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Quoting Image Processing, Analysis, and Machine Vision, 4th ed.

An image to be processed by computer must be represented using an appropriate discrete data structure, for example, a matrix. An image captured by a sensor is expressed as a continuous function f(x, y) of two co-ordinates in the plane

I can't quite understand how it's true that the sensors in a digital camera capture the image as a continuous signal $f(x, y)$ rather than just a discrete digital signal. It seems to me that the "resolution of a continuous image" is infinite. We can make the digitized image be in whatever resolution we want by sampling if this is true, right?

I'm still being introduced to image processing. Forgive me if this question seems naive to you.

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    $\begingroup$ The continuous function in that text refers to the image projected onto the sensor, before it is sampled and digitized. This image, by nature of the lenses doing the projection, is band-limited, meaning it has a limited resolution. Hopefully the sensor’s density, is sufficient to properly sample that signal without aliasing. Sometimes the sensor has an anti-aliasing filter in front of it, basically low-pass filtering the image before it hits the sensor. $\endgroup$ Oct 12 '21 at 1:53
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The image in a digital camera/sensor is represented as a matrix of discrete numbers, as you say.

The (spatial) digitization consists of an array of sensels each with some non-zero area that are sensitive to light, framed by non-sensitive area, possibly micro-lenses that increase the fill-rate (the effective light sensitive area), possibly optical lowpass filter (some glass arrangement that feeds spatially shifted copies of the image to the sensels, thus providing a bit of blur prior to sampling).

Seems to me that the text only have unfortunate/complex choice of words.

-k

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  • $\begingroup$ Nice "sensels" word! $\endgroup$ Oct 12 '21 at 12:40
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I agree with Knut Inge

In many image processing books an image is defined as a continuous 2D function (i.e $f(x,y)$). This is because sometimes the math is easier to do in a continuous function rather than a discrete one.

For example there is a whole area in partial differential equations, where the proofs and theorems and whatnot are made for a continuous image. And then the algorithms are applied to a discrete image that is thought as the discretization of said image.

The idea and the philosophy of this is thinking that the "real" image is not what the camera sees, the image is the real word scene that impacts on the camera sensor. Since in the real word there is not discretization (or the discretization is at an atomic level, i.e photons) that image is considered continuous and the image in the camera is nothing but that same image spatially discretized, quantizied and with some noise added.

It is specially useful to think on the image this way when you are trying to reduce noise. That's because you can think you'd want to restore the original image before the noise was added and sample that theoretical image. Sometimes besides noise there is a convolution of a function called the PSF (point spread function), but that's only relevant when modelling the camera moving (or objects moving in front of camera), or in microscopes or telescopes.

I hope this is useful for you.

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