improving on SPIHT-Haar (*) though purportedly impossible, we note that a pixel-quad base is less than its pixel-quad average, x- and y-tilt tend together, resolution is order-dependent: maybe with significant compressible redundancy |

[Topically related to Progressive Image Resolution; and Fully Interleaved Scanning]

* (The Haar Function was defined as one-dimensional, but we recognize the obvious two-dimensional extension, better known as the H-Transform in astronomy, but which tends to lossy by design compounding precision of quad-summations ... We also note that triangular arrayed pixels would be better suited for triads, than squares are for quads: as only the triad-average and two subdifferences would be measured, no next-order twist;- but also involve more-than-orthogonal-quadrature integer calculations.)

Considering--

SPIHT emits zeros for x- and y-tilts until either's top bit is nonzero, but x-y angle correlates the top bit over 2/π ... it would seem better putting out a single zero for both until one flags nonzero-'begin' ...

At the fixed-bandwidth lower limit, as for video, SPIHT suffers resolution-waffling where part of the picture has an extra bit, part does not, and part between, varies ... it would seem better time-interleaving the quad-selection processing order ...

Temporal correlation is very poor at the pixel level, as the camera moves (which is necessary for subpixel-interleaving fill-in) ... it would seem better to not process for temporal correlation at the pixel level ...

Luminance, tends reflective, multiplicative by its light source over large areas, more than additive as occurs along boundary edges ... it would seem better to use a logarithmic-like intensity value of pixels ... (compare also angular-lighting of round edges) ...

In general, color-luminance, is actually green: Red and blue appear luminant because they sit partway in ocular-green-receptivity ...

(In fact, maximum-definition-HDTV resolution is about 1-arcmin. intended, which is about 1/5th² of fovea-resolution at 12-arcsec.)

A partial remedy, is to take all images in double-resolution and compute on-registration and off-registration as alternate-images with equal resolution,- though it still results in double-vision with alternating pixel-widths, breathing, artifacts....

A better approach, slides off-registration a quarter-pixel, thickening pixel lines equally, relatively partially-filling adjacent pixels;- which as a simple 'averaging-mechanism', resolves twice-as-many, thinner-than-double-pixel-width, somewhat thicker-than-single....

Another approach, samples 4x8- or 16x16-subpixels in near-golden-ratio-interleaved order: to be displayed pointwise-subpixelwise... (4x8 uses 1,3-steps fully correlatively prime, and 16x16 uses 5,9-steps pushing nearer the middle each step; golden-ratio ensures that successive steps fill-in with the same ratio and also tend to fill nearer more-previous points sooner than the more-preceding).

* (Display Gamma adjusts this, as well as room-brightness, color sensitivities; and vertical and horizontal differ slightly by trace-overlap, and RGB/RGBG pixel placement, yet both are close about the median. Tolerance is apparently tight as unevenness is noticeable at ±10%, in either case: a third, of the web-standard six-cubed 8-bit color-scheme quantum of document-browsers.)

By comparison, On-pixel alignment exhibits alternating thickness 'breathing': where lines cross one-and-two pixels the half-bright double-wide lines single-width-equivalently bright about ~0.70, eg. 179/255, and fine-detail washout.

(Appraising the two results together, pixel-system-gamma is 2.00, or that is, the original-receptor pixel-system-gamma is 0.50, square root, equivalencing pixels as independent, orthonormalized vectors:-- A "digital box" pixel, uniform, slid to the halfway position, needs 0.50 = 0.71², as in the second result; Slid to the quarter position, needs 0.25/0.75 = 0.58², as in the first result. The 2-D sesquipixel roughly equivalences to spreading each original-definition pixel to [0.75 | 0.43 | 0.43 | 0.25], added gamma-correctly to the x-,y-,xy-adjacent pixels; and thence moving fine half-steps horizontally, vertically, diagonally, by column and row alternations.)

Gif-image-generation software then-needs support this smoothing by avoiding representing sharp edges as single-quantum steps.

