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Questions tagged [optimization]

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2
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3answers
4k views

Librosa stft + istft - Understanding my output (which always seems too perfect) at varying window lengths

I've just started to use Python with Librosa for a DSP project I'll be working on. First thing I've been trying to do is determine my preferred parameters for the FFT window size, and hop-distance. ...
2
votes
1answer
445 views

What Is the Difference between RLS, LMS and Wiener Filter? When Is One Preferred Over Another?

I'm dealing with a channel equalization problem where the channel is modeled as a WSS process. I understand LMS utilises a Wiener-like approach, ie it converges to the optimal (wiener) solution. I ...
0
votes
1answer
62 views

Efficient correlation of a low duty cycle training sequence

Is there a way to efficiently correlate a training sequence that is N samples long, framed at M samples where M >> N, with L occurrences of such frames (see below). For the pedants, the training ...
0
votes
0answers
44 views

On the transmit weight vector in MIMO-MRC systems

Recently, I am reading paper [1]. In this paper, the author wrote: In MIMO-MRC systems in the absence of interferers, the signal vector at the received $n_R$ antennas is given by $$\mathbf{r} = \...
1
vote
1answer
31 views

How to Formulate a Constraint Which Ensures All Variables Have the Same Sign

I'm trying to include a constraint in my problem (to be solved by any convex optimization solver). Let {a,b,c,d ...} be a finite set of continuous variables. How to formulate a constraint which ensure ...
0
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1answer
74 views

Constrained LASSO Problem - $ {L}_{1} $ Regularized Least Squares with Linear Equality Constraints

I have an optimization question. I want to solve the following problem: $$ \arg\min_S\frac{1}{2}\|s-c\|_2^2 +\lambda\|\Phi s\|_1 \mbox{ s.t. } As = 0 $$ in which $\Phi$ is the wavelet transform ...
-1
votes
1answer
120 views

Properties of Optimization Techniques in Filter Design

I have this design: Can you tell me about the type of filter and the algorithm just viewing this design, or should there be other information?
0
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2answers
64 views

Subgradient Method for K-Means Like Problem

I want to solve k-means-like problem. But at first, I wonder that we can apply subgradient method to k-means problem. We want to find the optimal clustering $S = (S_1,S_2,...,S_k)$ such that $$ {\...
1
vote
1answer
96 views

Sequential Non Linear Least Squares Problem

I have the the following non-linear function, $$s(x;A_k,\mu_k,\sigma_k)=\sum_{k=1}^2 A_k \exp\left(\frac{-(x-\mu_k)^2}{\sigma_k^2}\right)$$ with unknown (but deterministic) parameters $A_k,\mu_k,\...
0
votes
0answers
19 views

Linear equation set construction for Brox et al. optical flow optimization

I'm trying to implement an optical flow calculation program for two successive images based on this article by Brox et al. The Euler-Lagrange combined with a fixed point iteration loop yields two ...
0
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0answers
20 views

Bundle adjustment optimization parameters

While reading the Wikipedia article on Bundle adjustment, I came across the following objective function used to represent the bundle adjustment problem. I have two questions regarding this objective ...
0
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1answer
46 views

Wireless Body Area Networks with Minimum Energy Consumption [closed]

For adaptive compressive sensing(cs),the sensing matrix is related to the input signal. For example, in rakeness-based(cs), the sensing matrix is obtained by solving an optimization problem which ...
3
votes
1answer
292 views

Ideal Geometric Arrangement of Microphone Array

I'm going to have to give a bit of context for this question to make sense. I am working on a project which includes audio source localisation in 3-D space through TDoA (Time Difference of Arrival - ...
1
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0answers
29 views

Optimization Problem Involving Frobenious Norm, Linear Equality Constraints and Semi Orthogonal Matrix

I have a question, Suppose $ A, B, D \in \mathbb{R}^{n \times p} $, I want to solve the following optimization problem: $\arg\min_D\frac{1}{2}\|D + A\|_F^2 + \lambda\|BB^T + DB^T + BD^T\|_1 \mbox{ s....
0
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0answers
35 views

How to normalise data for iterative computation of extrinsic camera matrix?

Hartley & Zisserman defend that data normalisation is an essential step prior to the estimation of geometric image transformations. The reasoning behind this, as stated in their book [1, Section 4....
0
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1answer
66 views

Promote the Orthogonality between Rows of $ S $

I have a question. Suppose we want to solve an optimization problem: Consider $S \in \mathbb{R}^{N \times T}, T >> S$ $$\min_{S} f(S) \mbox{ s.t. } SS^T \mbox{is diagonal}$$ Which means each ...
2
votes
1answer
286 views

How to Use the DFT (FFT) to Solve a Least Squares Regularization Problem (Inverse Problem)?

