CN109067404A - A kind of compressed sensing signal blind reconstructing method based on single-bit quantification - Google Patents
A kind of compressed sensing signal blind reconstructing method based on single-bit quantification Download PDFInfo
- Publication number
- CN109067404A CN109067404A CN201810916606.8A CN201810916606A CN109067404A CN 109067404 A CN109067404 A CN 109067404A CN 201810916606 A CN201810916606 A CN 201810916606A CN 109067404 A CN109067404 A CN 109067404A
- Authority
- CN
- China
- Prior art keywords
- signal
- value
- iteration
- iterations
- supported collection
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
Classifications
-
- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
- H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
- H03M7/3059—Digital compression and data reduction techniques where the original information is represented by a subset or similar information, e.g. lossy compression
- H03M7/3062—Compressive sampling or sensing
Landscapes
- Engineering & Computer Science (AREA)
- Theoretical Computer Science (AREA)
- Compression, Expansion, Code Conversion, And Decoders (AREA)
Abstract
A kind of compressed sensing signal blind reconstructing method based on single-bit quantification, it is used for the reconfiguration technique field of compressed sensing signal.The problem of present invention solves current single-bit compressed sensing and has to pass through a large amount of calculating, could reconstruct source signal from the only measuring signal of stet position.The present invention quantifies the symbolized measurement value y of input signal, then optimal supported collection estimation is carried out using these symbol datas, and it is unanimously rebuild in optimal supported collection, to obtain the estimated value for updating input signal, by front and back, the difference of the signal amplitude estimation value of iteration and precision threshold are compared twice, to determine whether iteration terminates；To prevent iteration from falling into endless loop, setting also stops iteration when iterating to maximum number of iterations；The estimated value of degree of rarefication and signal amplitude is determined according to last iteration result, realizes the blind reconstruct of signal, and calculation amount is reduced 40% or more compared to existing method by the method for the present invention.Present invention could apply to the reconstruction field of compressed sensing signal use.
Description
Technical field
The invention belongs to the reconfiguration technique fields of compressed sensing signal, and in particular to a kind of compression based on single-bit quantification
Perceptual signal blind reconstructing method.
Background technique
Compressive sensing theory is equivalent to only there are two process: compression and reconstruct.Reconstruct is just combined with traditional sampling method
Two steps of middle sampling and coding.But in reality, the signal sampled must be quantified, that is, be AD converted, Cai Neng
Computer such as is stored and is calculated at the work.In order to further increase the speed of signal processing and the simplicity of actual circuit, need
Process is further processed in the measured value to obtain to traditional compressed sensing.Petros T.Boufouno indicates one
The new compact model of kind, the last observation signal of this model only have the symbolic information of 1 bit, it are referred to as: single-bit compression
Perception.This theory is considered as a kind of evolution of conventional compression perception.
Difference with compressed sensing is that the measured value that single-bit compressed sensing obtains it has carried out further fortune
It calculates, only remains the symbol of gained observation, so that it may the means reconstruction signal amplitude according to energy normalized.But single-bit
There is also some shortcomings for compressed sensing, that is, need very big calculation amount accurately could go out to reconstruct the amplitude of source signal, to be used for
Source signal is reconstructed from the only measuring signal of stet position.
Summary of the invention
It, could be from only retaining the purpose of the present invention is having to pass through a large amount of calculating to solve current single-bit compressed sensing
The problem of reconstructing source signal in the measuring signal of sign bit.