* (bandwidth-reduced Y-luminance)

** (signal amplitude inversion catches RF spikes as less-noticeable black streaks instead of white)

High-SNR signal-noise-ratio cable, satellite, DVD, technologies have increased the potential and actual resolution tenfold, signal levels to 4-5 bits (eg. QAM16/QAM32), pixel quantity 8× (esteemed commercial-35mm-film-equivalent, but film has its own improvements); deriving 6-7 bits from density (dither diminishes as SNR improves, and is inaccessible in most digital coding schemes * but modulation schemes utilize the noise reduction for signal-correction robustness).

* (An exception is OQAM64/OQAM128, Offset quad-interstitially compatible to QAM16/QAM32; cutely called, OQAM's shaver.)

But the technological shift from monotonic amplitude, analog, to digital, required revised methods of signal error detection-erasure-correction;- Especially digital signal coding required "smoothing-soothing" of code-errors that would otherwise result in irreverent, picturally unrelated temporal and spatial optical discompositions that looked more like TV-"ghosting" patching-in overriding channel discontent than TV-"snow" or motion aberrations. Simple save remedies involved stalling repeating the whole prior image or spotwise dark-outs (reduced-brightness image retention). But ideal smoothing-soothings were something like reduced-spatial-resolution "blur" and reduced-amplitude-resolution "snow";-- the blur was new and less noticeable than "snow" as its next image would restore detail. This lead to the selection of the sum-and-difference transform "blur" and bit-slicing "snow" where the channel could be bandwidth-truncated (as NTSC is bandwidth-fixed) and signal frames would each contain the most significant image-bits filled to the allotment.

Ideally also, photons are not pointwise bunched but faster quantum-refreshed, allowing for 'catching' flicker on the periphery.

(Nevertheless, Because the usual image viewing brightness photon shower is dense and rapid, pseudorandom works equally well on small scale, spotwise, as for whole images: An equivalent might then be a prime-ratio interleaving raster-scan in pixel groups, approximating golden-ratio area-fill, eg. 7x5-steps in 16x16-blocks ... retaining some local correlation, a few levels up, and timewise;- and might thus also adapt high-resolution to lower-bandwidth subsampling and non-microlensed pixelation, camera and, receiver: present possibility.)

The next-major application of image resolution is in third-dimensional travel, into the image, as with computer-generated imagery; and gave rise to the Haar approach (Haar Transform, useful as an approach for characterizing common image-source business): Consider a single pixel of given luminance: Travel into its depth requires resolving its subpixels. Haar wavelets do this, appending subdifferences Δx,Δy tilts and ΔΔxy saddle-twist, and third-dimension Δt and compound double and triple subdifferences, for motion compression.

Haar is used in astronomy where telescope lens and receptor systems have equal resolution, adjacent pixels are optically matched and usually spanned by single stars; But other applications, especially computer-generated imagery, eg. text/html document forms, where images are registered to pixel lines, should better differentiate subpixels directly by the smaller subpixel value and reconstruct the larger remainder ... we'll designate this, Haar-0 (zero), but a later scheme, switched compression, shows these are virtually the same.

THE HAAR TRANSFORM: (2-D spatial, still images)

The progressive effect (video):

- The over-pixel is as-is, the sum-amplitude of quads: (*)
- p-total = q
_{00}+ q_{01}+ q_{10}+ q_{11}

- p-total = q

Subsequent pixels are progressively differentiated as subquadrant summations:+ + + + + - + - + + - - + - - + - p-x-tilt = q
_{00}+ q_{01}- q_{10}- q_{11} - p-y-tilt = q
_{00}- q_{01}+ q_{10}- q_{11} - p-twist = q
_{00}- q_{01}- q_{10}+ q_{11} - differentials are limited by the amplitude: they can never be greater;

- p-x-tilt = q
- (The transform should equally resolve all values of data at every bit-level.)
- In practice the coefficients are halved, so that the consequent decode iteration (the transform is its own inverse) is to scale.

