From f46876930901a12cfa696e73e241a2a69fc3a8c7 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:03:41 +0100 Subject: [PATCH 01/13] diffusion RPCA --- Gemfile.lock | 59 ++--- ...5-11-16-Diffusion-Background-Supression.md | 208 ++++++++++++++++++ .../algorithm.jpg | Bin 0 -> 222348 bytes .../DiffusionBackgroundSupression/boxplot.jpg | Bin 0 -> 111352 bytes .../comparison.jpg | Bin 0 -> 117825 bytes .../DiffusionBackgroundSupression/example.jpg | Bin 0 -> 349143 bytes .../ks_statistic.jpg | Bin 0 -> 184551 bytes .../DiffusionBackgroundSupression/results.jpg | Bin 0 -> 901056 bytes 8 files changed, 228 insertions(+), 39 deletions(-) create mode 100644 collections/_posts/2025-11-16-Diffusion-Background-Supression.md create mode 100644 collections/images/DiffusionBackgroundSupression/algorithm.jpg create mode 100644 collections/images/DiffusionBackgroundSupression/boxplot.jpg create mode 100644 collections/images/DiffusionBackgroundSupression/comparison.jpg create mode 100644 collections/images/DiffusionBackgroundSupression/example.jpg create mode 100644 collections/images/DiffusionBackgroundSupression/ks_statistic.jpg create mode 100644 collections/images/DiffusionBackgroundSupression/results.jpg diff --git a/Gemfile.lock b/Gemfile.lock index bb4aed65..90a92219 100644 --- a/Gemfile.lock +++ b/Gemfile.lock @@ -1,23 +1,19 @@ GEM remote: https://rubygems.org/ specs: - 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securerandom (0.4.1) simpleidn (0.2.3) terminal-table (1.8.0) unicode-display_width (~> 1.1, >= 1.1.1) @@ -268,8 +251,6 @@ GEM tzinfo (2.0.6) concurrent-ruby (~> 1.0) unicode-display_width (1.8.0) - uri (1.0.3) - webrick (1.9.1) PLATFORMS x86_64-linux @@ -280,4 +261,4 @@ DEPENDENCIES tzinfo-data BUNDLED WITH - 2.4.19 + 2.3.25 diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md new file mode 100644 index 00000000..07178436 --- /dev/null +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -0,0 +1,208 @@ + +--- +layout: review +title: "Nuclear Diffusion Models for Low-rank Background Supression in Videos" +tags: rpca, diffusion-models, ultrasound-imaging, denoising +author: "Julia Puig" +cite: + authors: "Tristan S.W. Stevens, Oisin Nolan, Jean-Luc Robert and Ruud J.G. van Sloun" + title: "Nuclear Diffusion Models for Low-rank Background Supression in Videos" + venue: "preprint" +pdf: "https://arxiv.org/abs/2509.20886" +--- + +
+ +# Highlights + +- The authors propose to perform dehazing by combining the power of **Robust PCA** and **Diffusion Models**, which can model highly complex data distributions without having to rely on basic assumptions. +- +- + +
+ +# Background + +## Ultrasound image quality +During ultrasound acquisitions, phenomena such as aberration and reverberation produce unwanted echoes that degrade the image quality. In particular, **haze** is an artifact that occurs due to multipath reflections and produces a white haze on the echo image. + +{:refdef: style="text-align: center;"} +![](/collections/images/DehazingDiffusion/echo_images.jpg){: width="400" } +{:refdef} + +This phenomenon can **make diagnosis much more difficult**, both qualitative and quantitative. + +Existing dehazing techniques: +- **Harmonic imaging** receives echoes at frequencies that are multiples of the original frequency. It produces higher quality images, as multipath scatterers have less energy and therefore generate fewer harmonics. However, it results in reduced penetration depth. +- **Clutter filtering methods.** + - Block-matching and 3D filtering algorithm (BM3D) works by grouping similar patches of the image and then stacking and filtering them. It needs assumptions on the noise distribution. + - Temporal decompositions (PCA, SVD) allow to separate data correspoding to rapidly moving events (tissue) from data corresponding to stationary events (clutter). This assumption is not always true, leading to mistakes. + - etc. +- **Deep learning methods.** Supervised approaches have been implemented to supress reverberation haze. They require of a supervised dataset and may have difficulty to generalize across datasets. + +
