【Technology】LED brightness control using PWM (fade in/fade out)

I developed software (algorithm) to operate an LED in a gradient manner using a microcontroller.
The microcontroller (RENESAS: RX220) is connected to an LED driver IC (PCA9635) via the I2C bus to illuminate the LED.
To achieve a gradient illumination, while the brightness is linear, the PWM values form a curve, making the derivation of the calculation formula the biggest challenge. (Figure 1)
Since the method of determining the LED brightness through loop calculations cannot start from an intermediate brightness, I was able to achieve this by pre-calculating how the brightness should change after a certain number of milliseconds and combining it with sequential calculations. (Figure 2)
The RGB (red, green, blue) of the full-color LED is calculated individually.


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There is a video of the LED