DIGITAL PRECISION, REPRESENTATION: 1-D CASE:

Sum and difference increase the resulting representional precision of data necessary to recover the original, by an additional bit per; but only one of the two one-bits is needed as odd(a+b) = odd(a-b) is redundant information, conveniently in the difference coefficient, truncating the sum to average, at the original precision, so that progression likewise stays at the original precision, up the pyramid. At reconstruction, the odd(a+b) bit is recovered from odd(a-b). Haar leaves a trail of differences increased by one bit, unto the top average; and recovery is progressive and successive to lossless.DIGITAL PRECISION, REPRESENTATION: 2-D CASE:

Four-way sum-and-differentials increase the resulting representional precision by two additional bits per but can be truncated a bit: H-transform was defined without truncation, but all four LSB's are equal, and (a+b+c+d) = (a+b-c-d) + (a-b+c-d) - (a-b-c+d) mod 4 as -3d = d mod 4 ; and so also the 2-D Haar sum-average can be maintained constant precision up its pyramid: p-total 2-bit-truncated at the original precision, p-tilts at one bit more, and p-twist holds the full two-bits-more precision; Reconstruction simplifies to p-total-LSB#1 = oddsum(p-tilts-LSB#1's, p-twist-LSB#1). Haar leaves a slightly uneven field of differences increased by one bit, a third by two bits, unto the top truncated-average; and recovery is progressive and successive to lossless. (*)* (It is not certain in original on-web documentations, that SPIHT keeps precision this tight.)

From there, basic SPIHT compresses the coefficients.

[under reconstruction]

In binary, The more efficient Haar-0 subdivides a pixel-average by its subpixel-minimum ...

But because both Haar and Haar-0 resolve by progressively subdividing pixels by subcoefficients requiring four more bits per quad, the transition efficiency equator-crossing occurs exactly halfway where subdifference and subpixel amplitudes equal the pixel-average ... the middle half range (0.50 to 1.50) is most efficiently compressed by Haar, and the two outer quarter ranges (0.00 to 0.50; 1.50 to 2.00) by Haar-0.

And we might switch-between Haar and Haar-0 subcoefficients: a switch becomes useful when the image tends to extremes rather than middle coefficient values, as in the case of combined natural and artificial images typical in the modern Internet video information era (photographed objects and diagrammed constructions; real and virtual),- a choice favoring shallow slopes, or, sharp edges, low and high contrasts, over middling slow transitions which are not half as common in documents, astronomy, nor objects except at rolled edges.

But a switch bit shifts the transition efficiency equator-crossing to occur, above, the halfway mark,- toward the two-thirds depending on the entropic information content of the zero-bit, which, depends on the image (It is possible to partition the whole picture coarsely and switch only on partitions, costing each pixel a fraction of a bit); And we should saddle it on the lesser likely compression method.

Nevertheless, the highest-possible top-significant subcoeffient bit, one bit above the pixel-average top-significant bit, indicating the subcoeffient may-reach double the pixel-average, suffices as the switch bit, so that when a subcoefficient is out of optimal range, the higher value range is automatically the other type coefficient (*). (It would be in statistically high use itself in the Haar, were steep slopes common.) The Haar is more usably the prominent subcoefficient, as Haar-0 applies to high contrast which tends to take only its larger values (black-on-white, graphite-on-vellum, rather than dim-on-light gray, triple-on-unit dark); Also, in top-down bit-rastering, the top-significant bit already appears most orderly in Haar,- and, that facilitates progressive partial-decoding (called, "embedding"); (and may be best if we keep the transition at halfway).

* (For astronomy, a switched Haar/0 using the top-significant bit as the switch bit incurs no statistical loss, because all the values get used, whichever values get used first.)

However, it is now usefully apparent that the switch-selected smaller half Haar-0 is virtually identical to the unused larger half Haar by mere magnitude complementation (pixel-relative remainder) and doubling its magnitude by including its additional bit of arithmetic resolution; the Haar-difference sign is equivalent to the Haar-0 subchoice (by judicious pointer-sense selection); ... which in practical implementation means a, Justified-Haar (*), is the Haar with its upper-range subdifference bits below its top bit and sign, amplitude-reordered as a priority prediction: a reversal of its large-small ordering of amplitude graduations above halfway that presumes large-excursion adjacent subpixels are tending to higher contrast. (Justified, meaning, pulled tight to both limits,- of contrast.)

* (If you know the SPIHT bit-rastering algorithm already, the Justified-Haar must notify SPIHT of its top bit and sign as soon as it reaches its level, then later its choice of switch to magnitude-complementation, but which is, now, that top bit; and thereafter midhigh bits are mostly zeros when reordered, to be compressed by yet an intermediate SPIHT listing also leisurely spilling into the standard coefficients listing ... more discussion will ensue momentarily.)