+ +## Robust PCA for background supression +An ultrasound signal *Y* is acquired. where +$$Y \in \mathbb{C}^{N \times M \times T}$$ +can be decomposed in three signals: + +$$ Y = L + X + N, $$ + +where *L* is the background signal, *X* is the tissue signal and *N* is the noise signal. + +In the context of contrast-enhanced ultrasound imaging, and if the matrices are reshaped as +$$N \cdot M \times T$$ +such that the columns carry the temporal information, some assumptions can be made about *L* and *S*: +- *L* is low-rank because of the spatial coherence of background signal +- *X* can be considered sparse because tissue is not everywhere in the sector + +The decomposition of a matrix in low-rank and sparse components is called **Robust Principal Component Analysis** (RPCA) [[1]](https://arxiv.org/abs/0912.3599) and *L* and *S* can be found solving a convex minimization problem. + +To recover *L* and *X* the following minimization problem can be written, which promotes low-rank solutions for *L* and sparse solutions for *X*: + +$$ \min_{L,X} ||Y-(L+X)||^2_F + \lambda_1 ||L||_* + \lambda_2 ||X||_1, $$ + +where +$$||\cdot||_F$$ +is the Frobenius norm (square root of the sum of all squared elements), and +$$||\cdot||_*$$ +is the nuclear norm (sum of the singular values). This is then solved with methods such as ISTA or ADMM. + +Alternatively, the Bayesian version of this problem can be written. The joint distribution is given by: + +$$ p(Y, L, X) = p(Y | L, X) p(L) p(X). $$ + +Considering Gaussian noise *N* of variance +$\sigma^2$ +, the likelihood writes as: + +$$ p(Y | L, X) = N(Y; L + X, \sigma^2 I). $$ + +Enforcing *L* and *X* to be respectively low-rank and sparse the priors are: + +$$ p(L) \propto \text{exp}(-\gamma ||L||_*), $$ + +and + +$$ p(X) \propto \text{exp}(-\lambda ||X||_1). $$ + +Then, taking the negative log-likelihood and finding the maximum a posteriori (MAP) estimate is equivalent to solving the original RPCA optimization problem up to some constants: + +$$ (L^*, X^*) = \argmax_{L,X}p(L, X | Y) = \argmin_{L,X} (- \log p(L, X | Y)), $$ + +where + +$$ p(L, X | Y) \propto p(Y, L, X) = p(Y | L, X) p(L) p(X). $$ + +
+ +## Diffusion posterior sampling +Given a measurement model of the form: + +$$ y = f(x) + n, $$ + +where +$$ x $$ +is the signal of interest, +$$ y $$ +is a measurement, +$$ f $$ +is a forward operator, and +$$ n $$ +is noise, we usually want to find +$$ x $$ +given measurements +$$ y. $$ +One way to do this is to sample from the posterior +$$ p(x|y). $$ + +**Diffusion posterior sampling (DPS)** [[2]](https://arxiv.org/pdf/2209.14687) allows to sample from +$$ p(x|y). $$ +The idea is to interleave typical score-based denoising updates of the prior +$$ p(x) $$ +and guidance steps towards the measurement +$$ y: $$ + +1. Denoising step: + +$$ x_{0|\tau} := \text{E}[x_0 |x_{\tau}] = \frac{1}{\alpha_\tau} (x_{\tau} + \sigma_{\tau}^2 \nabla_{x_{\tau}} \log p(x_{\tau}) ), $$ + +where +$$ x_0 \sim p(x), \tau \in [0, \Tau], $$ +and +$$ \alpha_{\tau}, \sigma_{\tau} $$ +are predefined noise schedules. The score +$$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ +is parametrized by a neural network. + +2. Guidance step + +The solution is moved towards the measurements using the likelihood: + +$$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ + +
+ +# Proposed method: Nuclear Diffusion Posterior Sampling + +## Motivation +- The sparsity assumption of the RPCA model is too simple to properly model tissue. + +## Idea +- Hybrid RPCA and diffusion framework that integrates the low-rank component for background with diffusion posterior sampling. + +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/comparison.jpg){: width="400" } +{:refdef} + +
+ +## Steps of the algorithm +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/algorithm.jpg){: width="300" } +{:refdef} + +
+ +# Experiments + +## Data +- 4,376 clean samples from 75 easy-to-image subjects +- 2,324 noisy samples from difficult-to-image subjects +- Size of one sample: 60 x 256 x 256 + +## Configuration +- A pretrained 2D diffusion model is used for the prior on *X* +- 500 diffusion steps using SeqDiff [[3]](https://arxiv.org/html/2409.05399v1) (T=5000) with *Y* as the initialization + +## Metrics +- Generalized contrast-to-noise ratio (gCNR) +- Kolmogorov-Smirnov statistic (KS) + +