The technical solution adopted by the present invention to solve the above technical problem is:
A kind of compressed sensing signal blind reconstructing method based on single-bit quantification, method includes the following steps:
Step 1: the input signal x for being N for length is carved at the beginning, and the estimated value of initializing signal amplitude is x0
Step 2: the estimated value x of initializing signal amplitude is utilized0Come when obtaining the number of iterations l=1, corresponding input is believed
The negative element value of the estimated value of number amplitude；
Step 3: using the negative element value of the amplitude estimation value of step 2, calculate the number of iterations be corresponding to l=1 most
Excellent supported collection；
Step 4: calculating the number of iterations is the consistent reconstructed results of signal under optimal supported collection corresponding to l=1, is utilized
The consistent reconstructed results of obtained signal calculate the estimated value of updated signal amplitude；
Step 5: using the estimated value of signal amplitude after the corresponding update of the number of iterations l=1 as x1, calculate the number of iterations
The negative element value of the estimated value of input signal amplitude when for l=2；
Step 6: using the negative element value for the amplitude estimation value that step 5 calculates, calculating the number of iterations is corresponding to l=2
Optimal supported collection, and calculating the number of iterations is the consistent reconstructed results of signal corresponding to l=2 under optimal supported collection, is utilized
The consistent reconstructed results of obtained signal calculate the estimated value of updated signal amplitude；
Step 7: enabling l=l+1, repeats the operation of step 5 to step 6, until the front and back signal width that iteration obtains twice
When the difference of the estimated value of degree reaches the maximum number of iterations Q of setting less than or equal to precision threshold c or l, then stop iteration, after
It is continuous to execute step 8；
Step 8: the degree of rarefication of input signal is the corresponding normalized parameter b value of last iteration, i.e. degree of rarefication K=b, benefit
Input signal is reconstructed in the estimated value of the signal amplitude obtained with obtained degree of rarefication and last iteration.
The beneficial effects of the present invention are: a kind of blind reconstruct side of compressed sensing signal based on single-bit quantification of the invention
Method, the present invention first quantify the symbolized measurement value y of input signal, then carry out optimal support using these symbol datas
Collection estimation, then unanimously rebuild in the optimal supported collection of estimation, so as to deleted with normalization operation update it is defeated
Enter the estimated value of signal, finally the difference of the input signal estimated value of iteration and the precision threshold of setting carry out pair twice by front and back
Than to carry out the blind reconstruct of signal, and iteration falls into endless loop and can not stop in order to prevent, therefore setting iterates to maximum
Also stop iteration when the number of iterations；The array data model that the present invention uses Taylor expansion to obtain, is effectively utilized a period of time
The information of iteration is carved, and uses EM algorithm, signal power, noise function are realized by the parameter in estimated probability model simultaneously
Rate and DOA tracking, for method of the invention while realizing signal blind reconstruct, being compared to existing method can be by calculation amount
Reduce 40% or more.
Detailed description of the invention
Fig. 1 is the quality reconstruction figure of proposition method of the present invention；
Fig. 2 is the graph of relation of the reconstruct MSE and pendulous frequency of the method for the present invention；
Fig. 3 is the comparison diagram of method and the non-blind arithmetic reconstruction property of tradition of the invention.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawing, and however, it is not limited to this, all to this
Inventive technique scheme is modified or replaced equivalently, and without departing from the spirit and scope of the technical solution of the present invention, should all be covered
Within the protection scope of the present invention.
Specific embodiment 1: a kind of blind reconstruct of compressed sensing signal based on single-bit quantification described in present embodiment
Method, method includes the following steps:
Step 1: the input signal x for being N for length is carved at the beginning, and the estimated value of initializing signal amplitude is x0
(when initial time, x0=0)；
Step 2: the estimated value x of initializing signal amplitude is utilized0Come when obtaining the number of iterations l=1, corresponding input is believed
The negative element value of the estimated value of number amplitude；
Step 3: using the negative element value of the amplitude estimation value of step 2, calculate the number of iterations be corresponding to l=1 most
Excellent supported collection；
Step 4: calculating the number of iterations is the consistent reconstructed results of signal under optimal supported collection corresponding to l=1, is utilized
The consistent reconstructed results of obtained signal calculate the estimated value of updated signal amplitude；
Step 5: using the estimated value of signal amplitude after the corresponding update of the number of iterations l=1 as x1, calculate the number of iterations
The negative element value of the estimated value of input signal amplitude when for l=2；
Step 6: using the negative element value for the amplitude estimation value that step 5 calculates, calculating the number of iterations is corresponding to l=2
Optimal supported collection, and calculating the number of iterations is the consistent reconstructed results of signal corresponding to l=2 under optimal supported collection, is utilized
The consistent reconstructed results of obtained signal calculate the estimated value of updated signal amplitude；
Step 7: enabling l=l+1, repeats the operation of step 5 to step 6, until the front and back signal width that iteration obtains twice
When the difference of the estimated value of degree reaches the maximum number of iterations Q of setting less than or equal to precision threshold c or l, then stop iteration, after
It is continuous to execute step 8；
Step 8: the degree of rarefication of input signal is the corresponding normalized parameter b value of last iteration, i.e. degree of rarefication K=b, benefit
Input signal is reconstructed in the estimated value of the signal amplitude obtained with obtained degree of rarefication and last iteration.