(Magnitude-complementation of integer subdifference X across average Y, is simply 2Y+1-X -or- shiftup Y + 2s'complement X).

Now, implementation of a, "squeakey-clean Justified Haar," has the encoder checking the magnitude of the Haar subdifference against four-thirds of the Haar pixel-average (or three-quarters of the subdifference against the average), and when exceeding, jumps up and turns-on the subdifference's top-possible bit and fills with the magnitude-complement of the remaining amplitude bits --except that-- it also crops sooner when the Haar subdifference exceeds to the next bit by itself (squeaky-clean getting 87% numeric possibilities maximally justified; slightly less than 100% fully justified; a Nyquist-like corner). Nevertheless, the encoder needs only abide within its cropping rules, for the decoder already properly computes whatever it's given.

In total, the same amount of information per numeric coefficient ... just changed its preferred meaning along the way.

(Justified Haar also solves Haar's loss-of-compression-stroke problem in medium and high contrast limitary cases: middling values are already telling the compressor, the next-adjacent compression is something else as this is going for the limit in the pixel-step between; while higher values already disconnecting from slope compression are maximally effectually compressed.)

THE UN-HAAR TRANSFORM --almost Haar but we're now looking at its right-hand efficiency:-- While Haar and its extensions as herein described are ideal for progressive resolution, video compression would rather have the maximal edge-efficiency attainable:

To wit--

Because the subdifference requires the pixel-average to be of sufficient amplitude to not undershoot, a large subdifference, as in the high contrast case, restricts the minimum value of that pixel-average, even defines it, and we can take the reduction. (Cf progressive, Haar, where the pixel-average precedes its subdifferences in processing, and we will take the reduction in the subdifference instead).

1. In the initial bottom-up computation of Haar average-and-difference, the difference magnitude-only, subtracted from the average, suppresses the average to no less than zero;-- but which is just the pixel-minimum: and the difference-sign says, which pixel.

2. But, even more-different from Haar: The suppressed average of any one pair is not necessarily like that in the adjacent pair, -its difference is whatever it takes-on (especially unlike in the high contrast case),- but in fact, the full height, is like: That is to say, the edge value -alone, unaveraged- is very likely like the adjacent average ... whence the computation for the next higher level average-and-difference of lower averages in, Suppressive Un-Haar, includes the lower subdifferences to compute their edge values (but sends the suppressed average for maximum efficiency);- slightly more computation but to get at the compressive efficiency (And different from the present definition of SPIHT by other authors).

This differs from Progressive Haar because a pixel-difference processed on a Suppressive Un-Haar pixel-average, yields not two sub-pixel-averages to-be-resolved further, but a pair of yet-to-be-fully-defined sub-pixel-amplitudes which may be averages or the nearest edge, depending on the subpixel-difference being larger than half -[rechecking arithmetic]-... [under construction]

The Haar Transform is progressive binary subdetailing by differentiation: 2-D for imagery, though 3-D for object-stereoscopy may find use, and also used frame-to-frame 3.5-D, which is most effectual on very fast -continuum- cameras, or very slow scenes; Haar is ideal for database-movie interactive-representation of objects, but usually implemented on diced frames (*) subordinate to an initial "thumbnail sketch" miniature. In video applications, coarse details are needed more rapidly more often and fine details more slowly,- but that is essentially motion-detection, a continuum camera, or infinite resolution, a subsampling camera. For image-generation, Haar is computed top-down as successive differentiation; for image-compression, it is computed bottom-up.

* (For image compression the transform needs only be applied at the first few levels (8×8 in 2-D) as the bit-density fraction of each larger sum-average coefficient, included with even tiny but-nonzero nodes, becomes nilficient.)

Haar coefficients are all-but-one differences of adjacent pixel-amplitudes-and-averages; and one, sum-total-pixel-average atop:- At the first level, pairs of pixels are averaged (sum÷2) and sloped (differenced); At the second level, pairs of the averages are taken as larger pixels, and averaged-and-sloped ... the total computation is 2(n-1) for n initial pixels (typically a power-of-2);- simplicity that was fine for level backgrounds (astronomic black), but suffers on gradual slopes where the small slopes are all coded, uncompressed.

