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# A comprehensive method of non-uniformity correction for EMCCD

### EMCCD imaging model

The single pixel imaging model in EMCCD is shown in the equation. (2).

$${image} _ {out} = {k} _ {1} * left[left({k}_{0}*photons+{b}_{0}right)*Gright]+ {b} _ {1} + {image} _ {dark}$$

(2)

In this equation, (photons ) is the original single pixel signal entering the imaging area; ({k} _ {0} ) and ({b} _ {0} ) are the full linear photoelectric conversion coefficients before the photogenerated charge of a single pixel enters the multiplication register; (G ) is the true multiplication gain of the charge entering the current pixel channel; ({k} _ {1} ) and ({b} _ {1} ) are the full linear photoelectric conversion coefficients of the charge amplified by the multiplication register in the backbone; ({image} _ {dark} ) is the background signal superimposed on the load.

Coefficient ({b} _ {0} ) in eq. (2) is related to the dark current introduced during the charge transfer. Exposure time is usually controlled at the millisecond level when EMCCD is operating normally, so the effect of dark current can be ignored. Therefore, Eq. (2) can be simplified as follows:

$${image} _ {out} = {k} _ {1} * left[{k}_{0}*photons*Gright]+ {b} _ {1} + {image} _ {dark}$$

(3)

After developing the equation. (3), we obtain the following results:

$${image} _ {out} = k * photons * G + b + {image} _ {dark}$$

(4)

In this equation, (k ) and (b ) are the full linear photoelectric conversion coefficients of a single pixel in the entire EMCCD imaging process.

Equation (4) describes the single pixel imaging model in EMCCD. the (k ), (b ) and ({image} _ {dark} ) vary with pixels, and g varies by channel.

$${image} _ {out} (i, j) = k (i, j) * photons * G (i, j) + b (i, j) + {image} _ {dark} (i, j)$$

(5)

Equation (5) represents the imaging model of all pixels in EMCCD.

In this equation, (I) and (j ) are respectively the i-th row and the j-th column of the EMCCD output image; (G left (i, j right) ) is the true channel multiplication gain of the current pixel.

### Principle of global correction of non-uniformity

As shown in eq. (5), the difference of the full linear photoelectric conversion coefficients (k (i, j) ) , (b (i, j) ) and the background signal ({image} _ {dark} (i, j) ) between pixels and that of the actual multiplication gain (G left (i, j right) ) between channels will lead to the non-uniformity of the final EMCCD output image when the original signal (photons ) are consistent. Therefore, the reverse separation and direct recovery method is adopted to correct the non-uniformity of the whole EMCCD imaging process.

#### Calculation of the coefficient

(a) Run EMCCD in normal CCD mode and turn off the original signal (photons ) to get multiple background images, which is the background signal ({image} _ {dark} (i, j) ) of the current camera.

(b) Operate EMCCD in normal CCD mode and segmentally acquire the output image of the camera under a different original signal (photons left[{mathrm{n}}right])(n 20). Take the original signal (photons ) as the X axis and the difference between the output image and the background signal ({image} _ {dark} left (i, j right) ) as Y axis for linear adjustment to obtain the full linear photoelectric conversion coefficient (k (i, j) ) and (b left (i, j right) ) corresponding to each pixel.

(c) Run EMCCD in normal CCD mode to get the output image of the current camera when the multiplication register is free ((G = 1 )). The original signal ({photons} _ {0} ) is considered backwards as ({P} _ {0} ) using the results obtained ({image} _ {dark} left (i, j right), k left (i, j right) and b (i, j) ) according to eq. (5).

(d) Operate EMCCD in CCD multiplication mode and keep the original signal ({photons} _ {0} ) unchanged to get the output image under different gain voltages ( Delta v[m])( (m )30). Likewise, the signal after multiplication ({P} _ {1} ) could be calculated backwards to be ({photons} _ {0} * G (i, j) ).

(e) The actual multiplication gain under different voltages ( Delta v left[mright]) (m 30) could be calculated by dividing ({P} _ {0} ) in ({P} _ {1} ). Take the multiplication tensions ( Delta v ) as the x-axis and the corresponding real multiplication gain (G (i, j) ) as Y axis for the exponential fit to derive the function relation between the multiplication gain (G (i, j) ) and the tension ( Delta v ).

#### Reverse separation

Record the current EMCCD operating voltage ( Delta {v} _ {origin} ) to get an original uncorrected output image ({image} _ {origin} ). The original signal corresponding to the current image can be calculated inversely according to Eq. (6).

$${photons} _ {origin} (i, j) = frac {{image} _ {origin} left (i, j right) – {image} _ {dark} left (i, j right ) -b (i, j)} {k left (i, j right) * {G} _ {i, j} ( Delta {v} _ {origin})}$$

(6)

In this equation, ({G} _ {i, j} ( Delta {v} _ {origin}) ) is the true multiplication gain after substitution of the multiplication voltage ( Delta {v} _ {origin} ) in ({G} _ {i, j} ( Delta v) ); ({photons} _ {origin} ) is the original signal corresponding to the original output image ({image} _ {origin} ).

#### Forward recovery

The original signal obtained in the “Reverse split” section is recovered forward according to Eq. (7) to obtain an image with full correction of non-uniformity.

$${image} _ {modify} (i, j) = {k} _ {ave} * {photons} _ {origin} (i, j) * {G} _ {ave} + {b} _ {ave } + {image} _ {darkAve}$$

(seven)

In this equation, ({k} _ {ave} ) and ({b} _ {ave} ) are the mean value of (k left (i, j right) ) and (b (i, j) ), respectively, in the equation. (6); ({image} _ {darkAve} ) is the mean of the background signal ({image} _ {dark} ); ({Given}) is the average of the true multiplication gain of each channel of EMCCD under the current multiplication voltage ( Delta {v} _ {origin} ).

#### Determination of the global correction coefficient

The full correction equation for EMCCD is obtained by substituting Eq. (6) in Eq. (7) as follows:

$${image} _ {modify} left (i, j right) = {k} _ {modify} left (i, j right) * {G} _ {modify} left (i, j right left[{image}_{origin}left(i,jright)-{b}_{modify0}left(i,jright)right]+ {b} _ {modify1}$$

(8)

$${b} _ {modify0} left (i, j right) = {image} _ {dark} left (i, j right) + b (i, j)$$

$${G} _ {modify} (i, j) = frac {{G} _ {ave}} {{G} _ {i, j} ( Delta {v} _ {origin})}$$

$${k} _ {modify} (i, j) = frac {{k} _ {ave}} {k left (i, j right)}$$

$${b} _ {modify1} = {b} _ {ave} + {image} _ {darkAve}$$

In this equation, ({G} _ {modify} ) is the channel gain correction factor of the current pixel. 