Signed, Unsigned pixel values

Among the various image types proposed to import a gray-level image, even though you know the dynamic range of your image, you have to choose 'signed' or 'unsigned'...


In this series Import a raw image, let's see another option available in the Import dialog box: signed or unsigned numbers as shown in Fig.1. 

Fig.1: Several image types using (un)signed numbers are available in the Import dialog box.

1- A 'bit' of theory

Java (the programming language of ImageJ and its virtual processor, the JVM) encodes signed numbers with the method called  "two’s complement" [Wiki] and [Wiki].
The last bit (the most significant) is used for the sign (a value of 0 for positive and 1 for negative). For positive 16-bit numbers, no extra process are done, the bits #0 to #15 are used allowing the encoding of numbers from 0 (00000000 00000000₂) to +32767 (01111111 11111111₂). For negative numbers, a small calculation is done consisting of a bits inversion and an addition of +1 as shown in the following example ...

Example #1: -32767
+32767₁₀ = 01111111 11111111₂
inversion : 10000000 00000000₂
addition + 1 : 10000000 00000001₂ = -32767₁₀

Example #1: -1
1₁₀ = 00000000 00000001₂
11111111 11111110₂
11111111 11111111₂= -1

 Thus, signed 16-bit numbers are comprised in the range of [-32767;+32767] whereas unsigned 16-bit numbers between [0;65535].

2- Consequences for gray-level images

Now, we've seen the theory, what about gray-level images ?

If you import an image with the wrong 'sign', you get image with inverted areas as shown in Fig.2 (right panel). 

Fig.2: 16-bits image of Lena imported as unsigned (left) and signed (right).

Let's go further. Create an 16-bit image 512x50 with a ramp background (File > New > Image...) as shown in Fig. 3. If you look at the pixel values by hovering the image, values are comprised between 0 and 65535 (ImageJ always creates unsigned 16-bit image).

Fig. 3: Ramp image. In unsigned 16-bit  type, black is 0 and white 65535. In signed 16-bit, black is -32767 and white +32767. 0 (zero) is located at the middle of the ramp and corresponds to a medium gray.

If you save this ramp as a raw (16-bit unsigned, by default) image and import it as a '16-bit signed', you'll get the image of Fig. 4. What happens?

Fig.4: Ramp image saved as an unsigned 16-bit and imported as a signed 16-bit. The grays seem split in two parts.

Just look at some key numbers in the ramp image...

In unsigned 16-bit,
00000000 00000000₂ = 0₁₀
. . . . . .
01111111 11111111₂ = 32767₁₀
10000000 00000000₂ = 32768₁₀
10000000 00000001₂ = 32769₁₀
. . . . . .
11111111 11111111₂ = 65535₁₀

In signed 16-bit,

00000000 00000000₂ = 0₁₀
. . . . . .
01111111 11111111₂ = +32767₁₀
10000000 00000001₂ = -32767₁₀
. . . . . .
11111111 11111111₂ = -1₁₀

Now, if we compare the encoding of the signed and unsigned numbers, they are identical from 0 to +32767, but from +32768 to + 65535, depending of the encoding, they appear as negative (or positive) numbers (e.g. signed -1₁₀ has the same encoding as the unsigned 65535₁₀). That explains the strange pattern of the ramp of Fig. 4. The left part (medium gray to white) corresponds to the range [0;+32767] whereas the right part corresponds to negative numbers [-32767;-1].

To fix this problem, convert the 'wrong signed' image in floating-point numbers (Image > Type > 32-bit) and use the function Process > Math > Macro... to shift the negative values with the following formula:

if (v < 0) v=v+65536

Finally, convert again the fixed image to 16-bit.


3 - What about signed or unsigned 8-bit image?

In ImageJ, the only option called '8-bit' corresponds to unsigned 8-bit numbers (from 0 to 255) and there is no way to import a signed 8-bit image (-127 to +127). How can we deal with that?

Import the image as a 8-bit. Then the tricky part is to convert the image in floating-point numbers to make the calculation. Subtract 256 to all the pixels greater than 127 and finally change this fixed image to 8-bit. Automatically, the pixel values are shifted between 0 and 255.

if (v > 127) v=v-256

Your pixel values are back!!

Note: To create a signed 16-bit image, I use this formula '[if (v <= 32767) v+=32767; else v-=32767;]' and save it as a raw image (File > Save As > Raw Data...).

4- Links

• Series of posts Image File [TOC]
• Crazybiocomputing mini-games levels #8 and 9 [Link]