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,
cf 179/255, and fine-detail washout.
| -(improving on SPIHT-Haar)- we note pixel-quad x-and y-tilt tend together, resolution is order-dependent, unsent pixel- bits makes lossy-video, 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—
The 'ideal' bandwidth-constrainted video front-end would have stacked-3-color pixels atop instantaneous-sum-and-difference transform-processing and successive-approximation (top-down) bit-slice compression-transmission, so that-- picture-motion itself would be realtime, with lossless definition....
Various approximations may suffice technological applications by quad-adjacent RGBG-pixels (or RGBY), residue-retention at the compression-transmission level, 3D-and-motion-estimation at the picture-level (top-pixel-level), pixel-compression by subdivision-partitioning (rather than omnidirectional), and, adjacent-value-prediction, etc....
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-width-somewhat-thicker-than-single-width pixels.
For cgi computer-generated-imagery, a different criterion improves HDDV by taking vantage of "digital smoothing,"- a specialized concept better equalizing HDDV to 35mm, its touted film-equivalent, by spreading each single pixel onto an adjacent pixel, to a "digital quarter step"; The base value of this method is, that the smoothest-moving line of constant width by adjacent-pixel amplitudes (1,x) and (x,1), is about x ~ 0.60 (*), cf 153/255 ... Its successive offset half steps appear equal, Its apparent line thickness is roughly a sesquipixel, a half more than single-pixel, but half-as jumpy or discontinuous "half-moon-jogging," and still contains a hint of fine-resolution and-motion, and not as smeared -nor 'breathing'- as alternately straddling pixels which occur as the extreme in general pixel sampling ...
* (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,
cf 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.)
[2012 Note: On newer flat-panel screens, 136/255 sesquipixel, 162/255 pixelwash, looks almost smoothest: gamma may be 1.0]
Webpage-image-generation software then-needs support this smoothing by avoiding representing sharp edges as single-quantum steps.
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).
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 ...
* (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 (e.g. 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, e.g. 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, e.g. 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):
|
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(log2 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,--
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.
Full SPIHT-video, is additionally:
* (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,
e.g. 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
s1,s2, 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, (log2 n+1).
The average number of bits representing a range of numbers [0,N],
N≥n≥0, is ∑(log2 n+1)/N+1 = (log2 N+1!)/N+1.
Approximating n! ~ (n+.5/e)(n+.5)√2π,
this is about (N+1.5)(log2 N+1.5/e) + (log2√2π) / N+1,
or roughly ~ (log2 N+1) - (log2 e) + trim, that cannot exceed
that for its own largest value: thus, just over (log2 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
(log2 2N+1!/N!)/N+1, about (2N+1.5)(log2 2N+1.5/e) -
(N+1.5)(log2 N+1.5/e) / N+1, or roughly ~ (log2 N+1) + 2 -
(log2 e) - trim: thus, just under (log2 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 ~ (log2 N+1) + 0.56 +
(log2 N+1) - 1.44, and tiny trim: thus, just about 2(log2 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,
s1+ s2 > 0,s1,s2 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).
An illustration: progressively on samples; pairs; and pairs-of-pairs:
Haar slightly favors half-alignment (lower spectral frequency),
but significantly disfavors quarter-phase: e.g.
__
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. (Also had a similar concern
that could have been improved by smoothing at the single-quantum level: i.e. 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:
REF:--§ THEORY SPLIT §--
A "Theory Split" occurs here as it is now useful to separate two tracks of development
similar but distinct for efficiency; It means the metatheory or principle concept,
implementable efficiencies, are benefitted by choosing narrower discussion of general,
theory-elements (a properties-selection, a "design tuning;" cf a theory split occurred
in long-division on-the-left vs. on-the-right for prime factor
checking). We begin by cross-referring, till the discussion track gets dense:
REFERENCE: (differlets and data reduction; aka steplets, wavelets, sloplets, tiltlets)
Basically SPIHT [*] Set Partitioning Into Hierarchical Trees, is an efficient coding
routine of bitslices of list-sorted coefficients designed for signal-compression of
transformed data having a preponderance of near-zero coefficients ... however, SPIHT
video, is usually defined for the Haar Transform as that is facile in computational
encoding and decoding, essentially lossless and compatible with low-loss modes
(Low-loss provides efficient results suitable for an HDTV option). Haar gained fame
in astronomical telescopy imaging having sparse and pointillated singular stars
straddling four pixels at practical optical resolution (except planets, novae,
nebulae, galaxies; computer-generated-images).
THE HAARD TRANSFORM: (Haar-D, Ha'ard, Haar'd, Haar-D-Haar)
Coining the name to indicate the finish to the Haar (an extra Difference) and the
removal of the separation (dam) in the Ha(dam)ard ... Haard slopes are subtracted
down one level: the slope of the averages from the average of the slopes.
The computational cost is ≈3n, barely more than Haar ≈2n; and makes
possible reduction of slope-residuals to zero on long gradual slopes where slope
information is most redundant, --though average-slope is less widely effectual than
pixel-average because slope reaches upper and lower crop limits.
Also Haard directly defines line-doubling as an intermediate process during
successive differentiation and top-down reconstruction, smoothing the large
pixel-averages digitized stairsteps, before adding residuals.
--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 SPIHT-HAAR:
The essence of basic SPIHT, is:
* (Above a median total resolution, the optical resolution efficiency of each pixel
decreases about a bit-per; which is also true at about the DC-level, the total
brightness, is not very significant)
SPIHT-HAAR EFFICIENCIES:
Though SPIHT is efficient, its bandwidth depends on brightness, which for video is of
no consequence as the video channel is designed by regulatory convention
(Internet-computer imaging does take advantage of minimized transmission and storage);--
or if the camera has further luminance detail, SPIHT lossless-successive approximation
can bring it up. Alternatively, video contrast in dimmer areas can be raised by
preemphasis, e.g. taking the logarithm of pixel amplitude (except black-zero defaults),
and transforming that ... in fact, logarithm is somewhat more appropriate than straight,
because it represents the actual reflected or transmitted optical shade-proportionality
under any lighting: whence no detail is lost at lower lighting within the ability of the
camera: an object at hundredth lighting is coded nearly as efficiently as full lighting;
brightness can be turned-up at the receiver with no apparent fading (cf night-vision)
... the disadvantages are: less stability of slant-linearity along edges with coarser
quantization at brighter amplitudes; weaker compression at dimmer, as almost all values
have a higher bit turned on, indicating amplitude "range", and, the algorithm complexity
to ensure dimmer samples are not overly resolved; and loss of linearity approximation on
large-excursion slopes due to light-source size, e.g. sun, whose slopes are very linear,
less accurately compressed by the logarithmic "gamma" response ... rather, the logarithm
works better for the time-domain, as source-lights for a given scene vary over time, and
less for the spatial domain. (*)
NOW THE FUN:
Suppose we allow haardlets to build representational precision up to all available,
before switching to the Haar-mode knee (particularly improving larger smooth regions)
... then at the top, its bit is always "1" and we do not store it nor send it ... but
the indicator, which also contains some information (it gives us a choice between two
transforms, thus maximizing compression) ... and this may outdo SPIHT ... this ballooned
SPIHT may be the ultimate of all compressions: it locates and packs the coefficients.
(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:
In essence this computes the total (2n× average) sum, left-right drop, and the partwise
left-right drops,... the total average slope,-- thence the changes in slopes. (Inline
reordering is simple, too, to group temporal-frequencies in better order.) It is very
edge- and slope-detective-- even advantageously better than the Fourier.
and not easy to calculate; its major is that it squeezes to a difference sense of optimized
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
The sum-average and the difference-of-differences are constant in this high-contrast
example.
raw data 00000000 00100000 01100000 00000000 270° 1st-haar 00010000 -00100000 00110000 +01100000 2nd-haar 00100000 -00100000 -00100000 +01100000 haard 00100000 +001100000 -00100000 -010000000
Nanoscopic SPIHT:along lines temporally developed from "learned" linear motions
It is of interest to note that the base representation of a count of photons, is itself a
first-level signal reduction, essentially runlength: the total number of values possible
then fits into the arithmetic coding, which I showed was efficiently coded by entropic
choreonumeration,- which can be as arithmetic or not, as
needed. SPIHT typically uses a Huffman coding.
this may be smoothed by temporal averaging gathering more photons
* [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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