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 ...
- 6,080
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 $$\left| \frac{\textrm{...
- 14.9k
6
votes
Reconstructing a sine wave from an interval shorter than half its wavelength
If your signal is really as simple as
$$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 ...
- 88.9k
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 ...
- 14.9k
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 ...
- 13.5k
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 ...
- 25.2k
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 ...
- 35.2k
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 ...
- 19.3k
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 ...
- 35.2k
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 ...
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 ...
- 947
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 ...
- 700
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 ...
- 35.2k
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.
- 35.2k
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 $...
- 1,035
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 ...
- 5,465
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 ...
- 88.9k
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 + ...
- 7,520
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 ...
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 ...
- 42.5k
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 ...
- 1,736
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.
...
- 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 ...
- 88.9k
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.
- 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,...
- 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 ...
- 1,328
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 ...
- 106
2
votes
Accepted
What is the difference between image restoration and image reconstruction?
The use might depend on authors. From one of my colleagues, Jean-Christophe Pesquet, supported by the book Image Reconstruction: Algorithms and Analysis by Fessler, especially Chapter 1: Image ...
- 31.7k
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