9 votes
Accepted

Shannon interpolation formula for downsampled data with an "almost ideal" low pass filter

I don't get your downsample step when you downsampled by factor $M$. Let me go from scratch with the spectrum visualization below, with time domain, continuous frequency domain and discrete frequency ...
AlexTP's user avatar
  • 6,595
7 votes
Accepted

Signal values we will 'miss' between sampling instances during sampling of band limited signals

I don't have a real answer but I have the feeling that this result will help you out: Bernstein's inequality says that, if the signal $x(t)$ is bandlimited to $|f|\leq B$, then $$\left| \frac{\textrm{...
MBaz's user avatar
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6 votes

Reconstructing a sine wave from an interval shorter than half its wavelength

If your signal is really as simple as $$x(t)=A\sin(\omega_0t)\tag{1}$$ with known $\omega_0$, and you have observations $y(t_i)$, which are noisy samples of $x(t)$ at known time instances $t_i$, then ...
Matt L.'s user avatar
  • 90k
6 votes
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Why aren't negative frequencies folded in reconstruction of the aliased signal?

They are doing the same for the negative frequencies, implicitly. In a problem like this, all signals are real: the input, the sampled signal, and the reconstructed signal. As a consequence, all ...
MBaz's user avatar
  • 15.3k
5 votes

Signal values we will 'miss' between sampling instances during sampling of band limited signals

Observations I have used +1 and -1 in the sequence instead of your 1 and 0. With $\alpha=1$, the band-limited continuous function $f_m(T)$ in your first two figures (with the above mentioned ...
Olli Niemitalo's user avatar
5 votes

Random sampling vs uniform sampling

The key idea is that the random sampling approach enforces more constraints on the resulting signal than the uniform sampling approach does. The POCS (projections onto convex sets) algorithm used for ...
Peter K.'s user avatar
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4 votes
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Method of reconstructing a band-limited signal from discrete samples

Some recent cell phone models use something like a Cirrus Logic CS42xx series audio IO chip, which seems to use a digital polyphase interpolation filter, a sigma delta modulator, followed by a ...
hotpaw2's user avatar
  • 35.3k
4 votes

Proving Nyquist Sampling Theorem for Strictly Band Limited Signals (Whittaker Shannon Interpolation Formula)

Approaching The Sampling Theorem as Inner Product Space Preface There are many ways to derive the Nyquist Shannon Sampling Theorem with the constraint on the sampling frequency being 2 times the ...
Royi's user avatar
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4 votes

reconstruction filter - How does it actually work?

The sampling theorem requires a perfectly bandlimited signal, bandlimited to below twice the sampling frequency. The problem with this is that only an infinite length signal (e.g. exists before the ...
hotpaw2's user avatar
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4 votes
Accepted

Types of interpolation used for reconstruction in DSP?

Zero-order hold will result in a piecewise-constant waveform. Linear interpolation will result in a piecewise-linear waveform. If you want a piecewise-quadratic or piecewise-cubic or higher order ...
robert bristow-johnson's user avatar
4 votes
Accepted

Room Impulse Response Domain of Sparsity

That's tricky. RIRs are NOT sparse in any obvious physical sense (time, frequency, etc). In fact they are insanely complicated with thousands of degrees of freedom. The amount of relevant physical ...
Hilmar's user avatar
  • 44.6k
3 votes

What is the difference between image restoration and image reconstruction?

The introduction of this paper explains the difference and gives an example. In short: Image restoration techniques presume that data are acquired in the image space; that is, the raw data ...
anpar's user avatar
  • 957
3 votes

What algorithms can automatically determine a 3D scene from one or a few 2D images?

SLAM(Simultaneous Localization and Mapping) algorithms can be used to for 3D reconstruction. They offer solutions for both monocular as well as stereo cameras. With single camera they estimate depth ...
Navin Prashath's user avatar
3 votes

Does Zero Padding Work as Advertised?