+ +# Results +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/example.jpg){: width="300" } +{:refdef} + +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/ks_statistic.jpg){: width="300" } +{:refdef} + +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/results.jpg){: width="500" } +{:refdef} + +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/boxplot.jpg){: width="300" } +{:refdef} + +
+ +# Conclusions +- The proposed method better separates dynamic tissue fromlow-rank background compared to RPCA. +- The work shows the potential of combining classical low-rank priors with diffusion models for background supression in videos. + + diff --git a/collections/images/DiffusionBackgroundSupression/algorithm.jpg b/collections/images/DiffusionBackgroundSupression/algorithm.jpg new file mode 100644 index 0000000000000000000000000000000000000000..5acc466099c95604e7100230edb9c82d1ae3472c GIT binary patch literal 222348 zcmce;c{rEr`!@O^N~9>I2+>StGGt1q6q!S0%1|jXWR?si3N3^ZQ8LeCnF)o=V*^TN zGSAaKpYQwozH1-vAN#Mp_d1TXRw_QvbKm!MUFUV4=XF0(Q9e&g%|uNik!UYokW(X( zD8fjjEumCf@jGp1WOwjC3R~HW8dUhlmFk8szCK_luWhGpWoGAKWMfJ)ziV~Jl-t(C z#?y&YJNtT_NTfrgi*jc*93#hnI%qIlT@{@&4-4=mC+F0XsR}rut&+** zr(kmSZRRm1m#5q%22ZNPPgINeJ>h<7crG*6lV#j5;SlwSeH44i&ITQ0iRxur3wJP& zw=^>1ZXAjfd;Ge1vEjAR{lNx7he1U;`iFF~RP5gW>p#L5a`nmn{c>hHwr}@;|4f!j zm6MPjv!Z5RFt`L<%;_Jzi-K| zd1L0!m%g#F||yexB(jy^Ycs-i;HqUjA; 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zKasPiiLY(3n1{&O9eBHMApOXNb@=(!_qM`E5H%s8LgFrI8Dhukpk2LaCSqQ}E@pb` zmJH&%FPi&Po}2NrE7>g-d*TmS`h_t}b-wS135++oM&LJf^aQ$@`-KrtI-uJDuo{iq z+4QM=lzxK$3CN;AdYX-!!2t8ycS)%MBaHN`^~Y*IIPcE zYZTeAc?7M33xItf!thsA?%25%sE4z69I5%cyqIcjDj=hIw_kBB%|5p-a79;w}1WL0&GHyQBSxp^6mhjm&? z3%6P>$t%%6|v)?!@+ccBPsf&O*SSb{M>Ll>)%Ju3{huXf00g zJw$QDy*KaQzxPcnKque99E$e87M9J;I+II4w389}Y>?|A1w8wLB8rFUm+KyjTpFHm z>j<#w>?cWqxdwS8inzXsS+9^+kjPp4BeoUnmey|(;mLSaP$2(p4L6g*n;8g#$s`)& zL<0ak!1>2}NiB#}&#$f%Bo7Ey{DqxVD)sBwnEmTbWc%P1OTgL0nKT`{h9q;viWLS2 zi^d_x8S}uvN}`IX!Csv Pekm)c%Rf1B=Fa~DA+{5+ literal 0 HcmV?d00001 From c51436a338f8dd25075ebc8c56d8862f353ebc04 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:05:53 +0100 Subject: [PATCH 02/13] diffusion RPCA corrected --- collections/_posts/2025-11-16-Diffusion-Background-Supression.md | 1 - 1 file changed, 1 deletion(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 07178436..1a9b3fa2 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -1,4 +1,3 @@ - --- layout: review title: "Nuclear Diffusion Models for Low-rank Background Supression in Videos" From 774c6cac3a9b8c7bf6d3002021e62570523cbe51 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:12:39 +0100 Subject: [PATCH 03/13] diffusion RPCA corrections --- ...5-11-16-Diffusion-Background-Supression.md | 21 +++++++++---------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 1a9b3fa2..ef0b1404 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -13,7 +13,6 @@ pdf: "https://arxiv.org/abs/2509.20886"
# Highlights - - The authors propose to perform dehazing by combining the power of **Robust PCA** and **Diffusion Models**, which can model highly complex data distributions without having to rely on basic assumptions. - - @@ -26,7 +25,7 @@ pdf: "https://arxiv.org/abs/2509.20886" During ultrasound acquisitions, phenomena such as aberration and reverberation produce unwanted echoes that degrade the image quality. In particular, **haze** is an artifact that occurs due to multipath reflections and produces a white haze on the echo image. {:refdef: style="text-align: center;"} -![](/collections/images/DehazingDiffusion/echo_images.jpg){: width="400" } +![](/collections/images/DehazingDiffusion/echo_images.jpg){: width="500" } {:refdef} This phenomenon can **make diagnosis much more difficult**, both qualitative and quantitative. @@ -88,7 +87,7 @@ $$ p(X) \propto \text{exp}(-\lambda ||X||_1). $$ Then, taking the negative log-likelihood and