Reflections on Langlet’s APL Theory of Human Vision
Stuart Smith
[email protected]
Abstract
In the APL Theory of Human Vision [1,2], Langlet introduced the “neurobit system.” This system, it is
claimed, is able to simultaneously (1) detect periodic replications in boolean vectors and (2) compress
them, reducing their lengths without loss of information. In this paper, the neurobit network is shown to be
capable of detecting certain kinds of replicated patterns, but that this capability depends critically on the
data to be processed having dimensions that are powers of two. The success of data compression is also
shown to depend similarly on the dimensions of the data to be compressed; however, the key functions of
the neurobit-system give surprisingly simple spectral analyses of a whole class of fractal patterns. The
appendices contain the APL functions needed to allow experimentation with the ideas presented in this
paper. The workspace is also available from www.cs.uml.edu/~stu
Introduction
Fig. 1 is one of the more intriguing ones in Langlet’s APL Theory of Human Vision presentation [1].
Fig. 1 Langlet’s neurobit-system
What this figure seems to promise is almost too good to be true: an extremely simple system that
simultaneously detects periodic replications in boolean vectors and compresses them. The triangular
structure shown in fig. 1 is a “fanion.” [2,8] Given a length n boolean vector, S, the fanion function
generates successive rows of a fanion with n-1 iterations of S←(1↓S)≠ ¯1↓S, the initial value of S being
retained as the top row of the fanion. The top row of the fanion in this case is the character string ‘’'AA'’
rendered as the corresponding binary character codes. Reading down the left edge of the fanion we have
Langlet’s Helical transform (“helicon”) of the vector. Reading up the right edge of the fanion, we have
Langlet’s Cognitive transform (“cogniton”) of the vector [3,4,5,6,7]. The value of each element in a given
row is the exclusive-or (i.e., APL ≠) of the two elements immediately above it.1 As Langlet pointed out
[2], all of these relationships are retained when the fanion is rotated 120º in either direction, that is, the left
1
The neurobit system is thus a modulo-2 difference table, which, according to [9, p.1091], is equivalent to
the evolution of Wolfram’s simple cellular automaton using either Rule 60 or Rule 102. This connection is
discussed in [11] and [12].
edge is the Helical transform of the top row, the right edge is the Cognitive transform of the top row, and
each element in a given row is the exclusive-or of the two elements immediately above it.
Data Compression
The data compression claim will be considered first. The given vector of 16 items in fig. 1, representing the
binary codes for the string 'AA', is compressed to just the six topmost elements of both the left and right
edges of the fanion. All the 1’s are at the upper end of the result, followed by a tail consisting solely of 0’s.
To uncompress the result, the 0’s tail must be concatenated back onto one of the compressed segments—
left or right—and then the appropriate transform must be applied (Helical if the left edge of the fanion was
used, Cognitive if the right edge was used). It is therefore necessary to remember how many zeroes there
are below the compressed segment along each edge. The original string 'AA' requires two bytes, but the
compressed string also requires two bytes: one to hold the (encoded) replicated element 'A' and another to
hold a count of the zeroes that need to be tacked onto it in order to uncompress the string. Thus, no
compression is achieved; however, the transforms of strings of up to 255 identical characters still require
only two bytes, thus achieving a degree of compression that increases with the length of the string. This is
good, but the compression observed in fig. 1 disappears if only the seven low-order bits of the ASCII code
for 'A' are used to form the input vector, seven bits being all that are necessary to select any one of the
128 ASCII characters. The input to fanion in this case is a length 14 vector consisting of two identical 7bit codes. As can be seen, the output has no all-0’s tail; thus, no compression has been achieved.
fanion 1 0 0 0 0 0 1 1 0 0 0 0 0 1
1 0 0 0 0 0 1 1 0 0 0 0 0 1
1 0 0 0 0 1 0 1 0 0 0 0 1
1 0 0 0 1 1 1 1 0 0 0 1
1 0 0 1 0 0 0 1 0 0 1
1 0 1 1 0 0 1 1 0 1
1 1 0 1 0 1 0 1 1
0 1 1 1 1 1 1 0
1 0 0 0 0 0 1
1 0 0 0 0 1
1 0 0 0 1
1 0 0 1
1 0 1
1 1
0
Another, perhaps more serious, problem with the Helical and Cognitive transforms is their extreme
sensitivity to noise in the data they operate on. A single corrupt bit will force the use of another analysis
method, from which only an approximation of the repeated unit can be extracted.