Zero-padding data for a longer FFT is equivalent to interpolation by a (periodic) Sinc kernel. Interpolation by a (periodic) Sinc kernel can reconstruct points between samples of a signal that was ...
hotpaw2's user avatar
  • 35.3k
3 votes
Accepted

image type after an ifft reconstruction

FFT and IFFT are linear operators, and as such, the results only make a lot of sense in a linear intensity space, not if indexed into a non-linearly mapped space.
hotpaw2's user avatar
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3 votes

Method of reconstructing a band-limited signal from discrete samples

The general mathematical framework for interpolation is approximation theory. I guess the most important result is that for signals with bandwidth limitation, you can have perfect reconstruction via $...
Arnfinn's user avatar
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3 votes
Accepted

Multiple-image dense point cloud reconstruction with camera extrinsics/intrinsics

First, a warm welcome to SE! Basically, you have a calibrated 3D reconstruction problem. The typical approach follows a 5-stage pipeline: Identify 2D features in each image along with the associated ...
Tolga Birdal's user avatar
  • 5,465
3 votes
Accepted

What is Finite Rate of Innovation Signal?

If a signal can be exactly represented by $N$ real numbers per time interval, then its number of degrees of freedom for that time interval equals $N$. The most well-known example are band-limited ...
Matt L.'s user avatar
  • 90k
3 votes

Proximal Gradient Method (PGM) for a Function Model with More than 2 Functions (Sum of Functions)

Our goal is to obtain proximal operator of the following function $$ g \left( x \right) = {\left\| x \right\|}_{1} + \operatorname{TV}(x). $$ The involved optimization problem for any $z \in \mathbb{...
Mahesh Chandra Mukkamala's user avatar
3 votes

Reconstructing a sine wave from an interval shorter than half its wavelength

Build a basis set with your frequency and match your signal. It is straightforward linear algebra: $C$ is portion of cosine $S$ is portion of the sine $U$ is a vector of ones (DC) $$ X = a C + b S + ...
Cedron Dawg's user avatar
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3 votes
Accepted

Best parameter to estimate image reconstruction quality?

Since you say the π symbol will always be in the same place, you don't need to detect/locate it. You can compare per-pixel. You could calculate a correlation score against a model image. Since the ...
Christoph Rackwitz's user avatar
3 votes

Sampling frequencies calculated on paper and in MATLAB not matching

On paper, I've calculated a sampling frequency of at least 2 cycles/sec, since the frequency of my signal is bound between 0 and 1 cycles/sec. How did you figure out that one ? The Taylor expansion ...
Hilmar's user avatar
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2 votes
Accepted

Question about MRI signal construction

I would expect that you simply cut the outer parts of the image away. Certainly, there is always a hassle to figure out the proper pixel in the center, but that is a detail of the FT-algorithm ...
M529's user avatar
  • 1,736
2 votes
Accepted

MRI reconstruction using windowing based apodization

You say: I have a 128 point one dimensional k-space samples... The hanning window is the same size as the k-space vector (256)... Make sure that you have the appropriate sizes in your algorithm. ...
M529's user avatar
  • 1,736
2 votes
Accepted

How can a signal have two maximum frequency components?

$x(t)$ must be a band pass signal. Under certain conditions on the sampling frequency and its relation to the lower and upper band edges of the signal, $x(t)$ can be sampled at a frequency that is ...
Matt L.'s user avatar
  • 90k
2 votes
Accepted

Simulating noise in computed tomography reconstruction

The easiest, and probably most straight forward way, seems to add the noise in the measurement domain, hence the sinogram.
M529's user avatar
  • 1,736
2 votes

What algorithms can automatically determine a 3D scene from one or a few 2D images?

Let us first assume you can produce estimates of the camera state (position and attitude) via sensors, a filter like a Kalman Filter, and a (simple) model for the camera itself. Using this information,...
spektr's user avatar
  • 263
2 votes
Accepted

What is the general formula for radon back projection for a javascript implementation?

To implement projection the simplest way is to rotate your image then sum over a row or column. The simplest way to implement back projection is to take a line of your sinogram which is a projection ...
Mohammad M's user avatar
  • 1,327
2 votes
Accepted

Reconstructing Signal From Its Cyclic Autocorrelation

No, you cannot reconstruct the original signal from the cyclic autocorrelation. The fundamental reason is that it results from an averaging operation. Like the autocorrelation and PSD, the cyclic ...
Chad Spooner's user avatar

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