finding the maximum a posteriori (MAP) estimate is equivalent to solving the original RPCA optimization problem up to some constants: -$$ (L^*, X^*) = \argmax_{L,X}p(L, X | Y) = \argmin_{L,X} (- \log p(L, X | Y)), $$ +$$ (L^*, X^*) = \underset{L,X}{\operatorname{argmax}} \hspace{0.1cm} p(L, X | Y) = \underset{L,X}{\operatorname{argmin}} (- \log p(L, X | Y)), $$ where @@ -128,7 +127,7 @@ $$ y: $$ $$ x_{0|\tau} := \text{E}[x_0 |x_{\tau}] = \frac{1}{\alpha_\tau} (x_{\tau} + \sigma_{\tau}^2 \nabla_{x_{\tau}} \log p(x_{\tau}) ), $$ where -$$ x_0 \sim p(x), \tau \in [0, \Tau], $$ +$$ x_0 \sim p(x), \tau \in [0, T], $$ and $$ \alpha_{\tau}, \sigma_{\tau} $$ are predefined noise schedules. The score @@ -152,14 +151,14 @@ $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ - Hybrid RPCA and diffusion framework that integrates the low-rank component for background with diffusion posterior sampling. {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/comparison.jpg){: width="400" } +![](/collections/images/DiffusionBackgroundSupression/comparison.jpg){: width="500" } {:refdef}
## Steps of the algorithm {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/algorithm.jpg){: width="300" } +![](/collections/images/DiffusionBackgroundSupression/algorithm.jpg){: width="500" } {:refdef}
@@ -183,25 +182,25 @@ $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ # Results {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/example.jpg){: width="300" } +![](/collections/images/DiffusionBackgroundSupression/example.jpg){: width="500" } {:refdef} {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/ks_statistic.jpg){: width="300" } +![](/collections/images/DiffusionBackgroundSupression/ks_statistic.jpg){: width="500" } {:refdef} {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/results.jpg){: width="500" } +![](/collections/images/DiffusionBackgroundSupression/results.jpg){: width="700" } {:refdef} {:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/boxplot.jpg){: width="300" } +![](/collections/images/DiffusionBackgroundSupression/boxplot.jpg){: width="500" } {:refdef}
# Conclusions -- The proposed method better separates dynamic tissue fromlow-rank background compared to RPCA. +- The proposed method better separates dynamic tissue from low-rank background compared to RPCA. - The work shows the potential of combining classical low-rank priors with diffusion models for background supression in videos. From 74b5e398eae2ec844fa2e9ed30c10c5682980276 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:27:28 +0100 Subject: [PATCH 04/13] diffusion RPCA corrections --- .../_posts/2025-11-16-Diffusion-Background-Supression.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index ef0b1404..9d3b13bf 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -134,9 +134,7 @@ are predefined noise schedules. The score $$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ is parametrized by a neural network. -2. Guidance step - -The solution is moved towards the measurements using the likelihood: +2. Guidance step: The solution is moved towards the measurements using the likelihood: $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ From 51ca1216497f41a72670fdf25abb10561f344590 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:34:47 +0100 Subject: [PATCH 05/13] diffusion RPCA corrections --- .../_posts/2025-11-16-Diffusion-Background-Supression.md | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 9d3b13bf..18122149 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -13,9 +13,8 @@ pdf: "https://arxiv.org/abs/2509.20886"
# Highlights -- The authors propose to perform dehazing by combining the power of **Robust PCA** and **Diffusion Models**, which can model highly complex data distributions without having to rely on basic assumptions. -- -- +- The authors propose to perform background supression by combining the power of **Robust PCA** and **Diffusion Models**, which can model highly complex data distributions without having to rely on basic assumptions. +- They successfully apply this framework to dehazing of cardiac ultrasound images.