In present embodiment until the front and back signal amplitude that iteration obtains twice estimated value difference be less than or equal to essence
When degree threshold value c or l reach the maximum number of iterations Q of setting, then stop iteration；Refer to: if the number of iterations reaches greatest iteration time
Before number Q, the difference for having already appeared the estimated value of the front and back signal amplitude that iteration obtains twice is less than or equal to the feelings of precision threshold c
Condition then stops iteration, and time corresponding normalized parameter b value of (i.e. last) iteration is as degree of rarefication by after；If the number of iterations reaches
Before maximum number of iterations Q, the difference for not occurring the estimated value of the front and back signal amplitude that iteration obtains twice is less than or equal to precision
The case where threshold value c, when then reaching maximum number of iterations Q (last iteration i.e. maximum number of iterations Q at this time), stops iteration, will repeatedly
The corresponding normalized parameter b value of generation number Q is as degree of rarefication.
Specific embodiment 2: present embodiment is to a kind of compression sense based on single-bit quantification described in embodiment one
Know that signal blind reconstructing method is further limited, calculates the negative element value of the estimated value of the corresponding amplitude of the l times iteration
Detailed process are as follows:
The input signal x for being N for length, there are a calculation matrix Φ to meet RIP condition, by formula (1) to defeated
The symbolized measurement value y for entering signal is quantified:
Y=sign (Φ x) (1)
Wherein: sign indicates sign function；
Take each element of symbolized measurement value y be placed on diagonal line composition one diagonal matrix Y, diagonal matrix Y its
His element sets 0, i.e. Y=diag (y)；
Then when the l times iteration, the negative element value r of the estimated value of input signal amplitudelAre as follows:
rl=(diag (y) Φ xl-1)- (3)
Wherein: as l-1=0, xl-1=x0For the estimated value of the signal amplitude of initial time, as l-1 > 0, xl-1It is
The estimated value of signal amplitude, () after the update of l-1 iteration-Representative takes the function of negative value (both to retain negative element, other elements
It sets 0).
Specific embodiment 3: present embodiment is to a kind of compression sense based on single-bit quantification described in embodiment two
Know that signal blind reconstructing method is further limited, obtain the detailed process of the corresponding optimal supported collection of each iteration are as follows:
As the number of iterations l=1, the supported collection s of the corresponding signal estimation of the 1st iteration1For s1=ΦTdiag(y)r1,
From supported collection s1In select 1 optimal supported collection of the column as the l=1 times iteration of maximum absolute value
As the number of iterations l > 1, the supported collection s of the corresponding signal estimation of the l times iterationlAre as follows:
sl=ΦTdiag(y)rl (4)
The supported collection for the signal estimation that 2nd iteration obtains is denoted as s2, from supported collection s2Middle a effective element of selection, will
The optimal supported collection that the set of a effective element composition is obtained with the 1st iterationUnion is taken, the 2nd iteration pair is obtained
The optimal supported collection answered
Similarly, the supported collection s obtained from the l times iterationlMiddle a effective element of selection, the collection that a effective element is formed
The optimal supported collection closed and obtained after the l-1 times iterationUnion is taken, the corresponding optimal support of the l times iteration is obtained
Collection
Specific embodiment 4: present embodiment is to a kind of compression sense based on single-bit quantification described in embodiment three
Know that signal blind reconstructing method is further limited, calculates the estimated value of the corresponding updated signal amplitude of each iteration
Detailed process are as follows:
The optimal supported collection obtained using the l times iterationCalculate optimal supported collectionUnder signal
Consistent reconstructed resultsAnd signal is unanimously rebuild need to meet condition | | x | |2=1 He
Indicate optimal supported collectionThe corresponding x of supplementary set be 0,It is's
Supplementary set；||x||2It is 1 that=1 expression x, which takes the value of two norm operations,；Representative is worked asObtain the value of corresponding x when minimum value；
It is rightIt is normalized, obtains the amplitude Estimation of input signal x after the corresponding update of the l times iteration
ValueAre as follows:
For the 1st iteration, the value of normalized parameter b is 1；The number of iterations is every to be increased once, and normalized parameter b's takes
Value increases by 1.