Detection of Periodic Replication
For any theory of human vision, we will certainly want to know how effective the operations of the theory
are in detecting important features of two-dimensional scenes. The ability of the Helical and Cognitive
transforms to detect periodic replications is particularly well demonstrated when they are applied to highly
structured binary images. A very simple test case is a square image consisting of equal-width black and
white stripes. Fig. 2 (left), for example, is a 16×16 image consisting of sixteen stripes, each one unit wide,
and fig. 2 (right) is its Helical transform.
Fig. 2 Stripes pattern and its Helical transform
The 16×16 Helical transform of this image consists of a single black element (1) in an otherwise white
(0’s) matrix. The black element is located at (0,1), that is, in position 1 in row 0. If the striped pattern of
Fig.2 (left) is transposed, as in fig. 3 (left), the Helical transform is also transposed, as in fig. 3 (right):
Fig. 3 Horizontal stripes pattern and its Helical transform
Combining the vertical and horizontal stripes patterns element-wise with exclusive-or produces the
checkerboard pattern shown in fig. 4 (left). Because the combining operation is exclusive-or, the
intersections of black (1) stripes form white (0) squares:
Fig. 4 Checkerboard pattern and its Helical transform
Fig. 4 (right) is the Helical transform of the checkerboard pattern. The black square at (0,1) represents the
contribution of the vertical stripes, the black square at (1,0) the contribution of the horizontal stripes.
Because the Helical transform is its own inverse, applying it to the transformed image will reproduce the
original striped or checked image. Further investigation shows that the Helical transform of any binary
image that (a) is square, (b) has a side whose length is a power of two, and (c) is built only from alternating
equal-width black and white stripes will have a single 1 in the first row if the stripes are vertical, in the first
column if the stripes are horizontal, and in both the first row and first column for the checkerboard pattern.
Each 1 will be located in a row or column position whose index is a power of two, which will also be the
relative width of each stripe. We conclude, then, that the two-dimensional Helical transform can detect
simple cases of periodic replication.
A more searching test of the Helical transform’s ability to detect periodic replication is provided by 2×2
block-replacement L-system patterns. These fractal patterns, some quite pretty, are all built in the same
way: a 2×2 matrix of 1’s and 0’s is given as a start pattern, and then successively larger square matrices are
built by replacing each 1 with a copy of the start pattern and each 0 with a copy of another, usually
different, 2×2 matrix. Successively higher orders of Langlet’s G (“geniton”) matrix, for example, can be
built in exactly this way. The start pattern is G:
1 1
1 0
To generate the next higher order matrix, replace each 1 with G and each 0 with a 2×2 all-zeroes matrix.
The result is this 4×4 matrix:
1
1
1
1
1
0
1
0
1
1
0
0
1
0
0
0
This process can be continued indefinitely, producing larger and larger matrices, all of which will be square
with side a power of two. All 256 possible combinations of two 2×2 boolean matrices can be generated
with the tile function. Thus we can know in every case exactly how a given tiling pattern was generated.
Thue-Morse Plane
The next test case is an image that has a relatively complex visual appearance but, as it turns out, a very
simple Helical transform: the Thue-Morse plane, a familiar fractal image. Fig. 5 (left), for example, is the
16×16 Thue-Morse plane, and fig. 5 (right) is its Helical transform.
Fig. 5 Thue-Morse Plane and its Helical transform
As can be seen in the transformed image, 1’s are at positions 0, 1, 2, 4, and 8 in both the first row and first
column. This analysis of the plane yields a set of nine 16×16 “basic forms,” each corresponding to a
specific 1 in the transformed image. A particular basic form is obtained simply by applying the Helical
transform to a matrix containing 1 in the selected position and 0’s everywhere else. When the basic forms
are combined element-wise with ≠ the result is the original Thue-Morse plane. The first basic form,
corresponding to the 1 at (0,0) in the transformed matrix, is an all-1’s (i.e., completely black) square. The
effect of the all-1’s basic form is to complement an image, that is, reverse black and white; thus, the bit at
(0,0) functions as a sort of “sign bit.” The remaining basic forms correspond to 1’s in the first row and first
column. As expected, these images consist of black and white stripes whose relative widths are 1, 2, 4, and
8. The relative width of the stripes in a given basic form is the same as the row or column index of that
image’s 1 in the transform. Fig. 6 shows the set of basic forms of the 16×16 Thue-Morse plane:
Fig. 6 Decomposition of the 16×16 Thue-Morse plane into nine basic forms
The Helical transform reveals replication of both horizontal and vertical stripes at four different scales. This
is a global picture of the large-scale replicated elements from which the original image is built. This may be
a somewhat surprising result given that the construction of the Thue-Morse plane is usually described as a
process that gradually builds the plane out of smaller pieces.