@@ -134,7 +133,7 @@ are predefined noise schedules. The score $$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ is parametrized by a neural network. -2. Guidance step: The solution is moved towards the measurements using the likelihood: +2. Guidance step. The solution is moved towards the measurements using the likelihood: $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ From 28eb390a87543a3f8a1bd57f2fc683c78ed2cb3f Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 17 Nov 2025 17:46:56 +0100 Subject: [PATCH 06/13] diffusion RPCA corrections --- .../2025-11-16-Diffusion-Background-Supression.md | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 18122149..ed0c0796 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -154,9 +154,17 @@ $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$
## Steps of the algorithm +Obs: +- Each sample is a video of shape +$$ N \times M \times T $$ +- The denoising diffusion model is applied independently on channels of shape +$$ N \times M $$ +- Other steps are applied on reshaped matrices +$$ N \cdot M \times T $$ + {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/algorithm.jpg){: width="500" } -{:refdef} +{:refdef}
From 1771a74b41fd7eb4de486a6a30d30a276c0f7324 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Tue, 18 Nov 2025 09:35:39 +0100 Subject: [PATCH 07/13] diffusion RPCA corrections --- ...5-11-16-Diffusion-Background-Supression.md | 20 +++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index ed0c0796..74aff98c 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -71,7 +71,7 @@ Alternatively, the Bayesian version of this problem can be written. The joint di $$ p(Y, L, X) = p(Y | L, X) p(L) p(X). $$ Considering Gaussian noise *N* of variance -$\sigma^2$ +$$\sigma^2$$ , the likelihood writes as: $$ p(Y | L, X) = N(Y; L + X, \sigma^2 I). $$ @@ -123,19 +123,19 @@ $$ y: $$ 1. Denoising step: -$$ x_{0|\tau} := \text{E}[x_0 |x_{\tau}] = \frac{1}{\alpha_\tau} (x_{\tau} + \sigma_{\tau}^2 \nabla_{x_{\tau}} \log p(x_{\tau}) ), $$ + $$ x_{0|\tau} := \text{E}[x_0 |x_{\tau}] = \frac{1}{\alpha_\tau} (x_{\tau} + \sigma_{\tau}^2 \nabla_{x_{\tau}} \log p(x_{\tau}) ), $$ -where -$$ x_0 \sim p(x), \tau \in [0, T], $$ -and -$$ \alpha_{\tau}, \sigma_{\tau} $$ -are predefined noise schedules. The score -$$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ -is parametrized by a neural network. + where + $$ x_0 \sim p(x), \tau \in [0, T], $$ + and + $$ \alpha_{\tau}, \sigma_{\tau} $$ + are predefined noise schedules. The score + $$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ + is parametrized by a neural network. 2. Guidance step. The solution is moved towards the measurements using the likelihood: -$$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ + $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$
From 0433ca99e2b87999f6b1b37419850dd984e1031b Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Tue, 18 Nov 2025 10:39:20 +0100 Subject: [PATCH 08/13] diffusion RPCA corrections --- ...5-11-16-Diffusion-Background-Supression.md | 43 ++++++++++++++----- 1 file changed, 32 insertions(+), 11 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 74aff98c..6b7bb9cb 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -116,26 +116,47 @@ $$ p(x|y). $$ **Diffusion posterior sampling (DPS)** [[2]](https://arxiv.org/pdf/2209.14687) allows to sample from $$ p(x|y). $$ -The idea is to interleave typical score-based denoising updates of the prior +The posterior can be written in terms of the prior and the likelihood using Bayes rule: + +$$ p(x|y) \propto p(x) p(y|x). $$ + +Then, the idea of DPS is to represent the prior $$ p(x) $$ -and guidance steps towards the measurement -$$ y: $$ +with a diffusion model and to interleave denoising updats of the prior with likelihood-guided steps that move samples towards the masurements +$$ y. $$ -1. Denoising step: +1. Denoising step. The forward diffusion process is defined as: - $$ x_{0|\tau} := \text{E}[x_0 |x_{\tau}] = \frac{1}{\alpha_\tau} (x_{\tau} + \sigma_{\tau}^2 \nabla_{x_{\tau}} \log p(x_{\tau}) ), $$ + $$ x_{\tau} = \alpha_{\tau} x_0 + \sigma_{\tau} z, $$ where - $$ x_0 \sim p(x), \tau \in [0, T], $$ + $$ z \sim N(0, I), x_0 \sim p(x), \tau \in [0, T], $$ and $$ \alpha_{\tau}, \sigma_{\tau} $$ - are predefined noise schedules. The score - $$ \nabla_{x_{\tau}} \log p(x_{\tau}) $$ - is parametrized by a neural network. + are predefined noise schedules. + + Then, given a noisy sample + $$ x_{\tau}, $$ + the denoised estimate is given by: + + $$ x_{0|\tau} = \frac{1}{\alpha_\tau} (x_{\tau} - \sigma_{\tau} \epsilon_{\theta}(x_{\tau}, \tau) ), $$ + + where + $$ \epsilon_{\theta}(x_{\tau}, \tau) $$ + is a neural network trained to predict the Gaussian noise + $$z.$$ + +2. Guidance step. The denoised estimate is moved towards the measurement using the likelihood: -2. Guidance step. The solution is moved towards the measurements using the likelihood: + $$ x_{0|\tau} = x_{0|\tau} + \eta \nabla_{x_0} \log p(y | x_0) \Big|_{x_0 = x_{0|\tau}}, $$ - $$ x_{0|\tau} = x_{0|\tau} - \nabla_{x_{\tau}} \log p(y | x_{0 | \tau}) $$ + where + $$ \eta $$ + is a step size and + $$ x_{0|\tau} $$ + is used as a proxy for + $$ x_0 $$ + in the likelihood.
