The question can be construed as requesting a nonparametric estimator of the median of a sample in the form f(min, mean, max, sd). In this circumstance, by contemplating extreme (two-point) distributions, we can trivially establish that
$$ 2\ \text{mean} - \text{max} \le \text{median} \le 2\ \text{mean} - \text{min}.$$
There might be an improvement available by considering the constraint imposed by the known SD. To make any more progress, additional assumptions are needed. Typically, some measure of skewness is essential. (In fact, skewness can be estimated from the deviation between the mean and the median relative to the sd, so one should be able to reverse the process.)
One could, in a pinch, use these four statistics to obtain a maximum-entropy solution and use its median for the estimator. Actually, the min and max probably won't be any good, but in a satellite image there are fixed upper and lower bounds (e.g., 0 and 255 for an eight-bit image); these would constrain the maximum-entropy solution nicely.
It's worth remarking that general-purpose image processing software is capable of producing far more information than this, so it could be worthwhile looking at other software solutions. Alternatively, often one can trick the software into supplying additional information. For example, if you could divide each apparent "object" into two pieces you would have statistics for the two halves. That would provide useful information for estimating a median.
If you are willing to assume that $Y$ has a symmetric distribution within the two groups, then the medians of the two groups (i.e., $q_{21}$ and $q_{22}$) could be used in place of the means. Furthermore, if you are willing to assume that $Y$ is normally distributed within the two groups, then you could make use of the relationship between the IQR and the SD for the normal distribution, namely, $SD \approx IQR / 1.35$. So, you can compute the two IQRs with $IQR_1 = q_{31} - q_{11}$ and $IQR_2 = q_{32} - q_{12}$, transform them to SDs, pool those two SDs in the usual manner, and then you have all of the pieces to compute the standardized mean difference.
Example: For your example data, this would be $$IQR_1 = 174 - 58 = 116$$ $$IQR_2 = 158 - 31 = 127,$$ so $$SD_1 = 116 / 1.35 = 85.93$$ $$SD_2 = 127 / 1.35 = 94.07.$$ Therefore, $$SD_p = \sqrt{\frac{(80-1)85.93^2 + (46-1)94.07^2}{80+46-2}} = 88.97.$$ And finally: $$d = \frac{85-79}{88.97} = 0.07$$ Now you could use the usual equation to estimate the sampling variance of $d$ (Hedges & Olkin, 1985): $$v = \frac{1}{80} + \frac{1}{46} + \frac{0.07^2}{2(80+46)} = 0.034.$$
Remarks: Under normality, $d$ should be an okay estimator of the true SMD. However, the use of medians in place of means and the estimation of the SDs via the IQRs involves a loss of precision. The usual equation for the sampling variance of $d$ does not reflect that, so it yields values that are probably too small (on average).
Also, the appropriateness of this method hinges on the symmetry/normality assumption. Unfortunately, authors typically choose to report medians and IQRs whenever they suspect that $Y$ has a non-normal/symmetric distribution. So, I would regard this method only as a rough approximation.
References:
Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. Orlando: Academic Press.
Best Answer
You can check Wan et al. (2014)*. They build on Bland (2014) to estimate these parameters according to the data summaries available. See scenario C3 in their paper :
$$ \bar{X} ≈ \frac {q_{1} + m + q_{3}}{3}$$
$$ S ≈ \frac {q_{3} - q_{1}}{1.35}$$
or, if you have the sample size :
$$ S ≈ \frac {q_{3} - q_{1}}{2 \Phi^{-1}(\frac{0.75n-0.125}{n+0.25}) }$$
where $q_{1}$ is the first quartile, $m$ the median, $q_{3}$ is the 3rd quartile and $\Phi^{-1}(z)$ the upper zth percentile of the standard normal distribution.
So, in R :
* Wan, Xiang, Wenqian Wang, Jiming Liu, and Tiejun Tong. 2014. “Estimating the Sample Mean and Standard Deviation from the Sample Size, Median, Range And/or Interquartile Range.” BMC Medical Research Methodology 14 (135). doi:10.1186/1471-2288-14-135.