Repository: Lightning-AI/torchmetrics

Issue:
Error in ERGAS metric
In the definition of ERGAS, it should be `\ ratio` rather than `* ratio` 

For example see ERGAS defined in [Du et al](https://ieeexplore.ieee.org/abstract/document/4317530), [Meng et al](https://ieeexplore.ieee.org/document/9082183)

https://github.com/Lightning-AI/torchmetrics/blob/4230cfef3d2020fffff873565acea01ad883d3e4/src/torchmetrics/functional/image/ergas.py#L82


Discussion:
Hi! thanks for your contribution!, great first issue!

Resolving patch:
diff --git a/src/torchmetrics/functional/image/ergas.py b/src/torchmetrics/functional/image/ergas.py
--- a/src/torchmetrics/functional/image/ergas.py
+++ b/src/torchmetrics/functional/image/ergas.py
@@ -89,7 +89,7 @@ def error_relative_global_dimensionless_synthesis(
     ratio: float = 4,
     reduction: Literal["elementwise_mean", "sum", "none", None] = "elementwise_mean",
 ) -> Tensor:
-    """Erreur Relative Globale Adimensionnelle de Synthèse.
+    """Calculates `Error relative global dimensionless synthesis`_ (ERGAS) metric.
 
     Args:
         preds: estimated image
@@ -119,11 +119,6 @@ def error_relative_global_dimensionless_synthesis(
         >>> torch.round(ergds)
         tensor(154.)
 
-    References:
-        [1] Qian Du; Nicholas H. Younan; Roger King; Vijay P. Shah, "On the Performance Evaluation of
-        Pan-Sharpening Techniques" in IEEE Geoscience and Remote Sensing Letters, vol. 4, no. 4, pp. 518-522,
-        15 October 2007, doi: 10.1109/LGRS.2007.896328.
-
     """
     preds, target = _ergas_update(preds, target)
     return _ergas_compute(preds, target, ratio, reduction)
diff --git a/src/torchmetrics/image/ergas.py b/src/torchmetrics/image/ergas.py
--- a/src/torchmetrics/image/ergas.py
+++ b/src/torchmetrics/image/ergas.py
@@ -29,10 +29,18 @@
 
 
 class ErrorRelativeGlobalDimensionlessSynthesis(Metric):
-    """Calculate `Relative dimensionless global error synthesis`_ (ERGAS).
+    r"""Calculate the `Error relative global dimensionless synthesis`_  (ERGAS) metric.
 
     This metric is used to calculate the accuracy of Pan sharpened image considering normalized average error of each
-    band of the result image.
+    band of the result image. It is defined as:
+
+    .. math::
+        ERGAS = 100 * \frac{h}{l} \\sqrt{\frac{1}{N} \\sum_{k=1}^{N} \frac{RMSE(B_k)^2}{\\mu_k^2}}
+
+    where :math:`h` and :math:`l` denote the spatial resolution (pixel size) of the high and low resolution images,
+    often shorted to the ratio between them :math:`r=h/l`. :math:`N` is the number of spectral bands, :math:`RMSE(B_k)`
+    is the root mean square error of the k-th band between low and high resolution images, and :math:`\\mu_k` is the
+    mean value of the k-th band of the reference image.
 
     As input to ``forward`` and ``update`` the metric accepts the following input
 
@@ -45,7 +53,7 @@ class ErrorRelativeGlobalDimensionlessSynthesis(Metric):
       value over sample else returns tensor of shape ``(N,)`` with ERGAS values per sample
 
     Args:
-        ratio: ratio of high resolution to low resolution
+        ratio: ratio of high resolution to low resolution.
         reduction: a method to reduce metric score over labels.
 
             - ``'elementwise_mean'``: takes the mean (default)


Test patch: