In a mixed-signal system (
analog and
digital
Digital usually refers to something using discrete digits, often binary digits.
Technology and computing Hardware
*Digital electronics, electronic circuits which operate using digital signals
** Digital camera, which captures and stores digital ...
), a reconstruction filter, sometimes called an anti-imaging filter, is used to construct a smooth analog signal from a digital input, as in the case of a digital to analog converter (
DAC) or other sampled data output device.
Sampled data reconstruction filters
The
sampling theorem
Sampling may refer to:
* Sampling (signal processing), converting a continuous signal into a discrete signal
* Sampling (graphics), converting continuous colors into discrete color components
* Sampling (music), the reuse of a sound recording in a ...
describes why the input of an
ADC requires a low-pass analog
electronic filter
Electronic filters are a type of signal processing filter in the form of electrical circuits. This article covers those filters consisting of lumped electronic components, as opposed to distributed-element filters. That is, using components ...
, called the
anti-aliasing filter
An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruct ...
: the sampled ''input'' signal must be
bandlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandli ...
to prevent
aliasing (here meaning waves of higher frequency being ''recorded'' as a lower frequency).
For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter - because the ''output'' signal must be bandlimited, to prevent imaging (meaning Fourier coefficients being reconstructed as spurious high-frequency 'mirrors'). This is an implementation of the
Whittaker–Shannon interpolation formula.
Ideally, both filters should be
brickwall filters, constant phase delay in the pass-band with constant flat frequency response, and zero response from the
Nyquist frequency
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), it ...
. This can be achieved by a filter with a '
sinc' impulse response.
Implementation
While in theory a DAC outputs a series of discrete
Dirac impulses, in practice, a real DAC outputs pulses with finite bandwidth and width. Both idealized Dirac pulses,
zero-order held steps and other output pulses, if unfiltered, would contain spurious high-frequency replicas, "''or images''" of the original bandlimited signal. Thus, the reconstruction filter smooths the waveform to remove
image frequencies (copies) above the
Nyquist limit
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), i ...
. In doing so, it reconstructs the continuous time signal (whether originally sampled, or modelled by digital logic) corresponding to the digital time sequence.
Practical filters have non-flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere. The ideal
sinc waveform has an infinite response to a signal, in both the positive and negative time directions, which is impossible to perform in real time – as it would require infinite delay. Consequently, real reconstruction filters typically either allow some energy above the Nyquist rate, attenuate some in-band frequencies, or both. For this reason,
oversampling may be used to ensure that frequencies of interest are accurately reproduced without excess energy being emitted out of band.
In systems that have both, the
anti-aliasing filter
An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruct ...
and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment may be sampled at 44.1 kHz. In this case, both
audio filter
An audio filter is a frequency dependent circuit, working in the audio frequency range, 0 Hz to 20 kHz. Audio filters can amplify (boost), pass or attenuate (cut) some frequency ranges. Many types of filters exist for different audio a ...
s block as much as possible above 22 kHz and pass as much as possible below 20 kHz.
Alternatively, a system may have no reconstruction filter and simply tolerate some energy being wasted reproducing higher frequency images of the primary signal spectrum.
Image processing
In
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
, digital reconstruction filters are used both to recreate images from samples as in
medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to re ...
Project webpage
and for
resampling.
A number of comparisons have been made, by various criteria;
one observation is that reconstruction can be improved if the ''derivative'' of the signal is also known, in addition to the amplitude,
and conversely that also performing derivative reconstruction can improve signal reconstruction methods.
Resampling may be referred to as
decimation or
interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has ...
, accordingly as the sampling rate decreases or increases – as in sampling and reconstruction generally, the same criteria generally apply in both cases, and thus the same filter can be used.
For resampling, in principle the analog image is reconstructed, then sampled, and this is necessary for general changes in resolution. For integer ratios of sampling rate, one may simplify by sampling the impulse response of the continuous reconstruction filter to produce a discrete resampling filter, then using the discrete resampling filter to directly resample the image. For decimation by an integer amount, only a single sampled filter is necessary; for interpolation by an integer amount, different samplings are needed for different phases – for instance, if one is upsampling by a factor of 4, then one sampled filter is used for the half-way point, while a different sampled filter is used for the point 1/4 of the way from one point to another.
A subtlety in image processing is that (linear) signal processing assumes linear luminance – that doubling a pixel value doubles the luminance of the output. However, images are frequently
gamma encoded, notably in the
sRGB
sRGB is a standard RGB (red, green, blue) color space that HP and Microsoft created cooperatively in 1996 to use on monitors, printers, and the World Wide Web. It was subsequently standardized by the International Electrotechnical Commission ...
color space, so luminance is not linear.
Thus to apply a linear filter, one must first gamma decode the values – and if resampling, one must gamma decode, resample, then gamma encode.
Common filters
The most common day-to-day filters are:
*
nearest-neighbor interpolation
Nearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions.
Interpolation is the problem of approximating the value of ...
, with kernel the box filter – for downsampling, this corresponding to averaging;
*
bilinear interpolation, with kernel the tent filter;
*
bicubic interpolation
In mathematics, bicubic interpolation is an extension of cubic interpolation (not to be confused with cubic spline interpolation, a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regula ...
, with kernel a
cubic spline – this latter has a free parameter, with each value of the parameter yielding a different interpolation filter.
These are in increasing order of stopband suppression (anti-aliasing), and decreasing speed
For reconstruction purposes, a variety of kernels are used, many of which can be interpreted as approximating the sinc function,
either by windowing or by giving a spline approximation, either by cubics or higher order splines. In the case of windowed sinc filters, the frequency response of the reconstruction filter can be understood in terms of the frequency response of the window, as the frequency response of a windowed filter is the convolution of the original response (for sinc, a brick-wall) with the frequency response of the window. Among these, the
Lanczos window and
Kaiser window are frequently praised.
Another class of reconstruction filters include the
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
for various widths,
or cardinal
B-spline
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expresse ...
s of higher order – the box filter and tent filter being the 0th and 1st order cardinal B-splines. These filters fail to be interpolating filters, since their impulse response do not vanish at all non-zero original sample points – for 1:1 resampling, they are not the identity, but rather blur. On the other hand, being nonnegative, they do not introduce any overshoot or
ringing artifacts
In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "e ...
, and by being wider in the time domain they can be narrower in the frequency domain (by the
Fourier uncertainty principle), though at the cost of blurring, which is reflected in passband
roll-off ("scalloping").
In photography, a great variety of interpolation filters exist, some proprietary, for which opinions are mixed. Evaluation is often subjective, with reactions being varied, and some arguing that at realistic resampling ratios, there is little difference between them, as compared with bicubic,
Interpolation -- Part I
Ron Bigelow though for higher resampling ratios behavior is more varied.
Wavelet reconstruction filters
Reconstruction filters are also used when "reconstructing" a waveform or an image from a collection of wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
coefficients.
In medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to re ...
, a common technique is to use a number of 2D X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30&nb ...
photos or MRI scans to "reconstruct" a 3D image.
* Reconstruction algorithm
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johan ...
* Iterative reconstruction
Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques.
For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative recon ...
See also
* Signal reconstruction
In signal processing, reconstruction usually means the determination of an original continuous signal from a sequence of equally spaced samples.
This article takes a generalized abstract mathematical approach to signal sampling and reconstructi ...
* Signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
References
{{reflist
Digital signal processing
Linear filters
Electronic filter applications