Specific embodiment 5: present embodiment is to a kind of compression sense based on single-bit quantification described in embodiment four
Know that signal blind reconstructing method is further limited, the value that effective element number a is arranged in present embodiment is 8, precision
The value of threshold value c is 0.008, and maximum number of iterations is that Q is 200.
Embodiment
Fig. 1 is the quality reconstruction figure of proposition method of the present invention, and method proposed by the present invention has preferable as can be seen from Figure 1
Quality reconstruction；
The relationship of the length N of pendulous frequency M and input signal x meets following formula:
M=o (Klog2(N/K))
It can be seen that the increase of the length N with input signal x, pendulous frequency M also increase from formula.
Fig. 2 is the relationship of the reconstruct MSE and pendulous frequency of proposition method of the present invention, it can be seen that the present invention proposes method
Performance gradually improves with the increase of pendulous frequency；
Fig. 3 is the comparison of method proposed by the present invention and non-blind arithmetic (MSP) reconstruction property of tradition, it can be seen that the present invention
It is proposed method has the reconstruction property roughly the same with non-blind method.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto,
Anyone skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its
Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
Claims (5)
1. a kind of compressed sensing signal blind reconstructing method based on single-bit quantification, which is characterized in that this method includes following step
It is rapid:
Step 1: the input signal x for being N for length is carved at the beginning, and the estimated value of initializing signal amplitude is x0；
Step 2: the estimated value x of initializing signal amplitude is utilized0Come when obtaining the number of iterations l=1, corresponding input signal amplitude
Estimated value negative element value；
Step 3: using the negative element value of the amplitude estimation value of step 2, calculating the number of iterations is optimal branch corresponding to l=1
Support collection；
Step 4: calculating the number of iterations is the consistent reconstructed results of signal corresponding to l=1 under optimal supported collection, using obtaining
The consistent reconstructed results of signal calculate the estimated value of updated signal amplitude；
Step 5: using the estimated value of signal amplitude after the corresponding update of the number of iterations l=1 as x1, calculating the number of iterations is l=2
When input signal amplitude estimated value negative element value；
Step 6: using step 5 calculate amplitude estimation value negative element value, calculate the number of iterations be l=2 corresponding to most
Excellent supported collection, and calculating the number of iterations is the consistent reconstructed results of signal corresponding to l=2 under optimal supported collection, using obtaining
The consistent reconstructed results of signal calculate the estimated value of updated signal amplitude；
Step 7: enabling l=l+1, repeats the operation of step 5 to step 6, until the front and back signal amplitude that iteration obtains twice
When the difference of estimated value reaches the maximum number of iterations Q of setting less than or equal to precision threshold c or l, then stops iteration, continue to hold
Row step 8；
Step 8: the degree of rarefication of input signal is the corresponding normalized parameter b value of last iteration, i.e. degree of rarefication K=b is utilized
To the estimated value of signal amplitude that obtains of degree of rarefication and last iteration input signal is reconstructed.
