Understanding "IntensityData" coordinates

Question

Update - Playing around a bit, is it true that we need to process the list of "IntensityData" values $r$ as follows:

r = Partition[r,k];
r = Transpose@r;

And then apply:

vmapped= c + {v[[1]] - k/2 - 0.5, k/2 - v[[2]] + 0.5}

It looks like this properly maps the point $v$ to the larger image containing the morphological component with "IntensityData" $r$, however, it's difficult to tell if there's a small error here?

**

I would like to ask a follow-up to a previous question - Understanding the computation of IntensityCentroid? - since this question was answered well, but I am still a bit confused about the coordinates for "IntensityData".

Let $m$ be a square morphological component with $k \times k$ pixels. Let $c$ be the output of ComponentMeasurements[{m,img},"Centroid"], let $r$ be the output of ComponentMeasurements[{m,img},"IntensityData"]. For an example, we can set:

k = 7;
c = {157.5, 92.5};
r = {0.509752, 0.958211, 0.177492, 0.679586, 0.651634, 0.159438, 0.249294,      
     0.840233, 0.403427, 0.706836, 0.0924188, 0.523653, 0.907517, 0.219175, 
     0.134596, 0.599349, 0.320155, 0.417359, 0.767355, 0.280942, 0.960531, 
     0.332421, 0.756754, 0.186532, 0.188323, 0.457144, 0.80254, 0.252036, 0.66885, 
     0.0437862, 0.674661, 0.125856, 0.134035, 0.62525, 0.328863, 0.84693, 
     0.644811, 0.877901, 0.0685183, 0.307649, 0.481993, 0.695101};

We then partition $r$ in the following way to structure the data in the form of ImageData:

r = Partition[r,k];

I then do "something" to calculate a coordinate with respect to the coordinate frame in Image[r]. Call this coordinate $v$, where we might have, for example, v = {4.232,7.827} (perhaps this is an intensity centroid in the the coordinate system for $r$).

Using image coordinates rather than indices, i.e. the coordinate system used when outputting a value for ComponentMeasurements[{m,img},"IntensityCentroid"], how can I use only the component centroid $c$ to map $v$ to my original image, img? Note that I only provide you with $k$, $c$, and $v$ for this mapping, not the original image.

Ok, let's provide a small example using ten morphological components. Here, the first set of coordinates in each three-element array represents $v$, which is something similar to an intensity centroid computed with the output of r = Partition[r,7]; where $r$ is the "IntensityData" output for some box-like morphological component with a 7-pixel edge-length. The second set of coordinates in each array ({72.5,250.5}, {208.5,246.5}, etc.) represents the output of ComponentMeasurements[{m,img},"Centroid"] for the morphological component used to grab $r$. The final set of coordinates is the output of ComponentMeasurements[{m,img},"IntensityCentroid"] for the particular morphological component.

componentOneArray={{4.1209,2.39237},{72.5,250.5},{73.1307,250.376}}
componentTwoArray={{4.10345,2.51433},{208.5,246.5},{208.993,246.415}}
componentThreeArray={{2.74777,11.3037},{18.5,229.5},{16.6019,229.926}}
componentFourArray={{3.75292,2.4059},{92.5,228.5},{93.1028,228.757}}
componentFiveArray={{3.74452,3.1833},{109.5,224.5},{109.302,224.697}}
componentSixArray={{3.76436,2.40024},{137.5,223.5},{138.098,223.743}}
componentSevenArray={{3.52801,2.6132},{221.5,221.5},{221.89,221.992}}
componentEightArray={{4.10626,2.43366},{64.5,209.5},{65.0603,209.391}}
componentNineArray={{4.08676,3.06379},{98.5,208.5},{98.4326,208.419}}
componentTenArray={{3.90224,3.10008},{30.5,205.5},{30.3848,205.606}}

How can we use use the second element of each list, $c$, to map the first element of each list, $v$, to the coordinate system used for the computed intensity centroid of the morphological component?