Is 4K BS?

I hear that Sony & friends are going to start telling us that our HDTVs aren’t good enough and we all need to upgrade to 4K (which is twice the dimensions and 4 times the pixels of 1080p). NBC news says the experts are unconvinced, and quotes one of them: retina scientist, photographer, and blogger Bryan Jones. I thought I’d do the numbers and yeah, I think it’s probably BS.

[Update: Wow, this piece touched a nerve and generated a ton of interesting follow-ons, which I wrote up in More Things About TV.]

In Jones’ widely-quoted piece Apple Retina Display, he argues that the literature shows the human eye has an angular resolution of about an arcminute (1/60 degree). So, sitting in front of an NFL game, it occurred to me to wonder how far apart, in arcminutes, the pixels in my TV are. Let’s call Arc-Minutes per Pixel AMpP for short, and if it’s less then 1, that should mean that our eyes can’t distinguish pixels, which is the goal.

I’ve got a 42" screen and sit 7½ feet from it. Per this TV Size Chart, its height is 20.6 inches, and the total angle subtended, assuming my eye is somewhere near the vertical middle of the screen (this is all in radians):

angle = 2 × arctan(height/2, distance)

Then of course the AMpP is:

(angle / resolution) / (π / (60 × 180))

Which I think the following chunk of Ruby does.

def tv_height(diagonal)
  diagonal * 0.490261259680549
end

ARC_MINUTE = Math::PI / (60 * 180)

def pixel_arcminutes(diagonal, distance, resolution)
  height = tv_height(diagonal)
  tv_angle = 2 * Math.atan2(height / 2, distance * 12.0)
  (tv_angle / resolution) / ARC_MINUTE
end

diagonal_in_inches = ARGV[0].to_f
distance_in_feet = ARGV[1].to_f 
resolution_in_lines = ARGV[2] ? ARGV[2].to_f : 1080.0
p_in_a = pixel_arcminutes(diagonal_in_inches,
                          distance_in_feet,
                          resolution_in_lines)

puts "1 pixel in arcminutes: #{p_in_a}"

If I have the math right, then from where I sit, on my TV, my AMpP is 0.73. Yay!

But my 42" set, as big as we can fit in our video cave, is hardly state-of-the art. There’s a suggested-TV-size calculator at HDTVtest.co.uk, which suggests that a real video weenie ought to have a 55" screen at 7½ feet; which gives you around 0.95 AMpP. Hmm, I bet the math behind that site is designed to make the AMpP come out around 1.0.

At my distance, I’d need a 60" set before the AMpP hits 1.0 (at which point I’d be worrying about brain damage). So my take-away is that the 1080p TV standard is a pretty good match for our eyes.

The experts in the NBC article at the top suggest TV designers should worry more about color accuracy and dynamic range. I think a much bigger problem is that there’s often nothing on worth watching.

[Thanks to Bryan Jones for glancing at my math. He made a bunch of good points that deserve space of their own; I hope he writes them down.]

Contributions

Comment feed for ongoing:

From: Andrew Ducker (Dec 22 2013, at 15:16)

The chart on this page is pretty useful:

http://carltonbale.com/1080p-does-matter/

Given the size of your TV and distance you are from it, it'll tell you the maximum resolution you can see.

At 7.5ft you need to have a 60" TV to even begin to see a difference. Which, I think, would give me a headache...