Photo Editing – What does the gamma histogram mean?

A good way to analyze an image is to measure different locations (densities or intensities) and then draw these measurement points using graph paper. When done, the diagram resembles a half bell curve. We divide this curve into regions. The lower part is called "toe". This graph graphically indicates that the image is starting to form slowly. Next, the area of ​​the straight line will be referred to. This part of the graph graphically shows a proportional response to the exposure light. We can measure the angle of the straight line with a protractor. When the image contrast is low, the angle of the straight line is pressed. If the image has a high contrast, the straight line shows that fact. Its angle will be 45 ° or larger.
Historically, we have transformed this angle into a tangent (TAN) using trigonometry. Most images with sufficient contrast have a straight line with an angle of about 40 °. The tan of 40 = about 0.8. This is traditionally the target contrast for images that show a respectable contrast. We're talking about the science of sensitometry – taking test shots and the science of densitometry – measuring photographic images. It is common to call the TAN of the angle of the straight line "gamma".

The following graphic refers to the film – digital images are a subset of the film when it comes to the determination. we also draw these.

35 = gamma 0.7 = flat 45 = gamma 1.0 = contrast

Enter the image description here

Number theory – Stirling's approximation for normalized $ Gamma $

To let
$$
H (s) = frac {1} {2} s (1-s) pi ^ {- s / 2} gamma left ( frac {s} {2} right).
$$

With the approach of Stirling for the gamma function I would like to prove that
$$
frac {H (1/2 + it) overline {H} (1/2 + it + iu)} { left | H (1/2 + it) overline {H} (1/2 + it +) iu) right |} = left ( frac {2 pi} {t} right) ^ {iu / 2} left (1+ mathcal {O} left ( frac {u ^ 2 + 1)} {T} right) right)
$$

from where $ T <t <2T $ and $ | u | leq Delta $, Do you have any idea how to show it?
I think I should use the approach of the Stirling
$$
ln Gamma (s) = (s-1/2) ln ss + frac {1} {2} ln 2 pi + sum_ {m = 1} ^ { infty} frac {B_ { 2m}} {2m (2m-1) s ^ {2m-1}}
$$

What I thought might work, as we normalize our estimation function, we can use the fact
$$
z = | z | e ^ {i cdot arg (z)}
$$

So what we want to appreciate is basically
$$
e ^ {i cdot arg (H (1 / + it) overline {H} (1/2 + it + iu))}.
$$

We use that for that $ arg (z) = Im ( log z) $so $ arg ( Gamma (s)) = Im ( ln Gamma (s)) $ for which we use the Stirling approximation.
The contribution of the non-gamma factor is easier to estimate and should be
$$
( pi) ^ {iu / 2}
$$

Therefore, only an estimate of the gamma contribution remains.
For this purpose we use the approach of Stirling (in response to this question there are many useful approaches)
$$
arg left ( gamma ( frac {1} {4} + i frac {t} {2}) right) = In left[left(frac{1}{4}+ifrac{t}{2}-frac{1}{2}right)ln(frac{1}{4}+ifrac{t}{2})-frac{1}{4}-ifrac{t}{2}+frac{1}{2}ln(2pi)+mathcal{O}left(frac{1}{t}right)right]
$$

thus
$$
arg left ( gamma ( frac {1} {4} + i frac {t} {2}) right) = left[frac{tln(1/16+t^2/4)}{4}-frac{1}{4}arctan(2t)-frac{t}{2}+mathcal{O}(1/t)right]
$$

If I use only the first expression of such an extension, I get
$$
e ^ {i cdot arg left ( gamma ( frac {1} {4} + i frac {t} {2}) right)} sim left ( frac {1} {16} + frac {t ^ 2} {4} right) ^ {it / 4}
$$

and similar
$$
e ^ {i cdot arg left ( gamma ( frac {1} {4} -i frac {t} {2} -i frac {u} {2}) right)} sim left ( frac {1} {16} + left ( frac {t} {2} + frac {u} {2} right) ^ 2 right) ^ {- i (t + u) / 4}
$$