(This is a case example of basic computator elements: add-shift/subtract, as a functionally optimized process unit, hardware.)

Haar was great improvement over the Hadamard transform in compression efficacy and
computational scale efficiency:-- Hadamard coefficients [**] like Haar at the first
level, also computed averages-and-differences of the slopes,-- but, though
in potential cases might improve compression, did not often for practical imagery,
and its computational cost was n(log_{2} n), (where 2Mpx is 21-level).

Minus, Hadamard's compression of slope coefficients was ignorant (independent) of the obvious information available in the very-next processing of the averages: the slope of averages, in the next-level, was computed and rendered separate from the average of the slopes, in the lower level, --unused though containing a near-preeminent estimate, differenced only by curvature.... Haar, didn't go far enough to use it, and Hadamard ignored it; yet its inclusion completely extracts the slope, as Haar did for the averages ... we therefor remedy this by declaring and defining--

Averaging-up the slope adds no amplitude (no bits necessary for lossless reconstruction), however, luminance sensitivity favors its subsequent difference, and therefor the data compression scheme will need to look one bit deeper in that coefficient. The slope of adjacent averages is double-amplitude of the average of adjacent slopes,--

--which is just the Newton-weighting to a curve-fit-approximation (1,3,3,1). This is also intuitive as the average slope needs its corresponding order bit at the same encode-decode-time as the slope of adjacent averages already double-amplitude;-- cf at the coarsest resolution, the detail-advance in adding a next-level of interstitial pixels should result in increased detail immediately: a stairstep at a given resolution should not remain a stairstep at the next, if it is-not.

The Haard Transform is equally progressive, as successively extensible, as the Haar; its advantage is only to reduce two difference coefficients, both by average and difference and one by prediction. Further improvement may include the sum, of average and slope (3,1,1,3), making the higher levels smoother than polygonal. However, this particular sum affects the higher bitslice early;- and from the top-down constructive perspective where Haar resolved successive subpixels, Haard resolves successive subdifferences as well, and thence its two succession paths, the averages and differences paths, ladder-parallel,- and further process improvement involves and entangles that runging, and for decreasingly significant compression gain.

[under reconstruction] And a statistical item to consider: The slope-of-the-averages can be used half-efficiently, more or less, in predictive analysis, as it is less likely maintained across much of the image, especially where it is large, or either average, extreme, and the subdetail average-of-slopes more likely to roll-off, even cap-off ... which reduces the compression a bit less, rather than fully, but statistically better yet.

- high-to-low bit-slice successive approximation to precision (a particular method);
- total (average) first impression and progressive differential subdetailing;
- equilibrated resolution at each level: equally significant differlet bits together; (*)
- compress zeros in just-over-binary-order data (a consequence of entropically coding largest magnitudes first);
- lossless transform capability.

Full SPIHT-video, is additionally:

- simple encoding and decoding transform computations; differlets;
- lossless (n+1 bit) progessive Haar(d) transform.

* (The slope for the sun, or any disk or round hole, as a light source, is not linear; nor logarithmic; but typical edge-slopes occur at junctures of multiple obscurations, eg. leaves on a tree, where intersection of obscuration angles are, fairly linearly.)

SPIHT-Haar has intrinsic inefficiency on smooth increasing across small regions larger
than adjacent pixels,- which are expectably predominant in daylight scenes: SPIHT is
hard-programmed to compute and keep the smallest quad slopes, and codes them all rather
than include larger quad trend coefficients ... in other words: SPIHT might be improved
with haardlets,- or, by decidable-haardlets where that efficiency wanes:
requiring an additional bit of information at each coefficient,
whether it is a finished Haarwavelet, or a compounded haardlet in the next higher
quad ... however that bit contains significant information, that for sloplets
s_{1},s_{2}, taking their sum and difference, improves the
data-bit-compression, or not, over the two individually:

[under reconstruction]

BIT EFFICIENCY: The number of bits representing (the significance of) a number n≥0
(integer) is, (log_{2} n+1).

