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 ...
  • 5,865
8 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 ...
  • 42.6k
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{...
  • 13.9k
6 votes

What Are the Types of Deconvolution?

I would say you can classify using the following main properties: Blind Deconvolution. Non Blind Deconvolution. Then I'd follow: Linear Model. Time / Spatial Invariant Model. Time / Spatial ...
  • 42.6k
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 ...
  • 81k
6 votes
Accepted

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 ...
  • 13.9k
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 ...
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 ...
  • 23.1k
5 votes
Accepted

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

Indeed the model for the Proximal Gradient Method (Also see Proximal Gradient Methods for Learning) is in the form of: $$ F \left( x \right) = f \left( x \right) + g \left( x \right) $$ Where usually $...
  • 42.6k
4 votes
Accepted

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 ...
  • 34.1k
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 ...
  • 34.1k
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 ...
4 votes

Denoising a Grayscale Image Using Random Matrix Theory (RMT)

Usually the way this is done (Look for Low Rank models for Image Denoising) is something like: Decompose the image into d x d patches. Cluster patches which are ...
  • 42.6k
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 ...
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 ...
  • 34.1k
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 $...
  • 945
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.
  • 34.1k
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 ...
  • 927
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 ...
  • 5,255
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 ...
  • 81k
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{...
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 + ...
  • 6,911
2 votes
Accepted

optimization of Image Reconstruction Algorithm using Genetic Algorithm in Matlab

One simple approach would be taking the mean square error (MSE) by using fitness_1 = mean((inputimage(:) - reconstructedimage_1(:)).^2) though, as your image ...
  • 273
2 votes

What is done to minimize distortion due to the hold operation?

I agree with Jim Clay's answer, but I think it is important to point out two things. First of all, there are no phase distortions due to the hold operation, just a simple delay of half a sampling ...
  • 81k
2 votes

What is done to minimize distortion due to the hold operation?

What you are describing is the distortion introduced by an ideal digital-to-analog converter (DAC) in the analog domain. Two things are typically done to reduce this distortion: Analog filtering ...
  • 11.8k
2 votes
Accepted

Sample and reconstruct a real exponential (just one period)

First, since $t>0$ in your case, you can write your function as \begin{equation} C(t)=1.6925\left(\exp^{-0.136t}- \exp^{-1.192t} \right) u(t) \end{equation} where $u(t)$ is a unit-step function. ...
  • 320
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 ...
  • 81k
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 ...
  • 1,686
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. ...
  • 1,686

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