Hadamard matrix
The last simple test case, the Hadamard matrix, is interesting because it has a relatively complex visual
appearance but a very simple Helical transform. It is also of interest because it has applications in image
and signal processing and in cryptography, among other areas. Fig. 7 (left) is a 16×16 Hadamard matrix
represented as a binary image, and fig. 7 (right) is its Helical transform.
Fig. 7 Hadamard matrix and its Helical transform
In the transformed matrix, the 1’s lie on the main diagonal at positions (0,0), (1,1), (2,2), (4,4), and (8,8).
This analysis yields a set of five basic forms, each corresponding to a 1 in the transformed image. The first
basic form, corresponding to the 1 in the “sign bit” position (0,0) in the transformed matrix, is again an all1’s (i.e., completely black) square. Fig. 8 shows the basic forms corresponding to the 1’s on the diagonal.
The transform reveals replication of a squares-within-a-square pattern at four different scales.
Fig. 8 Decomposition of the 16×16 Hadamard matrix into five basic forms
Most of the other 2×2 tilings have Helical transforms in which the replication patterns are obvious but
involve a large number of basic forms, far too many to permit showing all of them as was done above with
the Thue-Morse plane and the Hadamard matrix. These patterns can be generated with the tile function
listed in Appendix A. Many of the patterns and their transforms produce a kind of visually appealing
“computer art.”
Beyond 2×2 Block Replacement Images
The Helical and Cognitive transforms appear to be exquisitely designed to detect multiple levels of
replication in the kinds of binary images represented by 2×2 block replacement images. This is not
surprising because both transforms were originally based on Langlet’s G matrix, which is itself a member
of that collection. However, we immediately encounter difficulties when we want to apply the transforms to
images that are not square with a side whose length is a power of 2. The solution in these cases is to pad the
image with zeroes and/or use a truncated transform matrix.2 The downsides are that (1) these versions of
the transforms are generally slower and (2) the results are typically much less informative.
2
These possibilities are discussed in Langlet transforms at www.cs.uml.edu/~stu
To illustrate the problem, consider fig. 9 (left), a 27×27 Sierpinski “carpet,” and fig.9 (right), its Helical
transform.
Fig. 9 27×27 Sierpinski Carpet and its Helical transform
The replication pattern is easily visible in the Carpet image, but the transformed image appears to reveal
nothing beyond mirror symmetry around the main diagonal. This situation arises out of the mismatch
between the Langlet transform matrices, which consist of elements that are replicated at scales that are
powers of two, and the 3×3 tiling patterns, which consist of elements that are replicated at scales that are
powers of three. While the rows and columns of a 2×2 tiling pattern are precisely aligned with the rows and
columns of Langlet transform matrices (which are themselves 2×2 tiling patterns), this can never occur
with 3×3 tiling patterns.
Discussion
As we already saw in the data compression example at the beginning of this paper, many of the nice
properties of the Helical and Cognitive transforms depend on the dimensions and periodicities of their
arguments being powers of two. In Langlet’s system the problems that arise when the dimensions and/or
periodicities are not powers of two cannot be dealt with through conventional image or signal processing
techniques (e.g., windowing, smoothing, filtering, etc.). The reason is that Langlet insisted that essentially
all operations in his system have exact inverses so that no information could ever be lost in the course of a
computation. To achieve the goal of completely reversible computation, it is necessary to follow Langlet’s
droll mini-manifesto and scrupulously respect the “the rights of a bit:” [6]
Déclaration des Droits du Bit (Antwerp, 1994)
0. All bits contain information, so they deserve respect.
1. Every small bit is a quantum of information.
Nobody has the right to crunch it.
