Jonah Patel
Shift variance is a phenomenon observed in various fields, including engineering, statistics, and image processing. It refers to the changes in the coefficients or properties of a system when there is a shift in the input or independent variable. In engineering, shift variance is observed in multirate systems and wavelet transforms, where a shift in the input signal results in corresponding changes in the output. In statistics, shift variance is related to changes in data sets, where shifting a data set by a constant value, k, can impact the mean and median while keeping the standard deviation, variance, and z-scores unchanged. Properly estimating variance in the presence of shifts in the mean is crucial, and certain procedures have been found to perform better than ordinary sample variance in such cases. In image processing, shift variance is addressed in image enhancement techniques, where methods such as linear transformations and adaptive thresholding are applied to improve image quality and details. Understanding and analyzing shift variance is essential for developing effective algorithms and systems in various domains.
| Characteristics | Values |
|---|---|
| Shift variance | Results from the application of subsampling in the wavelet transform |
| Shift invariant system | A shift in the independent variable of the input signal causes a corresponding shift in the output signal |
| Shift-variant linear periodically shift-variant (LPSV) systems | Both input and output spaces are assumed to be of continuous time |
| Multirate filter banks | Introduce periodic time-varying phenomena into their subband signals |
| Ordinary variance estimators | Perform poorly in the presence of shifts in the mean |
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What You'll Learn
Shift variance in data shifting
In statistics, shifting a dataset by a constant k results in a change in the mean and median, but the standard deviation, variance, z-scores, and percentile values remain the same. This is because every point in the dataset moves the exact same distance, and there is no change in their relations to each other.
In some cases, a shift in data may be accompanied by a rescaling. For example, when changing temperature readings from Fahrenheit to Celsius, the data must first be shifted by k = -32, and then rescaled by multiplying the mean and standard deviation by the rescaling constant 5/9. However, the standard deviation and variance remain unchanged during the initial shift.
In the context of wavelet transforms, shift variance can be problematic in applications like pattern recognition or image fusion. The DWT (Discrete Wavelet Transform) algorithm involves critically subsampling the output of the filtering stage, which can cause the coefficients to change when the input is shifted. This can lead to artifacts in fused images. One way to address this issue is by oversampling or removing the subsampling step and instead upsampling the filters at each scale.
In certain fields, such as remote gait impairment monitoring using wearable devices, shift-variance analysis plays a crucial role in enhancing medical images. By decomposing the initial image into different sub-bands, applying linear transformations, and utilizing techniques like adaptive thresholding and unsharp masking, researchers can improve image entropy, EME, and PSNR.
Additionally, multirate filter banks introduce periodic time-varying phenomena into their subband signals, resulting in aliasing that causes deterministic signals to become shift-variant. The behavior of these signals can be analyzed by comparing deterministic and wide-sense stationary (WSS) random signals in multirate filter banks.
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Shift-invariant systems
A shift-invariant system is one where a shift in the independent variable of the input signal causes a corresponding shift in the output signal. In other words, shift invariance means that we can commute the operations of shifting and filtering. Many of the systems encountered in digital signal processing (DSP) exhibit the properties of linearity and shift invariance.
In DSP, shift-invariant filters play an important role. A graph filter h is represented by a matrix H, and graph filtering or graph filtering convolution by h is matrix-vector multiplication.
The properties of time-shift invariance and frequency-shift invariance are common to all quadratic time-frequency distributions (TFDs). However, no proper DI kernel satisfies the frequency marginal, frequency support, or time delay property. Similarly, no proper LI kernel satisfies the time marginal property.
The DWT (Discrete Wavelet Transform) is not shift invariant since the wavelet coefficients of the DWT change when the signal is shifted. Shift variance results from the application of subsampling in the wavelet transform. The main step in all the wavelet transforms is convolving the signal (or image) with a filter bank to obtain the approximation and the detail images. In the DWT algorithm, the output of the filtering is critically subsampled, meaning the outputs of the filter banks are decimated by a factor of two (usually). This subsampling causes the coefficients to change when the input is shifted.
