Reconstruction filter
Encyclopedia
In a mixed-signal system (analog
and digital
), a reconstruction filter (or anti-imaging filter) is used to construct a smooth analogue signal from a digital input, as in the case of a digital to analogue converter (DAC
) or other sampled data output device.
requires a low-pass analog electronic filter
, called the anti-aliasing filter
: the sampled input signal must be bandlimited
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, as the output signal must be bandlimited, to prevent aliasing (here meaning Fourier coefficients being reconstructed as low frequency waves, not as higher frequency aliases), as in 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
. This is given by a filter with a 'sinc' impulse response.
In systems that have both, the anti-aliasing filter
and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment is sampled at 44.1 kHz. Both audio filter
s block as much as possible above 22 kHz and pass as much as possible below 20 kHz. Typically both filters are active op-amp filters, with exactly the same selection of resistors and capacitors.
Whilst in theory a DAC gives a series of impulse
s, in practice, the output of a DAC is more typically a series of stair-steps – thus the step response
of the filter (the integral of the impulse response) is of more interest. The low pass
reconstruction filter smooths the stair step (removes the harmonics above the Nyquist limit
) to (re)construct the analogue signal corresponding to the digital time sequence.
, digital reconstruction filters are used both to recreate images from samples as in medical imaging
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
, 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 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.
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 for various widths, or cardinal B-spline
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
, 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, though for higher resampling ratios behavior is more varied.
coefficients.
In medical imaging
, a common technique is to use a number of 2D X-ray
photos or MRI scans to "reconstruct" a 3D image.
Analog signal
An analog or analogue signal is any continuous signal for which the time varying feature of the signal is a representation of some other time varying quantity, i.e., analogous to another time varying signal. It differs from a digital signal in terms of small fluctuations in the signal which are...
and digital
Digital signal
A digital signal is a physical signal that is a representation of a sequence of discrete values , for example of an arbitrary bit stream, or of a digitized analog signal...
), a reconstruction filter (or anti-imaging filter) is used to construct a smooth analogue signal from a digital input, as in the case of a digital to analogue converter (DAC
Digital-to-analog converter
In electronics, a digital-to-analog converter is a device that converts a digital code to an analog signal . An analog-to-digital converter performs the reverse operation...
) or other sampled data output device.
Sampled data reconstruction filters
The sampling theorem describes why the input of an ADCAnalog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...
requires a low-pass analog electronic filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
, called the anti-aliasing filter
Anti-aliasing filter
An anti-aliasing filter is a filter used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem....
: the sampled input signal must be bandlimited
Bandlimited
Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...
to prevent aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
(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, as the output signal must be bandlimited, to prevent aliasing (here meaning Fourier coefficients being reconstructed as low frequency waves, not as higher frequency aliases), as in the Whittaker–Shannon interpolation formula
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...
.
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
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...
. This is given by a filter with a 'sinc' impulse response.
Implementation
Practical filters have non-flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere, as a 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 – it would require infinite delay.In systems that have both, the anti-aliasing filter
Anti-aliasing filter
An anti-aliasing filter is a filter used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem....
and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment is sampled at 44.1 kHz. Both audio filter
Audio filter
An audio filter is a frequency dependent amplifier circuit, working in the audio frequency range, 0 Hz to beyond 20 kHz. Many types of filters exist for applications including graphic equalizers, synthesizers, sound effects, CD players and virtual reality systems.Being a frequency dependent...
s block as much as possible above 22 kHz and pass as much as possible below 20 kHz. Typically both filters are active op-amp filters, with exactly the same selection of resistors and capacitors.
Whilst in theory a DAC gives a series of impulse
Impulse response
In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...
s, in practice, the output of a DAC is more typically a series of stair-steps – thus the step response
Step response
The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...
of the filter (the integral of the impulse response) is of more interest. The low pass
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
reconstruction filter smooths the stair step (removes the harmonics above the Nyquist limit
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...
) to (re)construct the analogue signal corresponding to the digital time sequence.
Image processing
In image processingImage processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
, digital reconstruction filters are used both to recreate images from samples as in medical imaging
Medical imaging
Medical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
and for resampling
Resampling
Resampling may refer to:* Resampling , several related audio processes* Resampling , resampling methods in statistics* Resampling , scaling of bitmap images* Sample rate conversion-See also:* Downsampling* Upsampling...
.
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
Decimation (signal processing)
In digital signal processing, decimation is a technique for reducing the number of samples in a discrete-time signal. The element which implements this technique is referred to as a decimator.Decimation is a two-step process:...
or interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
, 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
Gamma correction
Gamma correction, gamma nonlinearity, gamma encoding, or often simply gamma, is the name of a nonlinear operation used to code and decode luminance or tristimulus values in video or still image systems...
, notably in the sRGB 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, with kernel the box filter – for downsampling, this corresponding to averaging;
- bilinear interpolationBilinear interpolationIn mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a regular grid. The interpolated function should not use the term of x^2 or y^2, but x y, which is the bilinear form of x and y.The key idea is to perform linear...
, with kernel the tent filter; - bicubic interpolationBicubic interpolationIn mathematics, bicubic interpolation is an extension of cubic interpolation for interpolating data points on a two dimensional regular grid. The interpolated surface is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation...
, 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 for various widths, or cardinal B-spline
B-spline
In the mathematical subfield of numerical analysis, a B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. B-splines were investigated as early as the nineteenth century by Nikolai Lobachevsky...
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
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 "echos" near transients, particularly sounds from...
, 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
Roll-off
Roll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband...
("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, 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 waveletWavelet
A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have...
coefficients.
In medical imaging
Medical imaging
Medical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
, a common technique is to use a number of 2D X-ray
X-ray
X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...
photos or MRI scans to "reconstruct" a 3D image.
- Reconstruction algorithmReconstruction algorithmIn tomography, a variety of practical reconstruction algorithms have been developed to implement the process of reconstruction of a 3-dimensional object from its projections...
- Iterative reconstructionIterative reconstructionIterative 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...