The average number of bits representing a range of numbers [0,N],
N≥n≥0, is ∑(log_{2} n+1)/N+1 = (log_{2} N+1!)/N+1.
Approximating n! ~ (n+.5/e)^{(n+.5)}√2π,
this is about (N+1.5)(log_{2} N+1.5/e) + (log_{2}√2π) / N+1,
or roughly ~ (log_{2} N+1) - (log_{2} e) + trim, that cannot exceed
that for its own largest value: thus, just over (log_{2} N+1) - 1.44 bit (a
natural bit [nit] less than for its largest value N).

* (Cf typicals, for smallish N: 1.0-.50 bit, 2.0-.85 bits, 3.0-1.09, 4.0-1.23 [NTSC], 5.0-1.32; 6.0-1.38; 7.0-1.40; 8.0-1.42 bits ...).

The average number of bits in representing the sum, of two numbers N≥n≥0, is
(log_{2} 2N+1!/N!)/N+1, about (2N+1.5)(log_{2} 2N+1.5/e) -
(N+1.5)(log_{2} N+1.5/e) / N+1, or roughly ~ (log_{2} N+1) + 2 -
(log_{2} e) - trim: thus, just under (log_{2} N+1) + 0.56 bit.

The average number of bits in representing the sum and difference, of two numbers
N≥n≥0, both together: equals roughly ~ (log_{2} N+1) + 0.56 +
(log_{2} N+1) - 1.44, and tiny trim: thus, just about 2(log_{2} N+1) -
0.88 bit (not picking over the duplication at n=0); however the sum-average excludes
one bit in the Haar algorithm (as above).

Thus, at the first level, compared to the average number of bits in representing two raw numbers N,n≥0, the Haar sum-average-and-difference construction takes a half bit more on the average;- but subsequent differences of sum-averages also need a sign bit, not offset by an exclusion, and filling half the whole,- whence the total average is just over one and a half bit more.

(This appears anomalous, as the total information has not changed.)

VIDEO EFFICIENCY: The average 1.56-bit loss in the Haar is regained by two components of the video: Digital imaging technology, of fixed pixels not aligned with viewed objects, puts any high contrast edge straddling pixels, and any single-pixel-width line or star, across 2-4 adjacent pixels, for which average pixel-straddling ranges between edgelike, and half-equal, which latter Haar represents in nearly half the bits (Fine line drawing at the Nyquist frequency resolution is attainable only for aligned edges, necessitating computer-graphics-images); also, high density stark contrast (sterling gray) is not the broad usual in day scenes: slow contrasts abound on object-faces: whence the Haar takes advantage of the abundant lower-frequency components in usually similar adjacent pixels.

However this would be equally or more true in the lowest level haardlets, too:

First-stage haardlets, are the same Haar sum-average-and-difference construction; At the next stage the haardlets extend sum-and-difference to the differences, requiring one plus the average half bit, more precision, whichever is larger, sum or difference holding the surplus bit of its sum as reconstructed from the difference.

Localized switching between haarlets and haardlets may further improve video compression: the possibly simplest switch may be a rectolinear array of regional flags, each region 4×4, 8×8, or 16×16 pixels (an implementation-practical optimization), each flag the switch for the region ... an array that itself is highly compressible as image objects span multiple adjacent regions. Implementation precomputes Haard, which includes the Haar, counting the total of coefficient bits needed, by region, and setting the local regional flag to Haar or Haard, to minimize the compressed size.

SWITCHABLE HAARDLETS: The concept is: The high order bit of a coefficient becomes a decision that the original -haarlet- pair bit-space was larger than if compressed with the haardlet-step with the decision bit included, -and whence the haardlet is taken and demarked (switched)-... which occurs at -1+√17/4 (~0.7807764~~25/32~) for one bit, but is considerable above the small-golden coefficient -1 + √5/2 (~0.6180340~) equator-crossing, where the Haardlet gained reputation.

Consider again the Haar and Haard transforms as top-down constructions: The Haar takes
a single pixel, average,- and differentiates,

s_{1}+ s_{2} > 0,s_{1},s_{2} or negatively or the difference instead of sum. It
would be always true except for the efficiency of coding-out zeros and ones only at high
density (at equal density, there is no coding efficiency).