From 7e1a2f7eb3387b16cf1418da22cfc06606da6f1d Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Tue, 18 Nov 2025 10:59:54 +0100 Subject: [PATCH 09/13] diffusion RPCA corrections --- ...5-11-16-Diffusion-Background-Supression.md | 22 +++++++++++-------- 1 file changed, 13 insertions(+), 9 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 6b7bb9cb..68f06721 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -191,26 +191,30 @@ $$ N \cdot M \times T $$ # Experiments +## Configuration +- A pretrained 2D diffusion model is used for the prior on *X* +- 500 diffusion steps using SeqDiff [[3]](https://arxiv.org/html/2409.05399v1) (T=5000) with *Y* as the initialization + ## Data - 4,376 clean samples from 75 easy-to-image subjects - 2,324 noisy samples from difficult-to-image subjects - Size of one sample: 60 x 256 x 256 -## Configuration -- A pretrained 2D diffusion model is used for the prior on *X* -- 500 diffusion steps using SeqDiff [[3]](https://arxiv.org/html/2409.05399v1) (T=5000) with *Y* as the initialization +{:refdef: style="text-align: center;"} +![](/collections/images/DiffusionBackgroundSupression/example.jpg){: width="500" } +{:refdef} ## Metrics -- Generalized contrast-to-noise ratio (gCNR) -- Kolmogorov-Smirnov statistic (KS) +- Generalized contrast-to-noise ratio (gCNR) between ventricle and myocardium regions. +- Kolmogorov-Smirnov statistic (KS) to quantify agreement between the original *Y* and denoised *X* tissue distributions in the myocardial region. + +$$ \text{KS} = \underset{z}{\operatorname{sup}} | F_{\Omega_S(x)}(z) - F_{\Omega_S(y)}(z) |, $$ + +where *F()* is the empirical CFD of the respective regions of interest.
# Results -{:refdef: style="text-align: center;"} -![](/collections/images/DiffusionBackgroundSupression/example.jpg){: width="500" } -{:refdef} - {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/ks_statistic.jpg){: width="500" } {:refdef} From 1a48265ee7b83b23c98fa0c0590d5085ecf1e6f7 Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Tue, 18 Nov 2025 14:15:01 +0100 Subject: [PATCH 10/13] diffusion RPCA corrections --- .../_posts/2025-11-16-Diffusion-Background-Supression.md | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 68f06721..fd791e35 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -197,7 +197,7 @@ $$ N \cdot M \times T $$ ## Data - 4,376 clean samples from 75 easy-to-image subjects -- 2,324 noisy samples from difficult-to-image subjects +- 2,324 noisy samples from 40 difficult-to-image subjects - Size of one sample: 60 x 256 x 256 {:refdef: style="text-align: center;"} @@ -210,19 +210,24 @@ $$ N \cdot M \times T $$ $$ \text{KS} = \underset{z}{\operatorname{sup}} | F_{\Omega_S(x)}(z) - F_{\Omega_S(y)}(z) |, $$ -where *F()* is the empirical CFD of the respective regions of interest. +where +$$ F(\cdot) $$ +is the empirical CFD of the respective regions of interest.
# Results +- RPCA appears to greater modify tissue distribution. {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/ks_statistic.jpg){: width="500" } {:refdef} +- Both methods allow significant dehazing, but RPCA excessively attenuates tissue. {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/results.jpg){: width="700" } {:refdef} +- Nuclear diffusion acheives a better contrast while better preserving tissue distribution. {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/boxplot.jpg){: width="500" } {:refdef} From 9ba687d3ab46de4d79373524890937b2fa12c94c Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Tue, 18 Nov 2025 14:24:27 +0100 Subject: [PATCH 11/13] diffusion RPCA corrections --- .../_posts/2025-11-16-Diffusion-Background-Supression.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index fd791e35..0cfe9862 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -166,7 +166,7 @@ $$ y. $$ - The sparsity assumption of the RPCA model is too simple to properly model tissue. ## Idea -- Hybrid RPCA and diffusion framework that integrates the low-rank component for background with diffusion posterior sampling. +- Use a hybrid RPCA and diffusion framework that integrates the low-rank component for background with diffusion posterior sampling, thereby modeling tissue with a diffusion model. {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/comparison.jpg){: width="500" } From 4402d624e1be37ea84151924ab15eadafc4eac2d Mon Sep 17 00:00:00 2001 From: Julia Puig Date: Mon, 24 Nov 2025 15:03:43 +0100 Subject: [PATCH 12/13] typo corrections --- .../2025-11-16-Diffusion-Background-Supression.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md index 0cfe9862..0c6e7d61 100644 --- a/collections/_posts/2025-11-16-Diffusion-Background-Supression.md +++ b/collections/_posts/2025-11-16-Diffusion-Background-Supression.md @@ -33,14 +33,14 @@ Existing dehazing techniques: - **Harmonic imaging** receives echoes at frequencies that are multiples of the original frequency. It produces higher quality images, as multipath scatterers have less energy and therefore generate fewer harmonics. However, it results in reduced penetration depth. - **Clutter filtering methods.** - Block-matching and 3D filtering algorithm (BM3D) works by grouping similar patches of the image and then stacking and filtering them. It needs assumptions on the noise distribution. - - Temporal decompositions (PCA, SVD) allow to separate data correspoding to rapidly moving events (tissue) from data corresponding to stationary events (clutter). This assumption is not always true, leading to mistakes. + - Temporal decompositions (PCA, SVD) allow to separate data corresponding to rapidly moving events (tissue) from data corresponding to stationary events (clutter). This assumption is not always true, leading to mistakes. - etc. - **Deep learning methods.** Supervised approaches have been implemented to supress reverberation haze. They require of a supervised dataset and may have difficulty to generalize across datasets.
