About the "power spectral density" curve of an image

Especially in technical writings about various metrics for image sharpness (such as various kinds of acutance metrics), we often see a reference to the power spectral density function of a "scene" perhaps a test target or of the image generated from that scene by the lens (and perhaps by a subsequent digital sensor and associated processing algorithm.

The reference may seem mysterious because:

•• The concept of a "density function" is unfamiliar to many people.

• The name "power spectral density" is in fact not apt for what we are dealing with here anyway. (Oh, great!)

To work our way through both of these challenges, I will (as you might expect from me) start with the application of this concept to an electrical situation (in which it was first used anyway).

Suppose we have a finite length of an electrical waveform that is not repetitive (perhaps an ongoing audio waveform). We see it as an instantaneous voltage which varies with time - an instantaneous voltage that is a function of time. We often speak of this "original" form of the signal as a "time-domain representation".

If we take the Fourier transform of this function of voltage vs. time, we get a new "curve", with frequency, not time, as its x-axis, which (simplistically) we can think of as describing the composition of the waveform in terms of components at different frequencies. I will all this curve "AS", which I will explain later.

If we try to be more precise about what this curve means, we soon get rather entangled. So I will transform this curve to another form which can be explained more readily. I do this by, for every frequency in the range of our curve) just squaring the value of the AS curve.

It turns out that the unit of the y-axis here is V^2/Hz (volts squared per hertz of frequency). Wow. Don't fret over that - we will soon arrange to move beyond it!

To make what will follow easier to grasp, we must think in terms of the power created in some load (even an arbitrary one). The instantaneous power caused in the load by the signal at any instant of time is given by:


where p is the instantaneous power, e is the instantaneous voltage, and R is the resistance of the load.

In general we are not interested in some particular actual load - we just need to contemplate one so we can deal with the power implications of our signal. So we arbitrarily assume a load with a resistance of one ohm. Then the equation above becomes:


Having done that, the unit of the y-axis of our curve becomes W/Hz (watts per hertz of frequency). Wow! That's not much better, But it will make sense very soon.

Now, having taken care of that, we will look at this famous curve, which I will call PSD (why later):


Now it is tempting to say that the value of this curve at any frequency tells us the amount of power in the waveform at that frequency. But it doesn't. In fact, if the distribution of power by frequency is continuous (as we assume in that case), the power at any given frequency is - zero! This may seem startling, but we can easily see that it must be so.

There are an infinite number of possible frequencies over the range of our curve, and if there were some amount of power at each of them, the total amount of power in the signal would be infinite!

Rather, the value of the curve at any frequency tells us the amount of power in the signal for each hertz of frequency (in a band centered about the frequency of interest). If we imagine a frequency band of any width (even infinitesimal), then within that range of frequency there will be some power.

In fact, to be mathematically correct, the y-value of the curve, for any frequency, is the ratio of the amount of power contained in a band of some width, centered about that frequency, divided by the width of the band, in the limit as the width of the band approaches zero. We see that here:


The counterpart of this is that if we consider a specific band of frequencies (with a finite width), the amount of power in the signal with frequencies in that range will be the area under the curve between the limits of the range.

In fact, the name I gave this second curve, PSD, means power spectral density.

• power because this concept works with power (actually, the power implied by the signal voltage if we assume a load of resistance one ohm, but that is still power).

• Spectal because a presentation of the content of a signal or such by frequency as called a spectrum (it is evocative of what we see when we take sunlight and spread it by light frequency [wavelength] with a prism)

• Density because this curve is of the general class called density functions, where the vertical value does not tell us the quantum of something but rather the amount of the something per unit of the y-axis (as when we speak of the heaviness of a certain kind of reinforcing bar in "pounds per foot of length).

By the way, the area under the entire curve is the total power the signal would deliver to our hypothetical load.

Now lets go back to the first curve, the one I was so anxious to move behind. It obviously also shows the distribution of "something" in the signal by frequency, but what, and just how? It must be a density function. If so, then the area under this curve between two frequencies must be the amount of that something contained in the signal between those frequencies. What is that something?

Well, that area does not correspond to any "physical" quantity. So struggle as we might, all we can say about this curve is that is is the square root of the PSD curve.

Since, by assuming a hypothetical load with a resistance of one ohm, and since then power is the square of voltage (voltage is the square root of power), we might say think that this works just like the PSD but in volts rather than watts. But is doesn't, mainly because again the area under the curve does not correspond to anything. the total area under the curve does not, for example, correspond to the overall voltage of the signal (would that be in RMS terms or what, anyway?).