So I think it remains to be proved:
$$
left ( frac {1} {16} + frac {t ^ 2} {4} right) ^ {it / 4} cdot left ( frac {1} {16} + left ( frac .) {t} {2} + frac {u} {2} right) ^ 2 right) ^ {- i (t + u) / 4} = left ( frac {2} {t} right) ^ {iu / 2} left (1+ mathcal {O} left ( frac {u ^ 2 + 1} {T} right) right)
$$

and that all additional conditions in the serial extension of $ ln gamma (s) $ go also in the error expression.
Thanks in advance for any help!

multivariable calculus – Gradient of $ -y ^ T log left (f left (W x; gamma, beta right) right) $ w.r.t. $ left {W, gamma, beta right } $?

How to calculate the gradient of
begin {align}
L left (W, gamma, beta right): = -y ^ T log left (f left (W x; gamma, beta right) right)
end

in memory of $ left {W, gamma, beta right } $, from where $ x in mathbb {R} ^ n $, $ W in mathbb {R} ^ {m times n} $, and $ y in mathbb {R} ^ m $, but $ y_i in {0,1 } $, $ f (z; gamma, beta) $ is parameterized with $ gamma $ and $ beta $?

The definition of
$$ eqalign {
f (z; gamma, beta) & = gamma left (z- mu (z) right) \ left ( sigma (z) + epsilon right) ^ {- 1/2} + beta cr
mu (z) & = alpha 1 ^ Tz cr
sigma (z) & = alpha sum_ {k = 1} ^ m left (ex[k] – mu (z) right) ^ 2 equiv alpha 1 ^ T left[ left( z- mu(z) right) odot left(z – mu(z) right) right] cr
$$
from where $ 1 ^ T $ is a row vector with all, $ odot $ is an elementwise multiplication and $ alpha $ and $ epsilon $ are known scalars.

Thanks in advance for your help

Calculus – Understand how to integrate parts into the gamma function

The gamma function is defined as …

formula 1
,

I'll see how Gaussian representation The gamma function is derived and the first step is the integration of parts. No steps are shown and the following is the result of applying the integration of parts …

form2
,

I am confused how these values ​​were derived. That's my attitude …

u
,

dv
,

I imagine they chose it that way u and dv, which does ______________ mean…

v
,

you
,

I'm not sure how they came you, I tried to derive u and ended at …

$ e[n ln(1-t/n)]$

and then got another answer after trying to derive it. Can someone show me how you is derived or shows me where I am wrong so that I can complete the parts through integration?

Color Correction – How can gamma be applied correctly to linear raw files from an image processing camera?

Visually, this looks absolutely correct to me. In your unmatched image on my color-calibrated monitor, the steps between the gray spots appear at irregular intervals. In the adjusted image, they are more noticeable.

You ask:

Is color saturation a natural consequence of applying a gamma value, and if so, what can be done to compensate for this effect?

Short answer:

No. This is a consequence if correct black levels and white points are not set. You should do this before you apply the gamma curve.

Long answer:

The gamma curve that you used has the following form:

Original with histogram from linear version with applicable gamma curve

The light line applies to the values ​​displayed in the histogram. Once you've applied this curve, you'll get a histogram like this:

Curve applied

This is basically a boring, "flat" curve – that is, even though it has no linear value, it is perceptively basically like that. You can see, however, that the values ​​are all in the middle. This is very functional and maybe that's what you want for image processing, but in general it's not what we want visually Ones. You may just want to increase the contrast by dragging the black and white dots as follows:

Increase contrast

What gives a picture like this:

much less washed out

… much less washed out.

Your camera may have an adjustable black level, and you may want to raise it slightly. You must also set a suitable white level for your camera for conversion. You may want to check the Dcraw code to see how it normally does. (Well, spoiler: dcraw sets the white level to the 99th percentile of the histogram.)