Let $X$ and $K$ be an image and a Point Spread Function (PSF), respectively. The blur image $B$ is obtained as follows $$B = X * K$$ I want to solve the following general regularization problem $$...
2
votes
2answers
228 views

Best Metric to Compare Sparsity of Vectors

I solved the Basis Pursuit Denoising Problem looking for a sparse solution (I am in compressive sensing): $$ x^* = \text{arg min}_x \left\{\frac{1}{2} \lVert Ax-y\rVert_2^2 + \lambda \lVert x\rVert_1\...
0
votes
1answer
169 views

How to Smooth Gradient Estimates for Steepest Descent Optimization

In steepest descent methods of minimizing a function $f(x), x \in \mathbb{R}^d$, it's common to approximate the gradient by finite differences: $\qquad\qquad \nabla f(x) \approx gradest( x; h ) \...
0
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0answers
40 views

Optimal sensor placement for 3D TDoA positioning

Suppose there is a rectangle indoor area, we want to locate different positions within this area using TDoA estimations. 5 sensors are placed to obtain optimal 3D positions with TDoA errors, we only ...
1
vote
2answers
79 views

How to efficiently control an FIR's magnitude response by altering its phase spectrum

Question: Extensively searching the space of all possible vectors of length $n$ to satisfy a (non-overdetermined) requirement is possible in principle. Hence, there is a way to calculate a complex ...
0
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2answers
48 views

How do you properly organize data to compute multiple (independent) recursive filters at the same time taking advantage of SIMD instructions?

I'm processing multiple (independent) Exponential Moving Average 1-Pole filters on different parameters I have within my Audio application, with the intent of smooth each param value at audio rate: <...
0
votes
1answer
91 views

IMU speed tracking through known path

new to signal processing and KFiltering here. Thanks for your help. I working with an IMU for a tracking project where the IMU moves throw a known path but at an unknown speed (within limits), the ...
13
votes
5answers
19k views

Compressive Sensing Through MATLAB Codes

I am new to the topic of compressed sensing. I read a few papers about it by R.Baranuik, Y.Eldar, Terence Tao etc. All these papers basically provide the mathematical details behind it, i.e., Sparsity,...
0
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1answer
43 views

Control optimization problem

I am running into a problem where I have a control system $S[t]$ that takes a control $C[t]$, so that $$S[t+1] = H(C[t,t-1,...], S[t,t-1,...])$$ the response of the system is the history of controls ...
3
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0answers
89 views

How to implement the RLS for matrices

I need to implement the RLS algorithm but it's for matrices instead for vectors, I have made the below code, but still something wrong is not working well, EDIT: The code should be done as below, ...
1
vote
0answers
19 views

Improve NMF for data with partial overlaps in multiple groups?

I want to use NMF to separate true sources from data. My data is in group structure with overlap elements. For example (in the smaller version) group1: contains A,B,C,D,E,F,G patterns group2: ...
3
votes
1answer
74 views

How Is Mixed Norm ($ {L}_{1, 2 }$) Better than $ {L}_{1} $ Norm for Sparse Representation?

Using $ {l}_{1} $-norm regularization for the purpose of achieving sparsity of the solution has been successfully applied a lot. But many times, I found the paper using mixed-norm instead of $l_1$-...
1
vote
1answer
116 views

Automatic Image Enhancement of Images of Scanned Documents (Auto Whitening)

Dropbox have make a blog post about there automatic enhancement method for scanned document image - Fast Document Rectification and Enhancement. I followed the post and they mention a formula to make ...
2
votes
3answers
546 views

Derivative with respect to complex conjugate

I have a real function $C$ of a complex vector $x$. While taking the gradient of the function $C$ for minimising the same, why do we take the derivatives with respect to the complex conjugate of $x$, ...
3
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0answers
67 views

Designing a fast linear operator with $\pm 1$ entries with low condition number and low Hamming distance between consecutive rows

I need to design a matrix for compressive imaging where each row represents an $N$-pixel filter in a focal plane through which light is masked, summed, and measured (think of Rice's single-pixel ...
0
votes
1answer
50 views

Regularized Least Squares by Laplacian Operator - Optimal Value of the Regularization Factor (Lagrangian Multiplier)

Consider the cost function $$f(X,\lambda) = \|AX-b\|_2^2 + \alpha \|LX\|_2^2$$ $A:$Measurement matrix($R_{m\times n}$,$m \ll n$), $b:$observation vector($R_m$), $L:$Laplacian operator($R_{n \times n}...
4
votes
1answer
251 views