~~
(stepped Hadamard Transform)
Hadamard, by definition, is a "squarewave" block transform, involving only additions and
subtractions (one-bit multiplications; much simpler than multiplication-intense Fourier,
the sinewave transform). Hadamard can be decimated conveniently rather than computing
full hadamard-waves each process cycle ... bottom up, sum-and-difference
of pairs correspondingly compounded,- up to its top coefficients:
~~

~~
An illustration: progressively on samples; pairs; and pairs-of-pairs:~~

- s
_{1}; s_{2}; s_{3}; s_{4}; ... etc. - s
_{1}+s_{2}, s_{1}-s_{2}; s_{3}+s_{4}, s_{3}-s_{4}; ... etc. - s
_{1}+s_{2}+s_{3}+s_{4}, s_{1}+s_{2}-s_{3}-s_{4}, s_{1}-s_{2}+s_{3}-s_{4}, s_{1}-s_{2}-s_{3}+s_{4}; ... etc. - ++++++++ ++++---- ++--++-- ++----++ +-+-+-+- +-+--+-+ +--++--+ +--+-++- ... (shorthand) ... etc. ...

Haar slightly favors half-alignment (lower spectral frequency), but significantly disfavors quarter-phase: eg.

raw data | 00000000 | 10000000 | 00000000 | 00000000 | 000° |

1st-haar | 01000000 | -10000000 | 00000000 | +00000000 | |

2nd-haar | 00100000 | -10000000 | +01000000 | +00000000 | |

haard | 00100000 | -011000000 | +01000000 | -010000000 |

raw data | 00000000 | 01100000 | 00100000 | 00000000 | 090° |

1st-haar | 00110000 | -01100000 | 00010000 | +00100000 | |

2nd-haar | 00100000 | -01100000 | +00100000 | +00100000 | |

haard | 00100000 | -001100000 | +00100000 | -010000000 |

raw data | 00000000 | 01000000 | 01000000 | 00000000 | 180° |

1st-haar | 00100000 | -01000000 | 00100000 | +01000000 | |

2nd-haar | 00100000 | -01000000 | +00000000 | +01000000 | |

haard | 00100000 | +000000000 | +00000000 | -010000000 |

raw data | 00000000 | 00100000 | 01100000 | 00000000 | 270° |

1st-haar | 00010000 | -00100000 | 00110000 | +01100000 | |

2nd-haar | 00100000 | -00100000 | -00100000 | +01100000 | |

haard | 00100000 | +001100000 | -00100000 | -010000000 |

__

A general note: Expanding an image beyond its pixel-per-pixel resolution, yields an apparent blocky-digital image, unless the smallest slopes are recalculated ... this may result in speckly-like performance at edges between slopes. (Gif had a similar concern, but could have been improved by smoothing at the single-quantum level: ie. a difference of one quantum between adjacent pixels, means specially, they are not, different by one but the same with a capture-dither that must be smoothed--- a renderer responsibility.)

__

Ideally, original images consist of single photon emitter atoms less than unit-rate each:

A fully parallel nanoprocessor would:

- put out corresponding pulses one-per-count
- correlate adjacent sums and differences
- along lines temporally developed from "learned" linear motions

- rechannel the same pulses (without in/decrease, just reroute) ...the total output count bandwidth equaling its total input count, rerouted

- individual discrete photons appear equally in all four coefficients
- this may be smoothed by temporal averaging gathering more photons

- in a continuum, adjacent values are equal, sums, slopes, twists
- finite adjacent values are near equal; and distant are disparate
- better-than-SPIHT-wavelets would adapt for local image business, crossing more smoothly between Haard and Haard-like wavelets
- this is comparable to the twinning of x- and y-slopes nearer ±45° which coefficient-transitioning would be better as polar aiming: higher-resolution aim than mere single-bit coefficient selection: consider the improvement in taking x,y-selects as one 3-bit polar {0 no-coefficient, 100 x, 101 x+y, 110 x-y, 111 y} in calculation; this may require recto-polar conversion, or a new entropizing: consider the (old) plotter-stepping algorithm:

REF:

* [SPIHT was original work by other authors, its trademark now-abandoned but
generally retains its acronymic meaning]

** [An example of a full-Hadamard system would be the late-1970's NASA Telecomm
bit-per-pixel B&W system designed and built by Linkabit Corporation]

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