Nevertheless, noting that amplitude refers to the peak voltage of a waveform is a single frequency (a "sine wave"), the first curve is often called the "amplitude spectrum" (AS) of the signal. Sometimes people feel that they need to call it the amplitude spectral density function (ASD). But strictly it is not a "density" function (because the area under the curve, or a portion of it, has no meaning .

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In the next part, we will actually get to photographic imaging staff!

[continued]

[part 2]

Now, we will actually get to the real topic, in the field of photographic imaging.

To keep things simple:

• We will assume a monochrome system, in which luminance/illuminance is the "variable" of interest.

• We will assume a "one dimensional" situation, in which we only consider luminance/illuminance variation along a track across the object or image.

The variation of luminance (the luminance modulation) along a track across the image is very analogous to an electrical waveform. The difference is that here the x-axis is distance, not time, and the y-axis is illuminance, not voltage.

But just as for an electrical waveform, we recognize that this luminance modulation generally comprises components at different frequencies (but frequency here is spatial, not temporal).

So not surprisingly, we can take the Fourier transform of the luminance "waveform" and get a curve that is quite analogous to the amplitude spectrum (AS) of an electrical waveform.

But it has no consistent name. And like the AS, it is hard to really explain what the vertical axis means!

I will call it a luminance modulation spectrum (LMS).

If we take the luminance spectrum of the image on the focal plane, and divide it (that means at each spatial frequency) by the luminance spectrum of the object, we get a curve that shows, for each frequency, how the modulation (variation in luminance) at the object is transferred to modulation (variation in illuminance) at the focal plane. This is the (spatial) frequency response of the optical system. Yes, it is the modulation transfer function (MTF) of the optical system!

Now note that (unlike in the electrical situation) the square of luminance is not in any way a kind of "power" (or proportional to some kind of power).

But there are situations in which, because of the math involved, we need to work with the square of the luminance modulation spectrum. This is in fact a special kind of density function, and indeed the area under that curve across any range of frequencies has a significance (but it is very hard to describe and understand, and I will not attempt that here).

But it in no way relates to anything that can reasonably be called power.

So what do people call this curve? Well, often, the "power spectral density function" (PSD) of the object or image, because that is the thing it seems "most like".

Ugh.

What is a good name for it? "Luminance modulation spectrum squared."

Best regards,

Doug

What are examples of when we are interested in the square of a luminance modulation spectrum?

Well, it turns out that image noise components combine, in their visual effect, as the square root of the sum of the squares of the "modulation amplitudes" of the components.

If we want to assess all the noise on an image (it having been somehow separated from the "true" content), taking into account the varying sensitivity of the eye to noise components at different frequencies, we can do this (I take some liberties with rigor in favor of clarity of the concept):

• Determine the luminance modulation spectrum of the noise.

• Square its value at each frequency to get a new "spectrum" (one that is mathematically analogous to a power spectral density function - PSD - for an electrical signal, and which sadly is often spoken of as a PSD).

• Take the area under that curve.

• Take the square root of that value.

This number, which is of the same nature as a luminance modulation amplitude, will then be an indicator of total visual noise impact.

Now, in this situation, again because of the (distant) mathematical parallel of the "squared" function with the power spectral density function of an electrical signal, the squared function is often called the noise power spectrum function (NPS). But of course there is no "power" involved.

Then. because of that usage, in an image analysis situation not involving noise, the square of the luminance modulation spectrum (sometimes miscalled a "PSD" function) is often called the noise power spectrum (NPS) of the image. Of course here neither noise nor power are involved!

Double ugh!

Best regards,

Doug

As an example of this abuse of the term "noise power spectrum", this is a quotation from a CPIQ working group paper discussing the ongoing work to develop a sharpness metric that takes into account that the system MTF for actual texture detail may be "worse" than that measured with the slant edge technique:


Note that the matter of noise or a noise spectrum is not involved here at all. (And in any case, the concept of "power" is not apt, allegorical at best.) What they speak of as the "noise power spectrum" is in fact just the square of the luminance modulation spectrum for the test target or its image.

Why do they call this a "noise power spectrum"?

Because the noise power spectrum (which is the square of the luminance modulation spectrum for the noise components of an image) is a thing they are familiar with that is the square of a luminance modulation spectrum.

Ugh.

No, I don't think it is better to call it an NPS than to call it a PSD, just as I don't think it is better to call Buffalo "Cleveland" than to call it "Indianapolis".

In fact, this industry needs to craft a good name for the function that is the square of the luminance modulation spectrum function.

And in fact, it needs to craft a good name for the luminance modulation spectrum function. How about "luminance modulation spectrum" (or maybe, for conciseness, the "modulation spectrum")? It is quite parallel to its counterpart in the electrical real, the amplitude spectrum.

Best regards,

Doug