The Greatest Smallest PNG // -dealloc

2024-01-08 00:43:26

Just a few days in the past, my former coworker Evan Hahn posted “The world’s smallest PNG”, an article strolling by means of the minimal required components of the PNG picture format. He gave away the reply within the very first line:

The smallest PNG file is 67 bytes. It’s a single black pixel.

Nevertheless (spoilers!) he later factors out that there are a number of legitimate 67-byte PNGs, corresponding to a 1×1 all-white picture, or an 8×1 all-black picture, or a 1×1 grey picture. All of those exploit the truth that you’ll be able to’t have lower than one byte of pixel knowledge, so that you would possibly as nicely use all eight bits of it. Intelligent!

Nevertheless once more…are we actually restricted to 1 byte of pixel knowledge?

(At this level you must go read Evan’s article earlier than persevering with with mine.)

When “Compression” Isn’t

The shortest uncompressed picture knowledge chunk contents for a PNG is 2 bytes: one byte to introduce the present “scanline” (row) of pixels, and one byte for the pixel knowledge (which could be as much as 8 pixels when you’re encoding a black-and-white PNG). Evan represents this as 00 00, since he’s encoding a single black pixel with no filter. However then there’s a “compression” step, and even ignoring the extra connected header and checksum these two bytes flip right into a 4-byte “DEFLATE” block: 63 60 00 00.

What a nugatory compression algorithm, proper? Nicely, probably not. It’s mathematically not possible for each enter to a (lossless) compression algorithm to get shorter, for a similar motive that it’s not possible to assign each letter of the alphabet a singular quantity between 1 and 10: irrespective of how arduous you attempt when going the opposite method, you’re solely going to supply ten doable outputs. That is the pigeonhole principle, and it’s probably not an issue in follow: more often than not the belongings you wish to compress aren’t random or extraordinarily quick, and so there’s room for enchancment. We’re simply in a bizarre edge case the place that’s not true.

Anyway, Evan principally disregarded DEFLATE (which, completely honest, there’s lots to say about PNG), however left in a footnote about what was in these 4 bytes:

The primary bit signifies that that is the ultimate DEFLATE block. The following 2 say that that is block that makes use of hard-coded Huffman codes; no “dictionary” is included within the payload. The following 8 bits encode a literal zero, after which one other 8 bits encode the identical. The following 7 bits are the “finish of block” marker, and the ultimate 6 pad the info in order that it’s byte-aligned.

He additionally advisable the infgen instrument for enjoying with DEFLATE, which supplies the identical breakdown:

% infgen -dd evan-deflate-block
! infgen 3.2 output
!
final            ! 1
mounted           ! 01
literal 0       ! 00001100
literal 0       ! 00001100
finish             ! 0000000
                ! 000000

At which level I finished to ask myself: is that actually one of the best we are able to do?

A self-referential format

The LZ77 compression technique that DEFLATE makes use of saves house by encoding “backreferences” to earlier components of the string, somewhat than wastefully repeating these bytes. (A few years in the past Julia Evans did an extremely neat visualization of how these backreferences work by using poetry, which I extremely advocate having a look at to get some instinct for a way this works.) One considerably shocking property is that these backreferences can overlap with themselves; the explainer Evan discovered for DEFLATE makes use of the string “Blah blah blah blah blah!” for instance, which compresses to “Blah b[distance=5, length=18]!” (I’m not going to elucidate that right here, click on by means of to the DEFLATE explainer to see the way it works.) Can we use that to our benefit?

From the breakdown above, we all know the block goes to be a minimal of 18 bits: the final flag (1 bit) + the compression scheme mounted (2 bits) + no less than one literal 0 (8 bits) + the finish marker (7 bits). If we might squeeze out a second byte of output in 6 bits, we’d beat Evan’s document (as a result of our DEFLATE block would solely be three complete bytes lengthy), however sadly there’s no method to do this in DEFLATE. So as a substitute, we assume we’re going to match Evan’s 32 bits complete, and we’ve to determine tips on how to use these remaining 14 bits. Evan’s easy encoding used 8 bits to encode the second 00 byte, and left 6 bits as padding. What alternate options do we’ve?

