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I'm a cognitive neuroscience researcher designing a functional magnetic resonance imaging (fMRI) experiment based on the Flashed Face Distortion Effect (https://vimeo.com/27138942). The goal of the experiment is to compare neural activation while the participant looks at "distorted" versus "undistorted" faces. For the "undistorted" control condition, I need to alter the above video so that the faces are blurred, or otherwise have a bare minimum amount of detail to be recognizable as faces (sort of like this: http://bit.ly/38EZ9x0). In other words, I need to alter the face images so that they don't produce the distortion effect, but are still recognizable as faces. Importantly, the luminance of the images must be the same as the faces in the test condition. What software and method would achieve this?

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I would go with Davinci Resolve (free edition), which has one specific feature useful in this case, a waveform. https://duckduckgo.com/?q=davinci+resolve+waveform&t=h_&ia=videos

You could use the "general" waveform.

the luminance of the images must be the same as the faces in the test condition.

I do not know how strict this condition must be, especially if you want color images, let me explain.

"Luminance" as far as I know, does not say much about the "color" of the light being measured. what we could probably do is re-transform an RGB file to HSB, HSL or Lab components and use the B or L component, so probably the images need to be split into these channels beforehand, "normalize" the channel, and then recomposed with the other components. This would need additional software like Gimp and using Lab will be the safest bet.

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But you probably only mean "general brightness".

One option is to normalize the skin tone on its hue component, to minimize hue variations.

https://duckduckgo.com/?q=davinci+resolve+skin+tone&t=h_&ia=videos

You need to pay to the white point, the black point but mainly the middle tones.

You then can apply a blur effect. https://duckduckgo.com/?q=davinci+resolve+blur&t=h_&ia=videos

I do not know if it will make things easier applying it before the normalization or after. Play a bit with both approaches.

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All pro or advanced hobbyist level video editors have what you need. If you already are Adobe CC customer, the extra cost is mostly the time needed to learn one of its video editors. You can compose views, match colors, sizes and luminance levels plus add blur effect with standard functions.

The technique is quite basic: select the photos, place them to the timeline on different tracks as clips, adjust the the unwanted differences off, adjust the lengths, prepare the smaller than full-screen viewports, make copies, shuffle the display order, insert blur as a track effect.

Blurring can make the brightest point darker and the darkest point brighter. A contrast normalizing track effect fixes it.

A video editing hobbyist surely helps if you know one.

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