Approximating $ {L}_{0} $ Norm Minimization with Non Linear Convex Inequality Constraints using Reweighted $ {L}_{1} $ Minimization

I have an optimization problem consisting of the $ {\ell}_{0} $ norm as the objective and a nonlinear (convex) constraint as well as a linear constraint. I am wondering if the reweighted $ {\ell}_{1} $...
0
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2answers
337 views

Why Is Non Linear Least Squares Method from MATLAB and Alglib Gives Different Results on the Same Data?

i'm trying to rewrite my Matalab prototype for some DSP to C++ and encountering a displeasing problem. I'm trying to fit data to a function $y = a * (\pi / 2 + arctg(b * x))$. In Matlab it works well ...
3
votes
1answer
723 views

Design $ {L}_{2} $ Norm Optimal Infinite Impulse Response (IIR) Filters

It is widely known that matching a FIR filter of fixed length to a band model is an unconstrained QP-problem. The MATLAB function firls() implements a solution to ...
3
votes
1answer
445 views

How Can I Use MATLAB to Solve a Total Variation Denoising Problem?

The Total Variation Denoising Problem is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) $$ Where $ $ is the Total Variation ...
5
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1answer
358 views

Weighted Nuclear Norm Minimization for Image Denoising

Recently, I saw new published papers like Shuhang Gu, Lei Zhang, Wangmeng Zuo, Xiangchu Feng, Weighted Nuclear Norm Minimization with Application to Image Demonising [pdf]. about denoising images ...
1
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1answer
68 views

Sparse Recovery Best Algorithms

In the big data era, in order to control the cost, complexity, and bandwidth of collecting and processing high-dimensional data systems, it is critical to exploit models that ...
0
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1answer
569 views

Fastest Available Algorithm to Blur an Image (Low Pass Filter)

Iam working with a camera that produces ugly artifacts: by using ANY blur filter on the camera's output the visual quality improves drastically: The above image was created using OpenCV's cv::...
15
votes
1answer
1k views

What Does Make an Error Surface Convex? Is It Determined by the Covarinace Matrix or the Hessian?

I am currently learning about least-squares (and other) estimations for regression, and from what I am also reading in some adaptive algorithm literatures, often times the phrase "... and since the ...
5
votes
1answer
124 views

Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where $\...
0
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1answer
77 views

Iterative Blind Sinus Signal Suppression

There are two real signals in the form of $A_i sin(wt+p_i), i=1,2$. Suppose frequency $w$ of both the signals is the same and amplitude $A_i$ and phase $p_i$ are different. The first signal has ...
2
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1answer
2k views

$ {L}_{0} $ Pseudo Norm Minimization in Compressive Sensing

I have recently taken up studying compressive sensing related papers. Some things are not very clear to me or may be I am not able to visualize the scenario as is said. Like how $l_0$ norm ...
0
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2answers
241 views

Significance of $ \lambda $ in Basis Pursuit

In basis Pursuit, L1 minimization is done to perform compressed sensing. In the literature there is a $ \lambda $ parameter used as a regularizer. What is its significance?
2
votes
1answer
107 views

Adaptive Filter Gradient Descent

The quadratic performance surface of an adaptive filter is a paraboloid. Its minimum can be found wherever the gradient is zero. However, since there are two types of paraboloids (elliptical and ...
0
votes
1answer
49 views

The Gradient Operator of a Vectorized Image in Matrix Form

I have this optimization problem: $$ \arg \min_{ X \left( i, j \right) } \sum_{i, j} \left\| X \left( i, j \right) - 255 \right\|_{2}^{2} + \lambda \sum_{i, j} \left\| \nabla X \left( i, j \right) - \...
0
votes
2answers
368 views

Why Do Most of The Papers Use the Frobenius Norm for Denoising?

I have an noisy image and I want to remove noise from it; suppose $y$ is noisy image and $A$ is linear mask which makes my image noisy and $x$ is original image, so we have $$ Ax + \eta = y $$ and $\...
10
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0answers
592 views

Will an Unscented Kalman Filter Be “As Good” as Other Optimization Algorithms for This Problem?

I want to calibrate a tri-axis magnetometer when a tri-axis gyroscope is also available. I am fairly certain I can solve this problem using various optimisation algorithms, but I would prefer to use ...
2
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2answers
77 views

Orthonormal Dictionaries for Band Limited Signals

If $\mathbf{x} = [x_0, x_1, \ldots, x_{N-1}]^T$ is the time sampled input signal and $\mathbf{Y} = [Y_0, Y_1, \ldots, Y_{N-1}]^T$ is the Fourier transform of the input signal, then a linear ...
3
votes
0answers
74 views

When Does $ {L}_{1} $ Regularization Give a Sparse Solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...