by V. Bruni, E. Rossi, and D. Vitulano
To investigate about the existence of a formal relation between the entropy cost of an object of a real-world scene (an image I) and its visual cost.
We restrict to the class of linear distortions (α I(x,y) + β) and to the Michelson contrast  as effective and mathematically tractable tools. The formal link between the relative entropy, i.e., the Jensen-Shannon Divergence (JSD) , and the Michelson contrast relies on the ''relative'' nature of visual contrast --- an object is detected if it is different from its surrounding information.
A linear distortion of a certain region of the I changes image p.d.f. from p to q , as shown in the following Figure:
The behavior of the Michelson contrast (x) and the Jensen-Shannon divergence (solid) for different values of α and β, where '+' indicates the lower ( 1 - ε ) and upper ( 1 + ε ) bounds for the visual contrast of regions containing an unvisible stimulus, is depicted in the following figure
The absolute minimum of the JSD corresponds to a Michelson contrast (MC) equal to 1. However, a little tolerance ε can be left to the MC as the Human Visual System is not able to detect this stimulus.
Which is the corresponding cost in terms of bpp?
It can be proved that this cost corresponds to two thresholds TJS1 ( ≥ 1 ) and TJS2 ( α < 1 ), due to the asymmetry of JSD (see again the Figure above). But it can be also proved that
Hence, the least universal threshold TJS1 represents the bits budget granted to a given distortion to be not a stimulus for HVS without any additional information about the distortion (i.e. α <, = or > 1).
Some results can be found below while more details and proofs can be found in .
|1st synthetic distortion: α =.5, β=-17.5, m = 35.||Restored*: α=.997, β=-.1050, m = 35.
TJS1 = .0004, TJS2 = .0004.
|*Due to limitations of web browsers, all images are exported from matlab with JPEG at the highest quality for display in the browser. However, since the highest quality level is used, this should have a little effect on their perceived quality. Anyway, all experiments can be reproduced by the matlab codes at bottom of this page.|
|2nd synthetic distortion: α=1.3, β=3, m = 10.||Restored: α=.997, β=-.03, m = 10.
TJS1 = .0001, TJS2 = .0001.
|3rd synthetic distortion: α==1.5, β=20, m = 40.||Restored: α=.997, β=-.12, m = 40.
TJS1 = .0005, TJS2 = .0005.
|Synthetic distortion: α =.7, β=-1.5, m = 5.||Restored: α=.96, β=-.2, m = 5.
TJS1 = .0008, TJS2 = .0008.
|Synthetic distortion: α =.6, β=-6, m = 15.||Restored: α=.97, β=-.45, m = 15.
TJS1 = .0020, TJS2 = .0020.
|Synthetic distortion: α =1.3, β=4.5, m = 15.||Restored: α=.98, β=-.3, m = 15.
TJS1 = .0013, TJS2 = .0013.
|Synthetic distortion: α =.6, β=-2, m = 5.||Restored: α=.96, β=-.2, m = 5.
TJS1 = .0007, TJS2 = .0007.
|Original Image: Manuscript initially degraded by a semi-transparent Blotch.||Synthetic distortion: α =.7, β=-3, m = 10.|
|1st Restoration (unvisible): α =.97, β=-.3, m = 10.
TJS1 = .0014, TJS2 = .0014.
|2nd Restoration (slightly visible): α=.94, β=-.6, m = 10. |
TJS1 = .0028, TJS2 = .0029*.
Note that TJS1 is not equal to TJS2 in this case, while their behavior tends to coincide for a decreasing visibility of the distortion --- i.e. in most practical cases where an unvisible restoration is achieved.
|Empirical p.d.f.s. of 1st restoration||Empirical p.d.f.s. of 2nd restoration
|Region where to compute the distortion p.d.f. q.||Region where to compute the (supposed) original p.d.f. p. It has been achieved by dilating  the mask on the left and then subtracting it.|
|1st Restoration : DJS = .1064, TJS1 = .0048.||2nd Restoration : DJS = .1631, TJS1 = .0243.|
|3rd Restoration : DJS = .1545, TJS1 = .0250.|
|Region where to compute the distortion p.d.f. q.||Region where to compute the (supposed) original p.d.f. p.|
|1st Restoration : DJS = .0364, TJS1 = .0297.||2nd Restoration : DJS = .1070, TJS1 = .0538.|
|3rd Restoration : DJS = .0903, TJS1 = .0590.|