 15 unique classes;
only one member of each class is listed. The number in braces
next to each coordinate pair indicates the total number of pairs
in that class. For example, indicates the four pairs
, , , and
Tables (2)
Table 1.
Conversion between Block Probabilities and Coordinatesa
blocks
blocks
blocks
Tabulated coefficients are determined from the inverse transformation
[Eq. (8)] and the
definitions [Eqs. (13) through (16)].
Table 2.
Specification of Maximum-Entropy Ensembles from Pairs of
Coordinatesa,b
Value
Coordinate Method Pair and
{Multiplicity}
{2} 1DM
(P2)
{2} 1DM
(P1)
{4} 2DM
(P1)
{1} 2DM
(P2)
{1} 2DM
(P2)
0
0
0
0
0
{4} 2DM (P1)
0
0
0
0
0
0
0
{1} 2DMO
0
0
0
0
0
0
0
{8} 2DM (P1)
0
0
0
0
0
0
0
{4} 2DM (P1)
0
0
0
0
0
0
0
0
{4} 2DMT-DA
0
0
0
0
0
0
0
{2} 2DM
(P2)
0
0
0
0
0
0
0
0
{2} 2DM
(P1)
0
0
0
0
0
0
0
0
{2} 2DM (P1)
0
0
0
0
0
0
0
0
{4} 2DM-DA
0
0
0
0
0
0
0
0
{4} 2DM
(P1)
0
0
0
0
0
0
0
0
The row headed by each pair of coordinates indicates the
subspaces in which simple maximum-entropy ensembles may be
sampled by specific algorithms. Algorithms designated as
follows: 1DM: one-dimensional Markov process; 2DM:
two-dimensional Markov process; P1: One set of Pickard
conditions [Eq. (58) or Eq. (59)] hold; P2: Both sets of Pickard conditions hold;
2DMO: two-dimensional Markov process on oblique axes; 2DMT:
two-dimensional Markov process on a tee-shaped glider. For this
algorithm, * denotes that the parameter values obtained are
highly accurate approximations, but not exact—see Appendix C, Section C2. DA: donut algorithm.
, , and denote the roots of specific
cubic polynomials (see Appendix C, Section C3).
Because of symmetry, the 45 pairs that can be drawn
from the 10 coordinates constitute 15 unique classes;
only one member of each class is listed. The number in braces
next to each coordinate pair indicates the total number of pairs
in that class. For example, indicates the four pairs
, , , and