The shift variance of the DWT can be problematic in applications like pattern recognition or image fusion, causing artifacts in the fused images. This problem can be overcome by oversampling or removing the subsampling step at each scale in the DWT and instead upsampling the filters at each scale.
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Time-shift invariance
A simple shift operator can be defined as a discrete time point. In DSP, shift-invariant filters are important. Shift-invariant filters allow us to implement operations in the digital domain that cannot be implemented using discrete analog components.
The properties of time-shift invariance and frequency-shift invariance are common to all quadratic TFDs. However, no proper LI kernel satisfies the time marginal, time support, or frequency marginal property.
One example of a time-invariant system is system B, which can be defined as:
> {displaystyle f_{B}=10x(t)→{\frac{\partial f_{B}}{\partial t}}=0} so it is time-invariant.
System B's time-dependence is only a function of the time-varying input, which makes it a time-invariant system.
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Shift variance in image enhancement
In image enhancement, shift variance refers to the changes in wavelet coefficients when the input image is shifted. This occurs during the wavelet transform, a common image processing technique, where the output of the filtering step is subsampled, causing the coefficients to change when the image is shifted.
The problem of shift variance can be overcome by oversampling or removing the subsampling step entirely and instead upsampling the filters at each scale. This is important in applications such as pattern recognition and image fusion, where shift variance can cause artifacts in the fused images.
One example of shift variance in image enhancement is the Time-Shift Method, which is used to represent constant images in spacetime and enhance degraded images. This method assumes that images are moving objects on a z-plane with different velocities and uses the Lorentz factor to represent the movement of objects at a constant moment. The changes between the events and a reference frame are used to create adjacent z-plane events, providing statistical and pixel-based relationships to reconstruct a single frame for the common perspective of the reference image in time.
Another example of shift variance in image enhancement is the Color Shift Estimation and Correction method, which focuses on correcting exposure problems in images by adjusting image brightness and correcting color tone distortions. This method uses a UNet-based network to derive color feature maps of over- and under-exposed regions, creating pseudo-normal color feature maps, and then estimating and correcting the color shifts between the derived maps.
Additionally, convolutional neural networks have been proposed to address images with both over- and under-exposed regions through adaptive brightness adjustment and color shift correction. These networks have limitations in modeling geometric transformations due to their fixed kernel configuration.
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Shift variance in filter banks
Shift variance in the context of filter banks refers to the behaviour exhibited by certain multirate systems where the output signal changes when the order of the filter bank operator and shift operator is swapped. This phenomenon is observed in critically sampled multirate FIR filter banks, where non-ideal anti-aliasing filtering in the decimation stage causes periodic shift variant behaviour.
The discrete wavelet transform (DWT) is an example of a system that is not shift invariant. In DWT, the wavelet coefficients change when the input signal is shifted due to the subsampling step. This can create issues in applications such as pattern recognition or image fusion, leading to artefacts in the fused images.
To address the problem of shift variance in DWT, one approach is to remove the subsampling step and instead upsample the filters at each scale. This ensures that the coefficients remain consistent even when the input is shifted. Another technique is to oversample the data, which can also mitigate the shift variance issue.
In some cases, shift variance can be advantageous. For instance, in multirate systems, shift variance can be utilised to design causal controllers for stabilising plants. By exploiting the relationship between multirate systems and shift invariant systems, stabilising controllers can be parametrised and applied to control dynamic systems effectively.
Overall, shift variance in filter banks is an important consideration in signal processing and control systems. While it can pose challenges in certain applications, it also presents opportunities for innovative solutions and system optimisation.
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Frequently asked questions
Shift variance is the change in the coefficients of a system when the input is shifted.
Shift variance is associated with time-varying phenomena, particularly in the context of periodic shifts in linear systems with continuous-time inputs and outputs.
Shifting a data set by a constant value, k, does not change the variance. The variance remains the same because every data point moves the same distance relative to each other.