2. a kind of compressed sensing signal blind reconstructing method based on single-bit quantification according to claim 1, feature exist
In the detailed process of the negative element value of the estimated value of the corresponding amplitude of the l times iteration of calculating are as follows:
The input signal x for being N for length, there are a calculation matrix Φ to meet limited equidistant condition, by formula (1) to defeated
The symbolized measurement value y for entering signal is quantified:
Y=sign (Φ x) (1)
Wherein: sign indicates sign function；
Each element of symbolized measurement value y is taken to be placed in one diagonal matrix Y of composition, other yuan of diagonal matrix Y on diagonal line
Element sets 0, i.e. Y=diag (y)；
Then when the l times iteration, the negative element value r of the estimated value of input signal amplitudelAre as follows:
rl=(diag (y) Φ xl-1)- (3)
Wherein: as l-1=0, xl-1=x0For the estimated value of the signal amplitude of initial time, as l-1 > 0, xl-1It is the l-1 times
The estimated value of signal amplitude, () after the update of iteration-Represent the function of taking negative value.
3. a kind of compressed sensing signal blind reconstructing method based on single-bit quantification according to claim 2, feature exist
In obtaining the detailed process of the corresponding optimal supported collection of each iteration are as follows:
As the number of iterations l=1, the supported collection s of the corresponding signal estimation of the 1st iteration1For s1=ΦTdiag(y)r1, ΦTFor
The transposition of Φ, from supported collection s1In select 1 optimal supported collection of the column as the l=1 times iteration of maximum absolute value
As the number of iterations l > 1, the supported collection s of the corresponding signal estimation of the l times iterationlAre as follows:
sl=ΦTdiag(y)rl (4)
The supported collection for the signal estimation that 2nd iteration obtains is denoted as s2, from supported collection s2Middle a effective element of selection, by a
The optimal supported collection that the set of effective element composition is obtained with the 1st iterationUnion is taken, it is corresponding to obtain the 2nd iteration
Optimal supported collection
Similarly, the supported collection s obtained from the l times iterationlMiddle a effective element of selection, by a effective element composition set with
The optimal supported collection obtained after the l-1 times iterationUnion is taken, the corresponding optimal supported collection of the l times iteration is obtained
4. a kind of compressed sensing signal blind reconstructing method based on single-bit quantification according to claim 3, feature exist
In calculating the detailed process of the estimated value of the corresponding updated signal amplitude of each iteration are as follows:
The optimal supported collection obtained using the l times iterationCalculate optimal supported collectionUnder signal unanimously weigh
Build resultAnd signal is unanimously rebuild need to meet condition | | x | |2=1 He
Indicate optimal supported collectionThe corresponding x of supplementary set be 0,It isSupplementary set；
||x||2It is 1 that=1 expression x, which takes the value of two norm operations,；Representative is worked asIt obtains
The value of corresponding x when minimum value；
It is rightIt is normalized, obtains the amplitude estimation value of input signal x after the corresponding update of the l times iteration
Are as follows:
For the 1st iteration, the value of normalized parameter b is 1；The number of iterations is every to increase primary, the value increasing of normalized parameter b
Add 1.
