Revised March 20, 1997

The explanation of the hyperbolic conversion (Digital development) is somewhat confusing even in Japanese language! I hope the next explanation in English may help your understanding. A part of the English phrase of this text was checked (edited) by Rick Colman ( Your comment, question and advice are welcome. If you tell me kindly the error in my English, I am very happy.

First, I explain the theoretical background, then I describe how to process with Hidden Image. Finnaly, the 'color enhancement' is explained as an extention of this process.

Digital Development

1. CCD vs Film?

Good CCD sensor linearity is important for accurate measurement of magnitude. For most amateurs, however, the goal is obtaining beautiful pictures, not measuring magnitudes. If this is so, the next question is whether CCD linearity is also important for getting those beautiful pictures? The answer is usually no, it is not terribly important.
Two non-linear processes in film photography play an important role in making photographs look natural to the eye. The first is non-linear sensitivity ("gamma curve"), and the second is the edge enhancement effect.

1) Gamma curve non-linearity

CCD data has an extremely large dynamic range. The range is too large to be reproduced on print or in a video display. In the highlight region of an image, photographic film saturates gradually while a CCD image saturates suddenly, at a maximum point. This difference in saturation behavior expands the effective dynamic range of film, compared to CCD images (fig-1).
Keeping this behavior in mind, consider the "gamma value," which is the gradient of the linear region on the gamma curve. A larger gamma results in a higher contrast image. When we require the same gamma value for the major part of image, the effective dynamic range of the processed (and printed) CCD image will be "much narrower" than film. However, the down side of this non-linear expansion of the effective dynamic range is low contrast, which results in a "flat" appearance. Edge enhancement techniques can be used to compensate for low contrast.

2) Edge enhancement effect

Chemical processing of film also produces an edge enhancement effect. You can clearly see this effect in pictures of the moon, at the boundary (fig-2). This effect keeps image contrast from becoming too" flat" in the highlight regions.
Summarizing the above, it is a non-linear gamma curve plus edge enhancement effects in the highlight region that result in images with a wide dynamic range.

[The above part is checked by Rick. Thanks!]

2. Digital development (Hyperbolic conversion + edge enhancement)

2-1 Hyperbolic conversion

Let us define the original image data as X and the converted data as Y, and the pixel values of X and Y as xij and yij, respectively.

The digital development process, which is proposed here, is written as:

yij = k [ xij / ({xij} + a ) ] + b ....(equation-1)

where k is constant to control the maximum value of yij, which is not so important. 'a' and 'b' is artificial pedestals, which are the important parameters. {xij} means the unsharpened data of the original image X.

When {xij} = xij, i.e. no low pass filter was used, this is just a hyperbolic conversion of gamma curve. The relation between xij and yij is shown in fig-3. You can see that the curve is very similar to the gamma curve of conventional film.
The meanings of the key parameters 'a' and 'b' are clear in the figure.
1) When b < xij < a, the gamma is linear.
2) When xij > a, the gamma is gradually reduced. xij =a is threshold.
3) The minimum value of yij is b. This is similar to the base level of chemical film, which is never zero. The low gamma in the low level region will improve the visual S/N ratio in background level. This fact, which is usually overlooked, plays a very important role to attain the 'photo quality'.
4) The gamma value is unity at b < xij < a in this example.
If you replace xij by xij to the G power (xij^G), the gamma value will became G, while we assume G=1.0 througthout this report for simple discussion.

In fig-3, the curve of the log-conversion, which is available in most of IP software, is also shown. Note that there is no linear region in this curve. I believe that the linear region of gamma curve should be important to achieve the 'photo quality'.

2-2 Edge enhancement

The roll off of gamma curve at "xij > a" will result in a flattened image in the highlight region, without edge enhancement. The edge enhancement is possible when low pass filter is used to make the mask data {xij}. I am sure that the reason is easily understandable for you. This is very similar to the conventional unsharp-mask. At {xij} << a , this conversion makes no effect, because xij/({xij} + a) is nearly identical to xij / a in equation-1. The edge enhancement becomes conspicuous at {xij} >> a.
As described in Section 1, the edge enhancement + the non-linear gamma curve is essential in the 'photo quality'. Therefore, in my opinion, any types of histogram equalization and/or gamma conversion never reproduce the photo quality without edge enhancement.

Please note that, this digital development process does not require any specialized software. When you regard ({xij} + a ) as the flat frame for xij, the flat correction process is just identical to the process given by equation-1. Therefore, almost any CCD software are available to achieve this process.

3. How to process with Hidden Image

1) Load the original data X in left screen.
2) Choose [Utility] >> [Box]. Measure the average background intensity (=BG).
3) Choose [Process] >> [Sum/Offset]
Subtract BG from Y. For example, set the parameters as follows:

  ADD Frame 1 %   ADD constant
     [100]      [ '- BG' ]
  ADD Frame 2 %
    [ 0 ]

And save the right screen image as X, again.

4) Choose [Utility] >> [Linear Filter]
Check in Box for Low pass
Radius is, for example '100', order =1.
A smaller radius results in the strong blur effect.
Do not unsharpen too much.
This is {xij}.

5) Choose [Process] >> [Sun/Offset]
ADD the value 'a' to {xij}.
The choice of 'a' is very important. Remember the hyperbolic curve in fig-3. The roll off of gamma curve appears at xij > a. Use [Utility] >> [Box], and measure the middle intensity of the object. This is a good start point of 'a'. Try and Error will be required for skill.
Then the data ({xij} + a) is shown on the right screen. Save it as 'MASK'.

6) Do flat correction with MASK.
Choose MASK as the flat frame for X. And load X on left screen. The flat corrected X is shown.
This is the image of k [xij / ({xij} + a )]
The value of 'k' is automatically defined in the flat correction command.

7) Choose [Image] >> [Manual stretch]
Input the Max value and Min value.
Use a 'negative' value for a 'Min' value. For example, if you use ' - b ', then the displayed image just shows

k [xij / ({xij} + a )] + b

8) Choose [Image] >> [Manual Stretch] >> [from screen].
Stretched image is shown on the right screen.
Save it as 'Y'.

Now you completed the 'digital development process'.

4. Color enhancement

The color enhancement technique is an extension of the digital development. This technique results in dramatic effect for some object, especially for galaxies.
The process is very simple. When you process the Red image xij(R), use the mask from Blue image {xij(B)}

yij(R) = k [xij(R) / ({xij(B)} + a )] + b

When we process the "Red image with Blue mask", "Green image with Red mask" and "Blue image with Red mask", we call this process RGB/BRR.
The Hyakutake comet in my gallery was processed by RGB/BRR type, and Hale-Bopp was processed by RGB/BGR. The M13 image and most of galaxies were processed by RGB/GGG, because RGB/BGR or BRR type process causes a too intense color enhancement.
Fig-4a is the Hale-bopp image processed by RGB/RGB (i.e. the simple digital development without color enhancement ), while fig-4b is processed by RGB/BGR. You can see the dramatic effect of the color enhancement.

fig-4a without color enhancement

fig-4b with color enhancement

The digital development is very powerful tool to get 'Photo quality', while it will require some effort and skill. Please try it.
A typical example for galaxy is NGC2403 and another example for the globular cluster is M13 .

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