Project: 049_ProtComposer_Compositional_Protein_Structure_Gener | Function: blobify (iclr)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Ellipsoid segmentation algorithm that converts protein structures into compositional 3D ellipsoid layouts. Constructs a graph based on spatial proximity and secondary structure similarity, then extracts connected components and fits Gaussian ellipsoids to represent protein substructures.

**KEY TERMS:** {
  "distmat": "pairwise distance matrix",
  "dssp": "secondary structure labels",
  "covar": "covariance matrix of ellipsoid",
  "blobs": "ellipsoid representations"
}

**CONTEXT CODE:** # SPDX-FileCopyrightText: Copyright (c) 2025 NVIDIA CORPORATION & AFFILIATES. All rights reserved.
# SPDX-License-Identifier: LicenseRef-NvidiaProprietary
#
# NVIDIA CORPORATION, its affiliates and licensors retain all intellectual
# property and proprietary rights in and to this material, related
# documentation and any modifications thereto. Any use, reproduction,
# disclosure or distribution of this material and related documentation
# without an express license agreement from NVIDIA CORPORATION or
# its affiliates is strictly prohibited.


import numpy as np
import networkx as nx
import numpy as np
from scipy.special import logsumexp, softmax
import torch
from torch.distributions.multivariate_normal import MultivariateNormal
from scipy.stats import wishart

alpha_slope = 0.22213567936924072
alpha_intercept = 7.512307373103329

beta_slope = 0.2727479880396384
beta_intercept = 8.28561006547174


def blobify(pos, dssp, radius_thresh=5, size_thresh=5):
    raise NotImplementedError("Function implementation has been masked for testing")

def shannon_complexity(pos, dssp, thresh=5):
    blobs = blobify(pos, dssp, radius_thresh=thresh)
    counts = np.array([b["count"] for b in blobs])
    probs = counts / counts.sum()
    return -(probs * np.log(probs)).sum()


def score_blobs(centers, covars):
    energy = 0
    for pos, cov in zip(centers, covars):
        relpos = centers - pos
        d = np.einsum("li,ij,lj->l", relpos, np.linalg.inv(cov), relpos) ** 0.5
        d = d[d != 0]
        energy += (1 / d**2).sum()
    return energy


def sample_blobs(k, nu, psi, sigma):
    raise NotImplementedError("Function implementation has been masked for testing")

def gmm_ll(pos, centers, covars, probs):
    probs = probs / probs.sum()
    relpos = pos[:, None] - centers[None]
    ll = (
        -0.5 * np.einsum("lni,nij,lnj->ln", relpos, np.linalg.inv(covars), relpos)
        - 0.5 * np.linalg.slogdet(covars)[1][None]
        + np.log(probs)[None]
    )
    return logsumexp(ll, -1)


def dssp_gmm_ll(pos, centers, covars, probs, pos_dssp, centers_dssp):
    if np.any(centers_dssp == 1):
        gmm_ll_1 = gmm_ll(
            pos,
            centers[centers_dssp == 1],
            covars[centers_dssp == 1],
            probs[centers_dssp == 1],
        )
        gmm_ll_1[pos_dssp != 1] = -np.inf
    else:
        gmm_ll_1 = -np.inf * np.ones(len(pos))

    if np.any(centers_dssp == 2):
        gmm_ll_2 = gmm_ll(
            pos,
            centers[centers_dssp == 2],
            covars[centers_dssp == 2],
            probs[centers_dssp == 2],
        )
        gmm_ll_2[pos_dssp != 2] = -np.inf
    else:
        gmm_ll_2 = -np.inf * np.ones(len(pos))

    dssp_1_count = probs[centers_dssp == 1].sum() / probs.sum()
    dssp_2_count = probs[centers_dssp == 2].sum() / probs.sum()

    return np.log(np.exp(gmm_ll_1) * dssp_1_count + np.exp(gmm_ll_2) * dssp_2_count)


"""
Treats the blobs as a GMM and evaluates the average log-DENSITY at each pos.
Note that DENSITY is N * PDF where N is the total number of residues in the blobs.
Not suitable for partial blobs.
"""


def blob_likelihood(pos, dssp, blobs, structured_only=False, mean=True):
    if structured_only:
        pos = pos[dssp != 0]
        dssp = dssp[dssp != 0]
        if len(dssp) == 0:
            return -10
    if len(blobs) == 0:
        ll = -np.inf * np.ones(pos.shape[0])
    else:
        covars = np.stack([blob["covar"] for blob in blobs])
        counts = np.array([blob["count"] for blob in blobs])
        centers = np.stack([blob["pos"] for blob in blobs])

        ll = gmm_ll(pos, centers, covars, counts)
        ll += np.log(counts.sum())

    if not mean:
        return ll
    return ll.mean()


"""
Reblobs the protein and evaluates the number of residues per blob
"""


def blobs_per_res(pos, dssp, thresh=5):
    blobs = blobify(pos, dssp, thresh)
    return len(blobs) / len(pos)



Complete the function implementation replacing "raise NotImplementedError".