Nobody has the right to smooth it.
Nobody has the right to truncate it.
Nobody has the right to average it with its neighbour(s).
Nobody has the right to sum it, except modulo 2.
Given the severity of these requirements and the nature of the problems encountered, it is not clear how one
would proceed with the work described here. One of the practical problems Langlet sought to solve was to
somehow connect his system to the domains of image and signal processing. I had hoped that he had gone
further in this direction than what I knew from his publications; however, in conversations after his passing
with people who knew him I learned that he had never made the connection. By contrast, Wolfram’s
magisterial A New Kind of Science [9] has pointed in directions in which progress might be made in a style
of computing reminiscent of Langlet’s. One will recognize in this book many of the themes that Langlet
was developing, although there is no mention of Langlet in it. Meanwhile, however, the game has changed
significantly. The appearance of CNN (Cellular Neural Network) chips [10], which combine large numbers
of digital and analog processors on a single chip, has enabled real-time processing of large quantities of
optical sensor data. The main computational burden is carried by the analog units operating in parallel,
while the digital units are used principally for control and memory. The two-dimensional neurobit network
envisioned by Langlet resembles a CNN chip in some respects, but CNN’s constitute a more powerful and
comprehensive approach to machine vision, pattern recognition, and image processing.
References
[1] Gérard Langlet. The APL Theory of Human Vision (foils). Vector, 11:3.
[2] Gérard Langlet. The APL Theory of Human Vision. APL ‘94 - 9/94 Antwerp, Belgium
ACM 0-89791 -675-1 /94.
[3] Gérard Langlet. Towards the Ultimate APL-TOE. ACM SIGAPL Quote Quad, 23:1, July 1992.
[4] Gérard Langlet. The Axiom Waltz - or When 1+1 make Zero. VECTOR, 11:3.
[5] Gérard Langlet. Paritons and Cognitons: Towards a New Theory of Information. VECTOR, 19:3.
[6] Gérard Langlet. Binary Algebra Workshop. Vector, 11:2.
[7] Michael Zaus. Crisp and Soft Computing with Hypercubical Calculus: New Approaches to Modeling in
Cognitive Science and Technology with Parity Logic, Fuzzy Logic, and Evolutionary Computing: Volume
27 of Studies in Fuzziness and Soft Computing. Physica-Verlag HD, 1999.
[8] Gérard Langlet. The Fractal MAGIC Universe. Vector, 10:1, p.137, 1993.
[9] Stephen Wolfram. A New Kind of Science. Wolfram Media, 2002.
[10] Leon Chua and Tamás Roska. Cellular Neural Networks and Visual Computing.Cambridge University
Press, 2002.
[11] Stuart Smith. Wolfram’s Rule 60 and Extended Additivity.www.cs.uml.edu/~stu.
[12] Stuart Smith. In Search of the World Formula. www.cs.uml.edu/~stu.
Appendix A: Langlet Function Definitions
This section contains definitions of the functions needed to explore the ideas presented in
this paper. Appendix B contains definitions of the functions used to create figs. 2-8
above. Langlet’s Helical and Cognitive transforms can be implemented in several
different ways. The implementations here were designed to avoid WS FULL when applied
to big images (e.g., 512×512 and larger). The functions defined in Appendices A and B
are available in the Langlet tools workspace at www.cs.uml.edu/~stu.