Incidentally, this result has such a histogram:

increased contrast

what you can see extends to the extremes of the histogram. Since I am working with an 8-bit image in an 8-bit area, you can see that the colors are getting sparse. For "real work" you want to work with a higher bit depth (and probably only apply a transformation instead of a series).

This is definitely a bit boring. In the visual world, we may want to apply an s-curve to increase the power:

S-curve

yielding

punchy!

… or something like that.

raw – How to apply gamma correctly to the image processing camera

I'm using a machine vision camera that produces linear pixels, according to the manufacturer. As a result, I need to apply a gamma correction to the image before I can see the result. However, after applying a gamma correction to the image, the colors appear washed out.

Before each gamma correction:

Example of a raw gift

After applying 2.2 gamma:

Example port 2.2 gamma application

Overall, the brightness in the picture looks much better. The colors in the Color Checker, however, seem very oversaturated. Is color saturation a natural consequence of applying a gamma value, and if so, what can be done to compensate for this effect?

xfce – Redshift Gamma ramp size too small: 0 Error while starting the adaption method randr

I use Linux Mint 18.3, it is Ubuntu 16.04 base

in my thinkpad t480 (laptop has nvidia card), the redshift works fine

but in my dell optiplex 3060m (desktop pc has only intel gpu), the redshift does not work:

Gamma ramp size too small: 0
Failed to start fitting method randr.

How can I fix this?

Numerics – Calculation between gamma functions

I've calculated the gamma functions in Mathematica while there is no agreed answer.

By definition, $ Gamma[alpha]= int_0 ^ infty t ^ { alpha-1} e ^ {- t} dt $, $ Gamma[alpha,z]= int_z ^ infty t ^ { alpha-1} e ^ {- t} dt $ and $ Gamma[alpha,z_1,z_2]= int_ {z_1} ^ {z_2} t ^ { alpha-1} e ^ {- t} dt $, So we should have $ Gamma[alpha]-Gamma[alpha,z]$ should be the same $ Gamma[alpha,0,z]$, This is not the case. I'll just leave it $ alpha = 200 $ and $ z $ from 1 to 5 the numbers do not fit; see below.

    α = 200;
table[{Gamma[α] - gamma[α, z]gamma[α, 0, z]}, {z, 1, 5}]{{0th * 10 ^ 359, 0.00184859}, {0. * 10 ^ 359, 1.0983 * 10 ^ 57}, {0. * 10 ^ 359.6.71225 * 10 ^ 91},
{0th * 10 ^ 359, 2.41279 * 10 ^ 116}, {0. * 10 ^ 359, 2.14999 * 10 ^ 135}}

Why this?

complex analysis – Using the Cauchy formula to solve $ int_ gamma frac {e ^ {z ^ 2}} {z ^ 2-6z} =

Calculate the integral $ I = int_ gamma f (z) dz $,from where $ f (z) = frac {e ^ {z ^ 2}} {z ^ 2-6z} $ from where $ gamma = {z || z-2 | = 3 } $

I thought about using it $ f (z_0) = frac {1} {2 pi i} int frac {f (z_0)} {z-z_0} dz $

rewrite $ int_ gamma frac {e ^ {z ^ 2}} {z ^ 2-6z} = int_ gamma frac {e ^ {z ^ 2}} {z (z-6)} = int_ gamma frac { frac {e ^ {z ^ 2}} {z}} {(z-6)} $

The scope $ gamma $ determines that this is the domain in which we want the function $ frac {e ^ {z ^ 2}} {z} $ to use contains $ z-6 $,

So the application of the Cauchy formula:

$ int f (z) dz = pi i frac {e ^ {6 ^ 2}} {3} $

The solution, however, says that the value of integral is $ frac {- pi i} {3} $

Question:

What am I doing wrong?

Thank you in advance!

Thank you in advance!