I have yet to come across a method for quickly, or relatively quickly calculating the EV of calling a 4bet versus presumed aces. I've previously read articles where the math have been done, and it seems to be tedious work. Lazy
as I am, I kept pondering on a way to do it easier. A couple of years have gone by since last time I really thought about it, I even went so far as to ask my math-teacher at the university about a way to do this quickly, but he couldn't help me. Or maybe he was a fellow grinder, and simply wouldn't.

In this semi-serious article, I'll show you how to calculate the equity using Photoshop! Yes, thats right, Photoshop. And also Paint, since I seem to be unable to learn how to draw straight lines in Photoshop! 

I'm pretty certain that ProPokerTools.com soon (or maybe they already have) will make a way to do this even easier, but until then, this will have to do. For those of you who already understand the math behind it, you can probably skip directly to the .avi file at the bottom of the article, and check out my method for doing it.  

Edit after feedback from Tom: 
The only thing we are interested in is "average equity when you stack off". This can, as Tom points out be done by a best fit line, or by Simpsons method, as I have seen done previously. Whether or not this method is more/less accurate, or takes more/less time I can't comment on. What follows from here, is an introduction of how to calculate EV of calling a 4bet, and for those who already knows/understands the math behind it, it won't really state anything new. After the basic introduction, I show how to use Photoshop do to the integral, instead of using a best fit line/or doing it in another way.

I also figured out that one can use Oracle, and set number of trials to which-ever number one wishes, and then size the graphs as big as possible, and this should naturally produce a more accurate estimate, which might be interesting in close-spots.

Sightblinder integration using Photoshop

For us to do this, we need to make a good bit of assumptions. 

Stacksizes will be 100bb.

All bets and raises will be pot.

None of the players involved are in the blinds.

Calvin who's range is AAxx) will shove all flops. 

Django holds JdTd8c7h and will call when he has enough equity.

Calvin opens to 3.5 bb 

Django 3-bets to 12 bb 

Calvin 4-bets to 37.5 bb

Django is now facing a 4-bet, and wants to calculate the EV of calling the 4-bet, to stack off on good flops.

The pot on the flop will be (2 * 37.5 + 1.5) = 76.5 

Stacksize on the flop will be (100-37.5) = 62.5

SPR = 62.5 : 76.5 = 1:1.224

Equity needed for Django to stack off on the flop will be (1/3.224) = 0.31

Which is 31%.

Now for the more interesting and important parts.

How often will Django have enough equity to stack off? What is his average equity on the flops where he does stack off? The last question is where Photoshop comes into play. Where you would normally have to integrate "by hand" or manually, I found a way to do it quicker. As a funny little sidenote, I think most of you would be amazed on how difficult it was for me to simply "determine non-rectangular area of picture". I also bet after I post this, someone will instantly link to a program designed specificly for this, and I'll go mental!

This graph shows the flop-equity distribution for JT87ss versus aces. The picture is taken from www.propokertools.com. I've also marked Django's stackoff-treshold. We read from the graph that Django will have at least 31% equity on 63% of flops.

The EV of calling the 4-bet, and stacking off on good flops can be broken down like this.

On 37% of flops Django folds, and therefore looses his preflop investment of (37.5 – 12) = 25.5bb. 

On 63% of flops Django will stack off, and we need to calculate the average EV of his hand, on those flops.

EV = -(0.37 * 25.5) + 0.63*(AverageEQ*201.5 – 88)

Calculating average equity using Photoshop

I'm using Photoshop to count the pixels in the image. The only thing we are really interested in, is the average equity on flops where we do stack off, since this would normally have to be calculated by integration (atleast I do not know of any other way to do it today). As an accuracry-check, we can check JT87's equity versus aces, by using photoshop.

Entire image = EI = 213360 pixels 

Under graph = 85185 pixels 

Under graph/entire image should equal JT87's equity verus aces. 

85185/213360 = 0.399 = 40%

Now, it's a problem that it is not accurate, but it's a problem I can't seem to avoid. It would help greatly if propokertools.com would allow for bigger and more accurate pictures either on their website, or in their program (Oracle). It would also help greatly if propokertools would simple make a program/script themselfs to calculate the average equity on top X% of flops. For now, however, we'll just have to work with a little error-margin.

I continue to use PhotoShop, now counting the pixels under the graph, to the left of the threshold line. 

Part of image under the graph, to the left of threshold line = Total
threshold pixels = TTPTTP = 71937 pixels

(TTP / EI) = Equity = 0.337 

Equity/Fraction of flops we can stack off on = Average equity 

0.337 / 0.63 = 0.535

Calculating EV for calling 4bet

EV = -(0.37 * 25.5) + 0.63*((0.535*201.5) – 88) 

EV = - 9,435 + 12.475 

EV = +3.04BB

Link to video where I show how it's done with PhotoShop

http://imageshack.us/clip/my-videos/19/r7wkuzdhdqshgywiqqsyrb.mp4/

On the error-margin

A couple of things straight off the bat. I used a different picture when I did the calculations as mentioned above, and did it again when I made the video. Therefore, the number of pixels counted in the video, will not be accurate with regards to the calculations mentioned above. 

It annoys me to no end that the method is not accurate (enough). But the method is fast, and it's also a method which is very easy to make into an excel-sheet. One problem is that the flops where you have 100% (or very close to 100%) equity is not properly represented in the graph from ppt.com, because of its low resolution. This could however be quickly fixed if propokertools.com will allow for larger graphs, with higher resolution. 

Another thing about the error-margin is that the whole proccess is only a model. You persume he 4-bets only aces, when most will 4-bet a slightly different range like AAxx + AKKx as an example), so while you might be annoyed that the integration method is not 100% accurate, it kind of drowns in comparison to all the other assumptions. Also; I can't image it being that more accurate to do it by hand, since the graph will be equally low-resolution", and you'll have the same problems as I have when counting pixels with photoshop. 

Thank you guys for reading, and ofcourse, all comments will be greatly appriciated. I would love it if someone can think of ways to improve the accuracy of the model, which I will shamelessly call Sightblinder method. Maybe it will stick and I'll write myself into history, probably not though!

Edit: I made some updates to the paragraph under "calculating average equity with photoshop" after feedback from Tom