Helical transform of square boolean matrix ⍵, whose side is a power of 2
HEL2←{⍉↑HEL¨↓⍉↑HEL¨↓⍵}
Cognitive transform of square boolean matrix ⍵ , whose side is power of 2
COG2←{⍉↑COG¨↓⍉↑COG¨↓⍵}
Helical transform of boolean vector S, whose length is a power of 2
z←HEL S
:If (⍴S)=1
z←S
:Else
z←(HEL lhalf S),HEL(lhalf S)≠rhalf S
:EndIf
Cognitive transform of boolean vector S, whose length is a power of 2
z←COG S
:If (⍴S)=1
z←S
:Else
z←(COG(rhalf S)≠lhalf S),COG rhalf S
:EndIf
Auxiliary functions required by the transforms
lhalf←{((⍴y)÷2)↑⍵}
rhalf←{((⍴y)÷2)↓⍵}
Helical transform for general vector or matrix argument
y←hel x
:If 1=⍴⍴x
:If (2⍟⍴x)≠⌊2⍟⍴x
y←thel x
:Else
y←HEL x
:EndIf
:ElseIf 2=⍴⍴x
:If ∨/(2⍟⍴x)≠⌊2⍟⍴x
y←thel2 x
:Else
y←HEL2 x
:EndIf
:Else
⎕←'argument must be a vector or matrix'
z←⍳0
:EndIf
z←thel v;rv;lg;rm;mat
rv←⍴v
lg←⌈(⍟rv)÷⍟2
rm←(2*lg)-rv
mat←rm trunc P 2*lg
z←⌽v bmp mat
thel2←{⍉↑hel¨↓⍉↑thel¨↓⍵}
Cognitive transform for general vector or matrix argument
y←cog x
:If 1=⍴⍴x
:If (2⍟⍴x)≠⌊2⍟⍴x
y←tcog x
:Else
y←COG x
:EndIf
:ElseIf 2=⍴⍴x
:If ∨/(2⍟⍴x)≠⌊2⍟⍴x
y←tcog2 x
:Else
y←COG2 x
:EndIf
:Else
⎕←'argument must be a vector or matrix'
z←⍳0
:EndIf
z←tcog x
z←⌽thel⌽x
tcog2←{⍉↑cog¨↓⍉↑cog¨↓⍵}
z←n trunc mat
:If ((⍴⍴mat)≠2)∨((1↑⍴mat)≠(¯1↑⍴mat))
⎕←'argument must be a square matrix'
:EndIf
z←((-n),n)↓mat
Functions to view bitmap bits and write it out to .bmp file file
disp bits
'bm'⎕WC'bitmap'('Bits'bits)('Cmap'(2 3⍴255 255 255 0 0 0))
'f'⎕WC'form'('Picture' 'bm')
bits writebmp file;bm
'bm'⎕WC'bitmap'('Bits'bits)('Cmap'(2 3⍴255 255 255 0 0 0))
bm.File←mypath,file
2 ⎕NQ'bm' 'FileWrite'
Function to read .bmp file file into variable argb
argb←readbmp file;bmp
'bmp'⎕WC'Bitmap'(mypath,file,'.bmp')
argb←bmp.Bits
Appendix B: Functions to create the figures in the text
In each definition, n is the number of iterations of block replacement to be performed.
The side of the resulting matrix will be 𝟐𝒏+𝟏 . blk and wht specify the two 2×2 binary
patterns used to tile the image.
Sierpinski triangle (Langlet’s G matrix)
y←sierpinski n;blk;wht;⎕IO
⎕IO←0
blk←2 2⍴1 1 1 0
wht←2 2⍴0 0 0 0
tile_step←{⍎⍤1⍕(wht blk)[⍵]}
y←n{(tile_step⍣⍺)⍵}blk
Hadamard matrix
y←hadamard n;blk;wht;⎕IO
⎕IO←0
blk←2 2⍴1 1 1 0
wht←2 2⍴0 0 0 1
tile_step←{⍎⍤1⍕(wht blk)[⍵]}
y←n{(tile_step⍣⍺)⍵}blk
Thue-Morse plane
y←TMplane n;blk;wht;⎕IO
⎕IO←0
blk←2 2⍴1 0 0 1
wht←2 2⍴0 1 1 0
tile_step←{⍎⍤1⍕(wht blk)[⍵]}
y←n{(tile_step⍣⍺)⍵}blk
stripes
y←stripes n;blk;wht;⎕IO
⎕IO←0
blk←2 2⍴1 0 1 0
wht←2 2⍴1 0 1 0
tile_step←{⍎⍤1⍕(wht blk)[⍵]}
y←n{(tile_step⍣⍺)⍵}blk
checkerboard
y←checkerboard n;x;⎕IO
⎕IO←0
x←stripes n
y←x≠⍉x
Sierpinski carpet
y←carpet n;wht;blk;⎕IO
⎕IO←0
blk←3 3⍴1 1 1 1 0 1 1 1 1
wht←3 3⍴0
tile_step←{⍎⍤1⍕(wht blk)[⍵]}
y←n{(tile_step⍣⍺)⍵}blk