SPATIOTONAL PHOTOQUANTIGRAPHIC FILTERS

Most print and display media have limited dynamic range. Thus one might be tempted to argue against the utility of the Wyckoff principle based on this fact.

One could ask, for example, why bother building a Wyckoff camera that can capture such dynamic ranges if televisions and print media cannot display more than a very limited dynamic range? Why bother capturing the photoquantity q with more accuracy than is needed for display?

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Q (Quantity of light) (a )

f (response byte)

Estimated camera response function with ground truth data

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Relative certainty

f (Response byte) (b )

Estimated camera certainty function

Figure 4.32 The author’s simple two parameter model fits the response curve almost as well as the much more complicated 256 parameter model (e.g., the lookup table) of Figure 4.20a, and in some areas (e.g., near the top of the response curve) the fit is actually better. This method of using comparametric equations is far more efficient than the least squares method that produced the data in Figure 4.20a. Moreover, the result provides a closed-form solution rather than merely a lookup table. (b) This method also results in a very smooth response function, as we can see by taking its derivative to obtain the certainty function. Here the relative certainty function is shown on both a linear scale (solid line) and log scale (dashed line).

Compare this certainty function to that of Figure 4.20bto note the improved smoothing effect of the simple two parameter model.

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Quantity of light, Q = log(q )

Pixel integer

Response curves recovered from differently exposed images

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Quantity of light, Q = log(q )

Certainty

Certainty functions (derivatives of response curves) (a )

(b )

Figure 4.33 Response curves and their certainty functions: (a) Response functions shifted for each possibleK (e.g., each exposure that the imaging apparatus is capable of making).

(b) Amplitude domain filterbanks arise from the overlapping shifted certainty functions.

Some possible answers to this question are:

1. Estimates of q are still useful for machine vision and other applications that do not involve direct viewing of a final picture. An example is the wearable face recognizer [14] that determines the identity of an individual from a plurality of differently exposed pictures of that person, and then presents the identity in the form of a text label (virtual name tag) on the retina of an eye of the wearer of the eyeglass–based apparatus.

Since qˆ need not be displayed, the problem of output dynamic range, and the like, of the display (i.e., number of distinct intensity levels of the laser beam shining into a lens of the eye of the wearer) is of no consequence.

2. Although the ordinary dynamic range and the range resolution (typically 8 bits) is sufficient for print media (given the deliberately introduced nonlinearities that best use the limited range resolution), when performing operations such as deblurring, noise artifacts become more evident. In general, sharpening involves high-pass filtering, and thus sharpening will often tend to uncover noise artifacts that would normally exist below the perceptual threshold when viewed through ordinary display media. In particular, sharpening often uncovers noise in the shadow areas, making

dark areas of the image appear noisy in the final print or display. Thus in addition to the benefits of performing sharpening photoquantigraphically by applying an antihomomorphic filter as in Figure 4.3 to undo the blur of (4.5), there is also further benefit from doing the generalized antihomomorphic filtering operation at the point qˆ in Figure 4.4, rather than just that depicted in Figure 4.3.

3. A third benefit from capturing a true and accurate measurement of the photoquantity, even if all that is desired is a nice picture (i.e., even if what is desired is not necessarily a true or accurate depiction of reality), is that additional processing may be done to produce a picture in which the limited dynamic range of the display or print medium shows a much greater dynamic range of input signal, through the use of further image processing on the photoquantity prior to display or printing.

It is this third benefit that will be further described, as well as illustrated through a very compelling example, in this section.

Ordinarily humans cannot directly perceive the “signal” we process numer-ically but rather we perceive the effects of the “signal” on perceptible media such as television screens. In particular, in order to displayq(x, y), it is typicallyˆ converted into an image f (q(x, y))ˆ and displayed, for example, on a television screen.

Figure 4.31 is an attempt to display, on the printed page, a signal that contains much greater dynamic range than can be directly represented in print. To obtain this display, the estimate qˆ was converted into an image by evaluatingf (ˆˆ k₂q).ˆ Even though we see some slight benefit, over f₂ (one of the input images) the benefit has not been made fully visible in this print.

4.5.1 Spatiotonal Processing of Photoquantities

To fully appreciate the benefits of photoquantigraphic image processing, in Chapter 6 we will consider seemingly impossible scenes to photograph reason-ably (in a natural way without bringing in lighting equipment of any kind). There it will be shown how a strong high-pass (sharpening) filter can be applied to qˆ and sharpen the photoquantity qˆ as well as provide lateral inhibition similar to the way in which the human eye functions.

Because of this filtering operation we will see that there no longer needs to be a monotonic relationship between input photoquantityq and output level on the printed page. The dynamic range of the image will be reduced to that of printed media, while still revealing details over an extreme range of light quantities in the scene. This will answer the question, “Why capture more dynamic range than you can display?”