## Robust PCA for background supression -An ultrasound signal *Y* is acquired. where +An ultrasound signal *Y* is acquired, where $$Y \in \mathbb{C}^{N \times M \times T}$$ can be decomposed in three signals: @@ -48,13 +48,13 @@ $$ Y = L + X + N, $$ where *L* is the background signal, *X* is the tissue signal and *N* is the noise signal. -In the context of contrast-enhanced ultrasound imaging, and if the matrices are reshaped as +In the context of ultrasound imaging, and if the matrices are reshaped as $$N \cdot M \times T$$ such that the columns carry the temporal information, some assumptions can be made about *L* and *S*: - *L* is low-rank because of the spatial coherence of background signal - *X* can be considered sparse because tissue is not everywhere in the sector -The decomposition of a matrix in low-rank and sparse components is called **Robust Principal Component Analysis** (RPCA) [[1]](https://arxiv.org/abs/0912.3599) and *L* and *S* can be found solving a convex minimization problem. +The decomposition of a matrix in low-rank and sparse components is called **Robust Principal Component Analysis** (RPCA) [[1]](https://arxiv.org/abs/0912.3599) and *L* and *X* can be found solving a convex minimization problem. To recover *L* and *X* the following minimization problem can be written, which promotes low-rank solutions for *L* and sparse solutions for *X*: @@ -122,7 +122,7 @@ $$ p(x|y) \propto p(x) p(y|x). $$ Then, the idea of DPS is to represent the prior $$ p(x) $$ -with a diffusion model and to interleave denoising updats of the prior with likelihood-guided steps that move samples towards the masurements +with a diffusion model and to interleave denoising updates of the prior with likelihood-guided steps that move samples towards the measurements $$ y. $$ 1. Denoising step. The forward diffusion process is defined as: @@ -227,7 +227,7 @@ is the empirical CFD of the respective regions of interest. ![](/collections/images/DiffusionBackgroundSupression/results.jpg){: width="700" } {:refdef} -- Nuclear diffusion acheives a better contrast while better preserving tissue distribution. +- Nuclear diffusion achieves a better contrast while better preserving tissue distribution. {:refdef: style="text-align: center;"} ![](/collections/images/DiffusionBackgroundSupression/boxplot.jpg){: width="500" } {:refdef} From 723284b8c8cc30d0a60f99595f6043687bddeea5 Mon Sep 17 00:00:00 2001 From: Nathan Painchaud Date: Mon, 24 Nov 2025 15:11:17 +0100 Subject: [PATCH 13/13] Revert changes to `Gemfile.lock` --- Gemfile.lock | 59 ++++++++++++++++++++++++++++++++++------------------ 1 file changed, 39 insertions(+), 20 deletions(-) diff --git a/Gemfile.lock b/Gemfile.lock index 90a92219..bb4aed65 100644 --- a/Gemfile.lock +++ b/Gemfile.lock @@ -1,19 +1,23 @@ GEM remote: https://rubygems.org/ specs: - activesupport (7.1.4.2) + activesupport (8.0.2.1) base64 + benchmark (>= 0.3) bigdecimal - concurrent-ruby (~> 1.0, >= 1.0.2) + concurrent-ruby (~> 1.0, >= 1.3.1) connection_pool (>= 2.2.5) drb i18n (>= 1.6, < 2) + logger (>= 1.4.2) minitest (>= 5.1) - mutex_m - tzinfo (~> 2.0) + securerandom (>= 0.3) + tzinfo (~> 2.0, >= 2.0.5) + uri (>= 0.13.1) addressable (2.8.7) public_suffix (>= 2.0.2, < 7.0) base64 (0.3.0) + benchmark (0.4.1) bigdecimal (3.2.3) coffee-script (2.4.1) coffee-script-source @@ -23,7 +27,10 @@ GEM commonmarker (0.23.12) concurrent-ruby (1.3.5) connection_pool (2.5.4) - dnsruby (1.72.2) + csv (3.3.5) + dnsruby (1.73.0) + base64 (>= 0.2) + logger (~> 1.6) simpleidn (~> 0.2.1) drb (2.2.3) em-websocket (0.5.3) @@ -33,17 +40,21 @@ GEM ffi (>= 1.15.0) eventmachine (1.2.7) execjs (2.10.0) - faraday (2.0.0) - ruby2_keywords (>= 0.0.4) - ffi (1.17.2) + faraday (2.13.4) + faraday-net_http (>= 2.0, < 3.5) + json + logger + faraday-net_http (3.4.1) + net-http (>= 0.5.0) + ffi (1.17.2-x86_64-linux-gnu) forwardable-extended (2.6.0) gemoji (4.1.0) - github-pages (231) + github-pages (232) github-pages-health-check (= 1.18.2) - jekyll (= 3.9.5) + jekyll (= 3.10.0) jekyll-avatar (= 0.8.0) jekyll-coffeescript (= 1.2.2) - jekyll-commonmark-ghpages (= 0.4.0) + jekyll-commonmark-ghpages (= 0.5.1) jekyll-default-layout (= 0.1.5) jekyll-feed (= 0.17.0) jekyll-gist (= 1.5.0) @@ -80,9 +91,10 @@ GEM liquid (= 4.0.4) mercenary (~> 0.3) minima (= 2.5.1) - nokogiri (>= 1.13.6, < 2.0) + nokogiri (>= 1.16.2, < 2.0) rouge (= 3.30.0) terminal-table (~> 1.4) + webrick (~> 1.8) github-pages-health-check (1.18.2) addressable (~> 2.3) dnsruby (~> 1.60) @@ -95,9 +107,10 @@ GEM http_parser.rb (0.8.0) i18n (1.14.7) concurrent-ruby (~> 1.0) - jekyll (3.9.5) + jekyll (3.10.0) addressable (~> 2.4) colorator (~> 1.0) + csv (~> 3.0) em-websocket (~> 0.5) i18n (>= 0.7, < 2) jekyll-sass-converter (~> 1.0) @@ -108,6 +121,7 @@ GEM pathutil (~> 0.9) rouge (>= 1.7, < 4) safe_yaml (~> 1.0) + webrick (>= 1.0) jekyll-avatar (0.8.0) jekyll (>= 3.0, < 5.0) jekyll-coffeescript (1.2.2) @@ -115,9 +129,9 @@ GEM coffee-script-source (~> 1.12) jekyll-commonmark (1.4.0) commonmarker (~> 0.22) - jekyll-commonmark-ghpages (0.4.0) - commonmarker (~> 0.23.7) - jekyll (~> 3.9.0) + jekyll-commonmark-ghpages (0.5.1) + commonmarker (>= 0.23.7, < 1.1.0) + jekyll (>= 3.9, < 4.0) jekyll-commonmark (~> 1.4.0) rouge (>= 2.0, < 5.0) jekyll-default-layout (0.1.5) @@ -203,6 +217,7 @@ GEM gemoji (>= 3, < 5) html-pipeline (~> 2.2) jekyll (>= 3.0, < 5.0) + json (2.14.0) kramdown (2.4.0) rexml kramdown-parser-gfm (1.1.0) @@ -211,14 +226,16 @@ GEM listen (3.9.0) rb-fsevent (~> 0.10, >= 0.10.3) rb-inotify (~> 0.9, >= 0.9.10) + logger (1.7.0) mercenary (0.3.6) minima (2.5.1) jekyll (>= 3.5, < 5.0) jekyll-feed (~> 0.9) jekyll-seo-tag (~> 2.1) minitest (5.25.5) - mutex_m (0.3.0) - nokogiri (1.15.7-x86_64-linux) + net-http (0.6.0) + uri + nokogiri (1.18.10-x86_64-linux-gnu) racc (~> 1.4) octokit (4.25.1) faraday (>= 1, < 3) @@ -232,7 +249,6 @@ GEM ffi (~> 1.0) rexml (3.4.4) rouge (3.30.0) - ruby2_keywords (0.0.5) rubyzip (2.4.1) safe_yaml (1.0.5) sass (3.7.4) @@ -243,6 +259,7 @@ GEM sawyer (0.9.2) addressable (>= 2.3.5) faraday (>= 0.17.3, < 3) + securerandom (0.4.1) simpleidn (0.2.3) terminal-table (1.8.0) unicode-display_width (~> 1.1, >= 1.1.1) @@ -251,6 +268,8 @@ GEM tzinfo (2.0.6) concurrent-ruby (~> 1.0) unicode-display_width (1.8.0) + uri (1.0.3) + webrick (1.9.1) PLATFORMS x86_64-linux @@ -261,4 +280,4 @@ DEPENDENCIES tzinfo-data BUNDLED WITH - 2.3.25 + 2.4.19