I scanned by means of RFC 1951, the DEFLATE spec, and located this in part 3.2.6, “Compression with mounted Huffman codes (BTYPE=01)”:

Lit Worth    Bits        Codes
---------    ----        -----
  0 - 143     8          00110000 by means of
                         10111111
144 - 255     9          110010000 by means of
                         111111111
256 - 279     7          0000000 by means of
                         0010111
280 - 287     8          11000000 by means of
                         11000111

(Be aware that these “codes” are written backwards from how infgen shows them. The “endianness” drawback will get even worse whenever you begin speaking about bits grouped into bytes as a substitute of simply bytes grouped into multi-byte integers.)

To summarize a second desk, values 0 – 255 are literal bytes (just like the “literal 0” above), 256 is the finish marker, and 257 – 285 are “size codes” for backreferences. (286 and 287 are unused.) Which suggests we’re in enterprise! We will encode an overlapping backreference somewhat than a second “literal 0”, and repeat that first 00 N instances.

The best way backreferences are encoded is a bit difficult, and I’m not going to enter it right here; when you’re curious you’ll be able to poke by means of the tables within the RFC like I did. Thankfully, the longest doable backreference has a comparatively easy encoding: #285, i.e. 11000101, adopted by a hard and fast 5-bit “distance” encoding of 00000 (i.e. “zero bytes again from the newest byte”). Placing this collectively offers us a DEFLATE block of 63 18 05 00, or

% infgen -dd jordan-deflate-block
! infgen 3.2 output
!
final            ! 1
mounted           ! 01
literal 0       ! 00001100
match 258 1     ! 00000 10100011
finish             ! 0000000
                ! 0

This decompresses to a collection of 259 zeros, nonetheless encoded in simply 32 bits. (31 bits, even.)

What are you able to do with 259 zeros?

As Evan factors out, you need to spend one byte to introduce a brand new row in a PNG. If we’re attempting to maximise the variety of pixels in our 67-byte PNG, we must always subsequently put them multi function row, so we solely must pay this value as soon as. That leads to 258 bytes of pixel knowledge, which at 1 bit per pixel offers us a whopping 2064 pixels.

Which, after updating all of the CRC32 and DEFLATE (Adler-32) checksums, produces this PNG. Which I’m not even going to hassle embedding, as a result of there’s no good solution to present a 2064×1 picture on a vertically-scrolling web site structure. As a comfort prize, right here’s a 48×37 picture (1776 pixels), which is the closest to sq. you will get with 259 zeros:

(It's a black rectangle, you're not missing anything.)

Evan set a decrease sure on the dimensions of a PNG file, and now I feel I’ve set an higher sure on the contents of a sound smallest-possible PNG file. However I’d like to be flawed!

Appendix A: Do we’ve to make use of zeros?

Technically no! The byte needs to be a sound “filter kind”, which could be 0 “None”, 1 “Sub”, 2 “Up”, 3 “Common”, or 4 “Paeth Predictor”, however then we are able to repeat that as pixel knowledge as a lot as we would like. Encoded as bits, that’s 0000_0000, 0000_0001, 0000_0010, 0000_0011, or 0000_00100. Right here’s what the opposite 4 rectangles seem like:

1. A black rectangle with some oddly repeating vertical white lines.
2. A black rectangle with some odd fractal shapes repeating at regular intervals horizontally.
3. A black rectangle with some odd noise at the top and then white bands stretching down vertically.
4. A black rectangle with some odd fractal shapes, similar to #2, at a slight diagonal above horizontal.

See Also

*shrug* Not a lot use for these however there you go!

Appendix B: Instruments used within the creation of this publish

As famous, I used Mark Adler’s infgen instrument (the one Evan advisable) to play with and test my DEFLATE blocks. I used Python’s zlib.decompress() to test that I obtained the zlib checksums appropriate while not having to replace the PNG checksums.

For calculating the PNG checksums, I used crc32. (Be aware that these embrace the header identify however exclude the size discipline.) I did the Adler-32 checksums for zlib by hand (nicely, I constructed the arithmetic expressions, then let the pc consider them). This was a bit foolish as a result of I might have used Python’s zlib.adler32().

For hopping out and in of hex on the command line, I used xxd. xxd -r -p goes from textual content hex to binary, permitting areas, which was helpful for passing into crc32.

On a Mac, I like to recommend doing hex modifying in Hex Fiend. And I didn’t do something particular to open the ensuing PNGs; in the event that they labored, they displayed in Preview simply effective. I did additionally use pngcheck to look at what was within the file.

2024
and is filed beneath
Technical.

Tags:

Graphics,

Compression

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