5. a kind of compressed sensing signal blind reconstructing method based on single-bit quantification according to claim 4, feature exist
In the value that effective element number a is arranged is 8, and the value of precision threshold c is 0.008, and maximum number of iterations is that Q is 200.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810916606.8A CN109067404A (en) | 2018-08-13 | 2018-08-13 | A kind of compressed sensing signal blind reconstructing method based on single-bit quantification |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810916606.8A CN109067404A (en) | 2018-08-13 | 2018-08-13 | A kind of compressed sensing signal blind reconstructing method based on single-bit quantification |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN109067404A true CN109067404A (en) | 2018-12-21 |
Family
ID=64683634
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201810916606.8A Pending CN109067404A (en) | 2018-08-13 | 2018-08-13 | A kind of compressed sensing signal blind reconstructing method based on single-bit quantification |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN109067404A (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112906046A (en) * | 2021-01-27 | 2021-06-04 | 清华大学 | Model training method and device by using single-bit compression perception technology |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104333389A (en) * | 2014-10-23 | 2015-02-04 | 湘潭大学 | Adaptive threshold value iterative reconstruction method for distributed compressed sensing |
| CN105515585A (en) * | 2015-12-08 | 2016-04-20 | 宁波大学 | Compressed sensing reconstruction method for signals with unknown sparseness |
-
2018
- 2018-08-13 CN CN201810916606.8A patent/CN109067404A/en active Pending
Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104333389A (en) * | 2014-10-23 | 2015-02-04 | 湘潭大学 | Adaptive threshold value iterative reconstruction method for distributed compressed sensing |
| CN105515585A (en) * | 2015-12-08 | 2016-04-20 | 宁波大学 | Compressed sensing reconstruction method for signals with unknown sparseness |
Non-Patent Citations (2)
| Title |
|---|
| 丛爽等: "《量子控制***设计》", 31 August 2016, 中国科学技术大学出版社 * |
| 闫斌等: "一种基于盲运算的1比特压缩感知重建算法", 《西南交通大学学报》 * |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112906046A (en) * | 2021-01-27 | 2021-06-04 | 清华大学 | Model training method and device by using single-bit compression perception technology |
| CN112906046B (en) * | 2021-01-27 | 2024-04-19 | 清华大学 | Model training method and device using single bit compressed sensing technology |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN107147397B (en) | Wearable device-oriented rapid compressed sensing reconstruction method | |
| KR101958674B1 (en) | Actually-measured marine environment data assimilation method based on sequence recursive filtering three-dimensional variation | |
| CN103400402B (en) | Based on the sparse compressed sensing MRI image rebuilding method of low-rank structure | |
| CN105513026A (en) | Compressed sensing reconstruction method based on image nonlocal similarity | |
| CN102332153B (en) | Kernel regression-based image compression sensing reconstruction method | |
| CN111738448A (en) | Quantum line simulation method, device, equipment and storage medium | |
| CN101751697B (en) | Three-dimensional scene reconstruction method based on statistical model | |
| CN109584164B (en) | Medical image super-resolution three-dimensional reconstruction method based on two-dimensional image transfer learning | |
| CN109003229B (en) | Magnetic resonance super-resolution reconstruction method based on three-dimensional enhanced depth residual error network | |
| CN111667444B (en) | Image compressed sensing reconstruction method based on multi-channel residual error network | |
| CN104320144B (en) | Degree of rarefication self-adapting signal reconstructing method | |
| CN107547088A (en) | Enhanced self-adapted segmentation orthogonal matching pursuit method based on compressed sensing | |
| CN105184742A (en) | Image denoising method of sparse coding based on Laplace graph characteristic vector | |
| CN109067404A (en) | A kind of compressed sensing signal blind reconstructing method based on single-bit quantification | |
| Azghani et al. | Missing low-rank and sparse decomposition based on smoothed nuclear norm | |
| CN106301384A (en) | A kind of signal reconfiguring method based on splits' positions perception | |
| CN116341616B (en) | Electric load information acquisition method based on matrix reconstruction two-dimensional convolution network | |
| CN117333497A (en) | Mask supervision strategy-based three-dimensional medical image segmentation method for efficient modeling | |
| CN105846826B (en) | Compressed sensing signal reconfiguring method based on approximate smooth L0 norm | |
| CN104270158A (en) | A cooperative reconstruction method with adaptive sparsity | |
| CN106485014A (en) | A kind of 1 bit compression Bayes's cognitive method of strong robustness | |
| CN107564588B (en) | Physiological health data prediction device | |
| CN116189048A (en) | Non-contact heart rate detection method and system based on deep neural network and self-attention mechanism | |
| CN104392412A (en) | Compressed sensing signal recovery method based on orthogonal matching pursuit evolution | |
| Nagatsuka et al. | Parameter and quantile estimation for the three-parameter gamma distribution based on statistics invariant to unknown location |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| RJ01 | Rejection of invention patent application after publication |
Application publication date: 20181221 |
|
| RJ01 | Rejection of invention patent application after publication |